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Morgan Federal Reserve Ba\ nk of New York 33 Liberty Street Ne\ w York, N.Y. 10045; don.morgan@n y\ .f r b . org ) >> endobj 2388 0 obj << /S /P /T (Paragraph) /P 2382 0 R /Pg 2689 0 R /K 3 /Alt (Katherine A. Samolyk, Federal Depo\ sit Insurance Corporation 550 17th \ Street N.W. Washington, D.C. 2 0 4 \ 2 9 k s a m o l y k @f d i c .g o v\ ) >> endobj 2389 0 obj << /S /Sect /T (Section P2) /P 2687 0 R /K [ 2390 0 R 2392 0 R 2393 0 R 2394 0 R 2395 0 R ] >> endobj 2390 0 obj << /S /P /T (Paragraph) /P 2389 0 R /Pg 1 0 R /K 0 /Alt (Financial Intermediation and Commit\ ments to Optimal Investment Strategi\ es) >> endobj 2391 0 obj [ 2390 0 R 2392 0 R 2393 0 R 2394 0 R 2395 0 R ] endobj 2392 0 obj << /S /P /T (Paragraph) /P 2389 0 R /Pg 1 0 R /K 1 /Alt (Donald P. Morgan and Katherine A. S\ amolyk) >> endobj 2393 0 obj << /S /P /T (Paragraph) /P 2389 0 R /Pg 1 0 R /K 2 /Alt (Abstract) >> endobj 2394 0 obj << /S /P /T (Paragraph) /P 2389 0 R /Pg 1 0 R /K 3 /Alt (Reducing risks and enhancing liquid\ ity are functions long emphasized in\ financial intermediation literature\ . However, financial intermediaries\ may also help savers commit to opti\ mal investment strategies by limitin\ g liquidity when investment opportun\ ities warrant. We illustrate this s\ ort of commitment function using a v\ ariant of the Diamond and Dybvig \(1\ 983\) model in which the possibility \ that savers may need to liquidate th\ eir investments conflicts with long-\ run investment prospects. The asset\ s considered here are completely tra\ dable. Yet savers can do better if \ they delegate their investment decis\ ions to an intermediary as a piggy b\ ank of sorts that sets payments to o\ ptimize long-run returns, even thoug\ h claimants know that in some states\ they may incur penalties. Although\ this piggy-bank function may not ch\ aracterize any particular type of re\ al-world intermediary literally, in \ the abstract the function may apply \ to a number of them. By preventing \ savers from reneging on contracts in\ the short run, our intermediary all\ ows individuals to have better long-\ run outcomes. Thus, the function we\ emphasize may apply more particular\ ly to intermediation that is associa\ ted with long-term savings goals suc\ h as saving for retirement.) >> endobj 2395 0 obj << /S /P /T (Paragraph) /P 2389 0 R /Pg 1 0 R /K 4 /Alt (JEL Classification: G21, G23, E21 K\ eywords: Asymmetric information, f\ inancial contracting, microfoundatio\ ns of financial intermediaries ) >> endobj 2396 0 obj << /S /Sect /T (Section P3) /P 2687 0 R /K [ 2397 0 R 2398 0 R 2399 0 R 2402 0 R 2400 0 R ] >> endobj 2397 0 obj << /S /P /T (Paragraph) /P 2396 0 R /Pg 5 0 R /K 0 /Alt (1. Introduction) >> endobj 2398 0 obj << /S /P /T (Paragraph) /P 2396 0 R /Pg 5 0 R /K 1 /Alt (At least three basic functions have\ been identified with financial inte\ rmediaries, including risk reduction\ , liquidity provision, and informati\ on production. This paper illustrat\ es a fourth role, one in which inter\ mediaries enable savers to commit to\ optimal long-run savings strategies\ by limiting withdrawals at inopport\ une times. We would argue that this\ commitment has not been emphasized \ in the literature no doubt because c\ redit risk diversification, liquidit\ y provision, and information product\ ion are significantly more important\ functions of depository institution\ s \(i.e., banks\). Although banks ma\ y perform this fourth role to some e\ xtent, we view it as more broadly as\ sociated with intermediation earmark\ ed for long-term saving objectives s\ uch as retirement. Thus, our model \ is more illustrative of intermediati\ on through pension and mutual funds \ or insurance contracts. Indeed, ou\ r intermediary can even be thought o\ f as a sort of privatized social sec\ urity fund, optimizing intergenerati\ onal \(as opposed to intrapersonal\) \ transfers of wealth. With regard to\ savers  long-term investment goals,\ we would argue that this role is re\ levant and interesting.) >> endobj 2399 0 obj << /S /P /T (Paragraph) /P 2396 0 R /Pg 5 0 R /Alt (We illustrate this commitment funct\ ion using a variant of the liquidity\ -shock model used by Bryant \(1980\) \ and Diamond and Dybvig \(1983\). In \ the standard versions of the model, \ the random demand for liquidity forc\ es savers to trade off asset returns\ and liquidity; the long-term asset \ pays more, but the short-term asset \ pays sooner. Intermediaries emerge \ to provide a form of liquidity insur\ ance to savers that want to smooth i\ ntertemporal consumption. They offe\ r time-contingent deposit contracts \ that effectively shift some long-ter\ m asset returns to the unlucky saver\ s that must liquidate investments so\ as to consume early. Foonote callo\ ut 1.) /K [ 2 4 ] >> endobj 2400 0 obj << /S /P /T (Paragraph) /P 2396 0 R /Pg 5 0 R /K 3 /Alt (Footnote 1 Although this model w\ as used initially to explore the rea\ sons and effects for bank runs, it i\ s now used more widely as a model of\ intermediation. See also Bernanke a\ nd Gertler \(1987\), Wallace \(1990\),\ and Hellwig \(1994\).) >> endobj 2401 0 obj [ 2397 0 R 2398 0 R 2399 0 R 2400 0 R 2399 0 R 2402 0 R ] endobj 2402 0 obj << /S /Span /T (Span) /P 2396 0 R /Alt (Footnote 1) /Pg 5 0 R /K 5 >> endobj 2403 0 obj << /S /P /T (Paragraph) /P 2382 0 R /Pg 2689 0 R /Alt (The views stated here are those of \ the authors and do not necessarily r\ eflect those of their employers. Th\ e authors thank Michael Avery, Charl\ ie Himmelberg, Robert Moore, and \r\n\ Kei-Mu Yi for comment) /K [ 4 5 ] >> endobj 2404 0 obj << /S /Sect /T (Section P4) /P 2687 0 R /K [ 2405 0 R 2409 0 R 2406 0 R 2407 0 R 2420 0 R ] >> endobj 2405 0 obj << /S /P /T (Paragraph) /P 2404 0 R /Pg 8 0 R /Alt (Our version of the model is deliber\ ately altered to highlight the value\ of intermediaries as a means of com\ mitting to optimal investment strate\ gies. In our model, savers do face \ unobservable liquidity shocks; they \ know there is some probability that \ they will need to liquidate their ho\ ldings early. But risk sharing in t\ he sense of consumption smoothing is\ not as important for our savers; th\ ey are less risk averse regarding in\ vestment returns than the depositor\ s  considered in models emphasizing \ liquidity provision. Since our save\ rs can costlessly invest in and trad\ e short-term and long-term assets \(\ no information production is require\ d\), they can fund their own portfoli\ os and simply trade assets if they n\ eed to liquidate early. However, he\ rein lies a second key difference in\ our variant of the model. Our shor\ t-term assets can be reinvested at r\ eturns that are random when portfoli\ o decisions are made. And although \ expected returns are higher on long-\ term illiquid  projects, short-term\ project returns exceed those on lon\ g-term projects in some states. Foot\ note callout 2.) /K [ 0 3 ] >> endobj 2406 0 obj << /S /P /T (Paragraph) /P 2404 0 R /Pg 8 0 R /Alt (The commitment problem and piggy-ba\ nk solution to long-term investment \ strategies reflect how uncertainty a\ bout future returns can adversely in\ terplay with imperfect information a\ bout savers  true liquidity needs. \ Ex ante, before actual returns and l\ iquidity needs are realized, savers \ will optimally plan to reinvest some\ of their liquid assets in the high-\ return state. But, \r\nex post, the \ savers needing to liquidate, will re\ nege on this strategy consuming thei\ r short-term assets instead of rolli\ ng them over. Left to their own dev\ ices, savers violate the standard in\ tertemporal consumption condition, U\ prime of c sub 1 equals r times U p\ rime of c sub 2,by consuming too muc\ h, too soon when the interest rate r\ turns out high. The inability to c\ redibly commit to the optimal saving\ plan distorts savers  allocation of\ wealth across investment opportunit\ ies and lowers the return to savings\ and savers  welfare. Savers can ov\ ercome this commitment problem, or a\ t least reduce it, by  locking up  \ their funds in a piggy bank of sort\ s. This piggy-bank intermediary lim\ its withdrawals when short-term yiel\ ds turn out to be high, increasing r\ einvestment when it is most profitab\ le. Ex post, savers with liquidity \ needs lose out in this case, but ex \ ante, savers are better off limiting\ access to their funds and delegatin\ g investment decisions to a less myo\ pic intermediary: thus, they do.) /K [ 1 5 7 << /Type /MCR /Pg 12 0 R /MCID 3 >> 9 ] >> endobj 2407 0 obj << /S /P /T (Paragraph) /P 2404 0 R /Pg 8 0 R /Alt (The commitment function in our mode\ l is very different from the other r\ oles identified with intermediaries \ particularly banks. Our intermediar\ y is definitely not providing risk s\ haring in the sense of consumption s\ moothing. In fact, yields from this\ piggy bank are more volatile; and h\ ence its benefit is greater, the mor\ e willing savers are to take on grea\ ter risk for greater return. Nor is\ our intermediary producing informat\ ion about potential investments or d\ iversifying away idiosyncratic proje\ ct risks; the assets considered here\ are effectively securities that sav\ ers can hold directly and trade free\ ly as liquidity needs and interest r\ ates change. Nor, finally, is our i\ ntermediary providing liquidity; the\ problem here is really too much liq\ uidity in some sense, and the soluti\ on is to lock up funds in an interme\ diary that limits liquidity in some \ states.) /K [ 2 6 << /Type /MCR /Pg 12 0 R /MCID 4 >> ] >> endobj 2408 0 obj [ 2405 0 R 2406 0 R 2407 0 R 2405 0 R 2409 0 R 2406 0 R 2407 0 R 2406 0 R 2420 0 R 2406 0 R ] endobj 2409 0 obj << /S /Span /T (Span) /P 2404 0 R /Alt (Footnote callout 2) /Pg 8 0 R /K 4 >> endobj 2410 0 obj << /S /Sect /T (Section P5) /P 2687 0 R /K 2412 0 R >> endobj 2411 0 obj << /S /P /T (Paragraph) /K 2 /Pg 12 0 R /Alt (The commitment function in our mode\ l is very different from the other r\ oles identified with intermediaries \ particularly banks. Our intermediar\ y is definitely not providing risk s\ haring in the sense of consumption s\ moothing. In fact, yields from this\ piggy bank are more volatile; and h\ ence its benefit is greater, the mor\ e willing savers are to take on grea\ ter risk for greater return. Nor is\ our intermediary producing informat\ ion about potential investments or d\ iversifying away idiosyncratic proje\ ct risks; the assets considered here\ are effectively securities that sav\ ers can hold directly and trade free\ ly as liquidity needs and interest r\ ates change. Nor, finally, is our i\ ntermediary providing liquidity; the\ problem here is really too much liq\ uidity in some sense, and the soluti\ on is to lock up funds in an interme\ diary that limits liquidity in some \ states.) >> endobj 2412 0 obj << /S /P /T (Paragraph) /P 2410 0 R /Pg 12 0 R /K 1 /Alt (Of course, not all motives for savi\ ng require the sort of discipline pr\ ovided by our piggy- bank intermedia\ ry. However, for certain types of i\ nvestments at certain times, the lon\ g-run benefits of investing may be a\ t odds with short-term incentives to\ withdraw funds. Earmarking funds t\ o an insurance annuity or pension pl\ an may provide a commitment value to\ some savers quite apart from, or in\ addition to, the other benefits ass\ ociated with such intermediated cont\ racts. Other real-world intermediar\ ies may also help create incentives \ to stick to optimal investment strat\ egies, quite apart from or incidenta\ l to their more familiar roles. ) >> endobj 2413 0 obj [ null 2412 0 R 2411 0 R 2406 0 R 2407 0 R ] endobj 2414 0 obj << /S /Sect /T (Section P6) /P 2687 0 R /K [ 2415 0 R 2419 0 R 2416 0 R 2417 0 R 2421 0 R ] >> endobj 2415 0 obj << /S /P /T (Paragraph) /P 2414 0 R /Pg 15 0 R /K 0 /Alt (Even banks are now issuing callable\ CDs having withdrawal penalties tha\ t are effectively contingent on inte\ rest rates much like our conceptual \ intermediary. Footnote callout 3.) >> endobj 2416 0 obj << /S /P /T (Paragraph) /P 2414 0 R /Alt (Although the commitment function co\ nsidered here might seem new, the no\ tion of self-control or commitment p\ roblems among savers goes way back i\ n the literature. Thaler and Shefri\ n \(1981\) model the commitment probl\ em for two-sided savers the far-sigh\ ted planner struggling against the s\ hort-sighted consumer. They trace t\ his idea back to Adam Smith \(1759\).\ In his classic on investment, Stro\ tz \(1955\) observed that investment \ plans might not be implementable if \ savers had discount rates that decli\ ned over time. Laibson s \(1997\) hy\ perbolic discounting is a modern var\ iant of that idea, modeled as a game\ between selves over time. Barro \(\ 1997\) solves a neoclassical growth m\ odel with hyperbolic discounting and\ works out the implications for savi\ ngs, capital accumulation, and growt\ h. He notes in passing that these s\ orts of commitment problems seem to \ have important institutional implica\ tions as well, which are basically t\ he topic of this paper.) /Pg 15 0 R /K 5 >> endobj 2417 0 obj << /S /P /T (Paragraph) /P 2414 0 R /Pg 15 0 R /K 6 /Alt (The next section describes the savi\ ngs environment and optimal allocati\ on; we characterize the solution for\ general preferences and illustrate \ with linear \(i.e. risk-neutral\) pre\ ferences. Section three contrasts t\ hat optimal allocation with the allo\ cation achieved when savers invest d\ irectly and trade assets among thems\ elves; we highlight the divergence f\ rom the optimal allocation and show \ how the resulting portfolio distorti\ ons \(between short- and long-term a\ ssets\) decrease with the degree of r\ isk aversion. Section four shows ho\ w a financial intermediary can impro\ ve savers  welfare, even if the inte\ rmediary is not able to observe save\ rs  true liquidity needs. We contra\ st the intermediary contract with a \ more complicated option-like arrange\ ment in which savers hold long-term \ assets directly and buy options to s\ ell these assets contingent on their\ liquidity needs. ) >> endobj 2418 0 obj [ 2415 0 R 2419 0 R null null null 2416 0 R 2417 0 R 2421 0 R 2421 0 R ] endobj 2419 0 obj << /S /Span /T (Span) /P 2414 0 R /Pg 15 0 R /K 1 /Alt (Footnote callout 3) >> endobj 2420 0 obj << /S /P /T (Paragraph) /P 2404 0 R /Pg 8 0 R /K 8 /Alt (Footnote 2 While this assumption \ is key for our result, it seems more\ plausible than the constant returns\ often assumed.) >> endobj 2421 0 obj << /S /P /T (Paragraph) /P 2414 0 R /Pg 15 0 R /Alt (Footnote 3 Issuers can call the \ CD over a specified interval, either\ by paying off the CD holder or by r\ olling it over at the new market rat\ e. All else equal, banks will exerc\ ise the call when rates have fallen \ and will pass when rates have risen,\ so the reinvestment decision by ban\ ks and spending decisions by savers \ will resemble those of our model sav\ ers to some extent.) /K [ 7 8 ] >> endobj 2422 0 obj << /S /Sect /T (Section P7) /P 2687 0 R /K [ 2423 0 R 2425 0 R 2426 0 R 2427 0 R 2428 0 R 2429 0 R ] >> endobj 2423 0 obj << /S /P /T (Paragraph) /P 2422 0 R /Pg 19 0 R /K 0 /Alt (We contrast the intermediary contra\ ct with a more complicated option-li\ ke arrangement in which savers hold \ long-term assets directly and buy op\ tions to sell these assets contingen\ t on their liquidity needs. We conc\ lude in section five. Proofs are in\ the appendix.) >> endobj 2424 0 obj [ 2423 0 R 2425 0 R 2426 0 R 2427 0 R 2428 0 R 2429 0 R ] endobj 2425 0 obj << /S /H2 /T (Heading Level 2) /P 2422 0 R /Pg 19 0 R /K 1 /Alt (2. The Savings Environment) >> endobj 2426 0 obj << /S /P /T (Paragraph) /P 2422 0 R /Pg 19 0 R /K 2 /Alt (The savings and investment environm\ ent in our model is simple: in perio\ d zero, savers invest their endowmen\ t in short-term assets and long-term\ assets to support consumption in pe\ riods one and two. Like others, we \ assume savers are uncertain about th\ eir investment horizons. The main w\ rinkle in our setup is that long-ter\ m investors may want to roll over th\ eir short-term assets.) >> endobj 2427 0 obj << /S /P /T (Paragraph) /P 2422 0 R /Pg 19 0 R /K 3 /Alt (We model uncertainty about investme\ nt horizons in the same way that Bry\ ant \(1980\) and others do. In perio\ d one, a fraction rho of savers will\ realize a privately observed liquid\ ity shock that forces them to consum\ e immediately, while the remaining 1\ minus rho will wait until period tw\ o to consume. If consumption in per\ iod t is c sub 1, utility in period \ zero is ) >> endobj 2428 0 obj << /S /P /T (Paragraph_Equation) /P 2422 0 R /Pg 19 0 R /K 4 /Alt (U of c sub t equals U of c sub 1 wi\ th probability rho, or U of c sub 2 \ with probability \(1 minus rho\)) >> endobj 2429 0 obj << /S /P /T (Paragraph) /P 2422 0 R /Pg 19 0 R /Alt (where U prime of dot is less than o\ r equal to 0 and U double prime of \ dot is less than or equal to 0. The\ liquidity shocks could be thought o\ f as an irresistible impulse to cons\ ume, or as a drastic event, like ill\ ness or death, that forces savers to\ consume early. They provide a key \ friction in our setting \(as in othe\ r models\) as they are a tractable me\ ans of modeling withdrawal decisions\ that cannot be reversed simply by t\ he adjustment of a market-clearing p\ rice. Allowing late consumers to su\ bstitute between period-one and peri\ od-two consumption would not change \ our basic result, but as in the rest\ of the literature using these types\ of liquidity shocks, the assumption\ that early consumers cannot postpon\ e consumption until period two is im\ portant. As Hellwig \(1994\) states,\  These [consumption] needs are ine\ xorable ...there is no question of s\ ubstitution between dates 1 and 2.  \ ) /K [ 5 << /Type /MCR /Pg 23 0 R /MCID 8 >> ] >> endobj 2430 0 obj << /S /Sect /T (Section P8) /P 2687 0 R /K [ 2454 0 R 2455 0 R 2456 0 R 2457 0 R 2458 0 R 2459 0 R 2460 0 R 2641 0 R ] >> endobj 2431 0 obj << /S /Sect /T (Section P9) /P 2687 0 R /K [ 2461 0 R 2462 0 R 2463 0 R 2464 0 R 2465 0 R ] >> endobj 2432 0 obj << /S /Sect /T (Section P10) /P 2687 0 R /K [ 2466 0 R 2467 0 R 2468 0 R 2469 0 R 2470 0 R 2471 0 R 2472 0 R 2473 0 R 2474 0 R 2636 0 R 2637 0 R 2638 0 R ] >> endobj 2433 0 obj << /S /Sect /T (Section P11) /P 2687 0 R /K [ 2475 0 R 2476 0 R 2477 0 R 2478 0 R 2479 0 R 2480 0 R 2481 0 R 2482 0 R ] >> endobj 2434 0 obj << /S /Sect /T (Section P12) /P 2687 0 R /K [ 2483 0 R 2484 0 R 2485 0 R 2486 0 R 2487 0 R 2488 0 R 2489 0 R 2490 0 R ] >> endobj 2435 0 obj << /S /Sect /T (Section P13) /P 2687 0 R /K [ 2491 0 R 2492 0 R 2493 0 R 2494 0 R ] >> endobj 2436 0 obj << /S /Sect /T (Section P14) /P 2687 0 R /K [ 2495 0 R 2496 0 R 2497 0 R 2498 0 R 2499 0 R 2500 0 R ] >> endobj 2437 0 obj << /S /Sect /T (Section P15) /P 2687 0 R /K [ 2501 0 R 2502 0 R 2503 0 R 2504 0 R 2505 0 R 2506 0 R 2507 0 R 2627 0 R 2628 0 R 2629 0 R 2630 0 R ] >> endobj 2438 0 obj << /S /Sect /T (Section P16) /P 2687 0 R /K [ 2508 0 R 2642 0 R 2509 0 R 2510 0 R ] >> endobj 2439 0 obj << /S /Sect /T (Section P17) /P 2687 0 R /K [ 2511 0 R 2512 0 R 2513 0 R 2514 0 R 2515 0 R 2516 0 R 2517 0 R 2518 0 R 2519 0 R 2520 0 R ] >> endobj 2440 0 obj << /S /Sect /T (Section P18) /P 2687 0 R /K [ 2521 0 R 2522 0 R 2523 0 R 2524 0 R 2525 0 R ] >> endobj 2441 0 obj << /S /Sect /T (Section P19) /P 2687 0 R /K 2526 0 R >> endobj 2442 0 obj << /S /Sect /T (Section P20 PROBLEM) /P 2687 0 R /K 2649 0 R >> endobj 2443 0 obj << /S /Sect /T (Section P21) /P 2687 0 R /K [ 2536 0 R 2537 0 R 2538 0 R 2539 0 R 2540 0 R 2541 0 R 2542 0 R 2543 0 R ] >> endobj 2444 0 obj << /S /Sect /T (Section P23) /P 2687 0 R /K [ 2544 0 R 2545 0 R 2657 0 R 2547 0 R 2548 0 R ] >> endobj 2445 0 obj << /S /Sect /T (Section P24) /P 2687 0 R /K [ 2549 0 R 2610 0 R 2550 0 R 2551 0 R 2552 0 R 2656 0 R ] >> endobj 2446 0 obj << /S /Sect /T (Section P25) /P 2687 0 R /K [ 2553 0 R 2554 0 R 2555 0 R 2556 0 R 2557 0 R ] >> endobj 2447 0 obj << /S /Sect /T (Section P26) /P 2687 0 R /K [ 2546 0 R 2558 0 R 2559 0 R 2607 0 R 2560 0 R 2561 0 R 2653 0 R 2562 0 R ] >> endobj 2448 0 obj << /S /Sect /T (Section P27) /P 2687 0 R /K [ 2563 0 R 2564 0 R 2565 0 R 2566 0 R ] >> endobj 2449 0 obj << /S /Sect /T (Section P28) /P 2687 0 R /K [ 2567 0 R 2568 0 R 2569 0 R 2570 0 R 2571 0 R 2572 0 R ] >> endobj 2450 0 obj << /S /Sect /T (Section P29) /P 2687 0 R /K [ 2573 0 R 2574 0 R 2575 0 R 2576 0 R ] >> endobj 2451 0 obj << /S /Sect /T (Section P30) /P 2687 0 R /K [ 2577 0 R 2578 0 R 2579 0 R ] >> endobj 2452 0 obj << /S /Sect /T (Section P31) /P 2687 0 R /K [ 2580 0 R 2581 0 R 2582 0 R 2583 0 R 2584 0 R 2585 0 R 2586 0 R 2587 0 R 2588 0 R 2589 0 R 2590 0 R 2591 0 R 2599 0 R 2600 0 R 2601 0 R ] >> endobj 2453 0 obj << /S /Sect /T (Section P32) /P 2687 0 R /K [ 2592 0 R 2593 0 R 2594 0 R 2595 0 R 2596 0 R ] >> endobj 2454 0 obj << /S /P /T (Paragraph) /P 2430 0 R >> endobj 2455 0 obj << /S /P /T (Paragraph) /P 2430 0 R /Pg 23 0 R /K 1 /Alt (Consumption is supported by the ret\ urns from short-term and long-term i\ nvestments made in period zero. Sho\ rt-term assets yield one unit of out\ put in period one per unit invested \ in period zero, and this output can \ be consumed or reinvested until peri\ od two. Long-term assets yield noth\ ing in period one and one unit of ou\ tput in period two \(per unit invest\ ed in period zero\).Footnote Callout \ 4. Here, long-term assets are illi\ quid only in the physical sense that\ they yield no output in the short t\ erm; individuals can trade long-term\ assets for short-term assets, so th\ ey are liquid in the financial sense\ . This contrasts with models that a\ ssume long-term projects can be phys\ ically liquidated, at a loss, but ar\ e not tradable. ) >> endobj 2456 0 obj << /S /P /T (Paragraph) /P 2430 0 R /Pg 23 0 R /K 2 /Alt (A key feature of our setup, however\ , is that in some states, long-term \ investors will want to roll over the\ ir short-term assets. We model this\ by assuming that the period-two ret\ urn on the short-term asset is rando\ m:) >> endobj 2457 0 obj << /S /P /T (Paragraph) /P 2430 0 R /Pg 23 0 R /K 3 /Alt (r sub i equals r sub l less than 1 \ with probability pi, or r sub h with\ probability 1 minus pi) >> endobj 2458 0 obj << /S /P /T (Paragraph) /P 2430 0 R /Pg 23 0 R /K 4 /Alt (Savers learn this return in period \ one, at the same time they learn the\ ir investment horizon. In terms of \ expected returns, long-term assets d\ ominate short-term assets,) >> endobj 2459 0 obj << /S /P /T (Paragraph) /P 2430 0 R /Pg 23 0 R /K [ 5 9 ] /Alt (pi times r sub l plus \(1 minus pi\)\ times r sub h is less than 1) >> endobj 2460 0 obj << /S /P /T (Paragraph) /P 2430 0 R /Pg 23 0 R /Alt (Nonetheless, savers need to hold so\ me short-term assets in case they mu\ st consume early. Whereas in much o\ f the literature this is the only re\ ason that savers hold short-term ass\ ets, in our setup savers will want t\ o reinvest their short-term assets w\ hen their return is high. The optio\ n to reinvest is valuable only if th\ e short-term asset return dominates \ the long-term asset return in the hi\ gh-return state:) /K [ << /Type /MCR /Pg 26 0 R /MCID 13 >> 6 ] >> endobj 2461 0 obj << /S /P /T (Paragraph) /P 2431 0 R /Pg 26 0 R /K 1 /Alt (r sub h is greater than 1. \(Equati\ on 1\)) >> endobj 2462 0 obj << /S /P /T (Paragraph) /P 2431 0 R /Pg 26 0 R /K 2 /Alt (Assumption \(1\) is a fundamental di\ fference in our setup; others genera\ lly assume the opposite inequality \(\ or equivalently, that pi equals 1\).\ Here we merely allow some probabil\ ity that short-term assets can pay m\ ore than long-term assets if short-t\ erm rates rise sufficiently. ) >> endobj 2463 0 obj << /S /P /T (Paragraph) /P 2431 0 R /Pg 26 0 R /K 3 /Alt (Privately observed liquidity shocks\ in tandem with the uncertainty abou\ t future returns on short-term, rein\ vestable projects generate the comm\ itment problem  that our financial i\ ntermediary helps to mitigate. We t\ ake asset returns as primitive facto\ rs determined outside the model. In\ vestment and trading among agents, h\ owever, determine the relative price\ of the two types of assets and the \ equilibrium consumption allocations.\ We present three general equilibri\ um solutions: first we consider the \ planner s solution, then the allocat\ ions yielded by direct investment an\ d trading in decentralized markets, \ and finally the intermediated soluti\ on.) >> endobj 2464 0 obj << /S /P /T (Paragraph) /P 2431 0 R /Pg 26 0 R /K 4 /Alt (The timeline below illustrates the \ sequence of investment, project retu\ rns, and consumption realizations.) >> endobj 2465 0 obj << /S /P /T (Paragraph) /P 2431 0 R /Pg 26 0 R /K [ 5 6 7 8 9 10 11 12 ] /Alt (A timeline illustrating 3 periods, \ Period 0, Period 1 and Period 2 Unde\ r period 0 the following are listed:\ Bullet invest in short-term and lon\ g-term projects Bullet return on sho\ rt-term asset in period one is uncer\ tain and Bullet consumption needs un\ certain Under Period 1 are listed: B\ ullet short-term projects yield 1 pe\ r unit; long-term projects yield zer\ o output Bullet learn period 2 yield\ on short-term projects reinvested i\ n period 1 time r sub h greater than\ 1 greater than r sub 1 Bullet learn\ consumption needs: eaerly consumers\ swap long-term projects for output \ from short-term projects of late con\ sumers and consume this output Bulle\ t late consumers reinvest any remain\ ing short-term projects Under Period\ 2 are listed Bullet long-term proj\ ect pays 1 per unit invested in peri\ od 0 Bullet short-term projects yiel\ d r sub j per unit reinvested in per\ iod one time 1 Bullet late consumers\ eat  and returns from long-term p\ rojects and reinvested short-term pr\ ojects) >> endobj 2466 0 obj << /S /H2 /T (Heading Level 2) /P 2432 0 R /Pg 29 0 R /K 0 /Alt (The First Best ) >> endobj 2467 0 obj << /S /P /T (Paragraph) /P 2432 0 R /Pg 29 0 R /K 1 /Alt (In period zero, the planner collect\ s each saver s unit endowment and in\ vests s sub zero per saver in short\ -term assets and 1 minus s sub zero \ per saver in long-term assets. In p\ eriod one, when the short-term rate \ r sub I is realized, the planner can\ reinvest some fraction s sub I in s\ hort-term assets until period two. \ Reinvestment will be state-contingen\ t; hence the subscript. The planner\ also chooses consumption in each pe\ riod, and consumption will be state-\ dependent as well. The period-one c\ onsumption constraint in state i is \ ) >> endobj 2468 0 obj << /S /P /T (Paragraph) /P 2432 0 R /Pg 29 0 R /K 2 /Alt (C sub 1 i equals \(s sub zero minus\ s sub i\) all over rho) >> endobj 2469 0 obj << /S /P /T (Paragraph) /P 2432 0 R /Pg 29 0 R /K 3 /Alt (Early consumers get the output from\ short-term investments less the amo\ unt that is reinvested. \(Consumptio\ n is scaled up by 1 over rho because\ the planner invests S sub zero, but\ only rho consume early.\) Late cons\ umers get the returns from long-term\ assets and from any short-term asse\ ts rolled over from period one; henc\ e, period-two consumption in state i\ is) >> endobj 2470 0 obj << /S /P /T (Paragraph) /P 2432 0 R /Pg 29 0 R /K [ 4 5 6 ] /Alt (C sub 2 i equals \(1 minus s sub ze\ ro plus r sub i s sub i\), all over \(\ 1 minus rho\)) >> endobj 2471 0 obj << /S /P /T (Paragraph) /P 2432 0 R /Pg 29 0 R /K 7 /Alt (The only other constraint facing th\ e planner is that reinvestment in bo\ th states must be nonnegative:) >> endobj 2472 0 obj << /S /P /T (Paragraph) /P 2432 0 R /Pg 29 0 R /K 8 /Alt (S sub i is greater than or equal to\ zero) >> endobj 2473 0 obj << /S /P /T (Paragraph) /P 2432 0 R /Pg 29 0 R /K 9 /Alt (Subject to these three constraints,\ the planner chooses s sub zero, s s\ ub i, c sub 1i, c sub 2 i. to maximi\ ze the representative saver s expect\ ed utility in period zero:) >> endobj 2474 0 obj << /S /P /T (Paragraph) /P 2432 0 R /Pg 29 0 R /K [ 10 11 ] /Alt (expected utility for c sub t i equa\ ls pi times \(rho utility c sub 1 l \ plus \(1 minus rho\) utility c sub 2 \ l\), plus \(1 minus pi\) times \(rho u\ tility c sub 1 h plus \(1 minus rho\)\ utility c sub 2 h\).) >> endobj 2475 0 obj << /S /P /T (Paragraph) /P 2433 0 R /Pg 34 0 R /K [ 11 12 ] /Alt (U prime of c sub 1 h minus r sub h \ times U prime of c sub 2 h equals la\ mbda sub h divided by \(1 minus pi\) \ \(Equation 4\)) >> endobj 2476 0 obj << /S /P /T (Paragraph) /P 2433 0 R /Pg 34 0 R /K [ 8 9 10 ] /Alt (\(1 minus pi\) times \(U prime of c\ sub 1 h minus U prime of c sub 2 h\)\ equals pi times \(U prime of c sub \ 2 l minus U prime of c sub 1 l\). \(\ Equation 5\)) >> endobj 2477 0 obj << /S /P /T (Paragraph) /P 2433 0 R /Pg 34 0 R /K 7 /Alt (where lambda sub i is the multiplie\ r on the constraint that s sub I is \ greater than or equal to zero. Condi\ tions \(3\) and \(4\) are the combined\ first-order conditions for c sub t \ I and s sub i while \(5\) is the fir\ st-order condition for s sub zero.) >> endobj 2478 0 obj << /S /P /T (Paragraph) /P 2433 0 R /Pg 34 0 R /K 6 /Alt (Naturally enough, the planner s rei\ nvestment strategy for the short-ter\ m asset is contingent on the rate re\ alized in period one: if the rate is\ low, optimal reinvestment is zero; \ if high, reinvestment is positive \(\ see appendix for proof\). Since s s\ ub h is greater than zero implies la\ mbda sub h minus zero, condition \(4\ \) becomes) >> endobj 2479 0 obj << /S /P /T (Paragraph) /P 2433 0 R /Pg 34 0 R /K [ 3 4 5 ] /Alt (U prime of c sub 1 h divided by U p\ rime of c sub 2 h equals r sub h. \(\ Equation 6\)) >> endobj 2480 0 obj << /S /P /T (Paragraph) /P 2433 0 R /Pg 34 0 R /K 2 /Alt (in the high interest-rate state. T\ his standard intertemporal efficienc\ y condition has the planner transfer\ goods from period one to period two\ \(by rolling over short-term assets\ \) until the marginal \(social\) rate \ of substitution between the early an\ d late consumption equals the margin\ al rate of transformation. Since r \ sub h is greater than 1, c sub 1 h i\ s less than c sub 2 h, late consumer\ s benefit more when the return on th\ e short-term asset is high. On the \ other hand, s sub l implies lambda s\ ub l is greater than zero, hence c s\ ub 1 l is greater than c sub 2 l, ea\ rly consumers are compensated for th\ eir sacrifice in the high interest-r\ ate state by more consumption in the\ low-rate state. Below, we show tha\ t when savers invest on their own, w\ ithout benefit of a planner or an in\ termediary, they may violate \(6\); C\ sub 1 h is too high, C sub 2h is to\ o low, and reinvestment is inadequat\ e when short-term project returns ar\ e high.) >> endobj 2481 0 obj << /S /P /T (Paragraph) /P 2433 0 R /Pg 34 0 R /K 1 /Alt (Substituting \(6\) into \(5\) yields\ ) >> endobj 2482 0 obj << /S /P /T (Paragraph) /P 2433 0 R /Pg 34 0 R /K 0 /Alt (Footnote 5 Reinvestment in the low-\ return state is at least conceivable\ . If r sub h minus 1 were large en\ ough relative to 1 minus r sub l, th\ e planner might invest only in the s\ hort-term asset in order to maximize\ reinvestment in the high-return sta\ te. With zero long-term investment,\ the planner must also reinvest in t\ he low state to support late consump\ tion. That strategy is suboptimal, \ however, because the short-term asse\ t has a lower expected return. Like\ wise, zero reinvestment in the high-\ return state is also conceivable. C\ arrying over short-term assets means\ the planner could have invested mor\ e in the long term at time zero, whi\ ch has the higher expected return. \ Nevertheless, the possibility of the\ high short-term return, and the sav\ er s willingness to reallocate consu\ mption, lead him to reinvest.) >> endobj 2483 0 obj << /S /P /T (Paragraph) /P 2434 0 R /Pg 40 0 R /K [ 7 8 ] /Alt (pi times \(U prime of c sub 2 l min\ us U prime of c sub 1 l \) equals \(1\ minus pi\) times \(r sub h minus 1\) \ times U prime of c sub 2 h \(Equatio\ n 7\)) >> endobj 2484 0 obj << /S /P /T (Paragraph) /P 2434 0 R /Pg 40 0 R /K 0 /Alt (The left-hand side is the net margi\ nal opportunity cost of investing in\ short-term assets when their return\ is low. In this state, the planner\ does not reinvest; he gives the uni\ t return in period one to early cons\ umers, which raises their utility by\ U prime \(c sub 1 l\) Had the planne\ r invested in the long-term asset in\ stead, he would have paid the unit r\ eturn to the late consumers, which w\ ould have raised their utility by U \ prime \(c sub 2 l\) Since c sub 2 l i\ s less than c sub 1 l the opportuni\ ty cost of investing in short-term a\ ssets on the downside \(when their r\ eturn is low\) will be positive. Foot\ note callout 6. The right side o\ f \(7\) is the net marginal benefit o\ f investing in short-term assets whe\ n their return turns out to be high.\ In this state, the planner reinves\ ts in short-term assets and earns r \ sub h. Had he invested more in the \ long-term asset in period zero, he w\ ould have earned only 1, so an extra\ unit of short-term investment raise\ s late consumers  utility by r sub h\ minus 1 times U prime \(c sub 2h\). \ The optimal portfolio equates the m\ arginal cost of short-term investmen\ ts on the downside with the marginal\ benefit on the upside dash when the\ ir return is high.) >> endobj 2485 0 obj << /S /P /T (Paragraph) /P 2434 0 R /Pg 40 0 R /K 1 /Alt (The optimal portfolio and consumpti\ on allocations are determined in the\ following manner. Since s sub l eq\ uals 0 and s sub h is greater than \ 0, consumption in the low short-term\ rate state depends only on s sub 0,\ whereas consumption in the high-rat\ e state depends on reinvestment as w\ ell:) >> endobj 2486 0 obj << /S /P /T (Paragraph) /P 2434 0 R /Pg 40 0 R /K 2 /Alt (c sub 1 l equals s sub 0 divided by\ rho.) >> endobj 2487 0 obj << /S /P /T (Paragraph) /P 2434 0 R /Pg 40 0 R /K 3 /Alt (c sub 2 l equals \(1 minus s sub 0\)\ divided by \(1 minus rho\)) >> endobj 2488 0 obj << /S /P /T (Paragraph) /P 2434 0 R /Pg 40 0 R /K 4 /Alt (c sub 1 h equals \(s sub 0 minus s \ sub h\) divided by rho.) >> endobj 2489 0 obj << /S /P /T (Paragraph) /P 2434 0 R /Pg 40 0 R /K 5 /Alt (c sub 2 h equals \(1 minus s sub 0 \ plus r sub h times s sub h\) all over\ \(1 minus rho\).) >> endobj 2490 0 obj << /S /P /T (Paragraph) /P 2434 0 R /Pg 40 0 R /K 6 /Alt (Footnote 6 The low short-term rate\ , r sub l does not enter the equatio\ n because the planner never invests \ at that rate; he pays out all short-\ term asset returns in that state to \ early consumers. ) >> endobj 2491 0 obj << /S /P /T (Paragraph) /P 2435 0 R /Pg 44 0 R /K 0 /Alt (Substituting these equations into \(\ 6\) and \(7\) determines the first-be\ st amount of short-term investment a\ nd the optimal amount of reinvestmen\ t when short-term asset returns are \ high. This portfolio and reinvestme\ nt strategy then determines the firs\ t-best consumption allocation.) >> endobj 2492 0 obj << /S /H3 /T (Heading Level 3) /P 2435 0 R /Pg 44 0 R /K 1 /Alt (Risk-Neutral Savers: An Example ) >> endobj 2493 0 obj << /S /P /T (Paragraph) /P 2435 0 R /Pg 44 0 R /K 2 /Alt (As we show below, the role for an i\ ntermediary depends on how the consu\ mption allocation achieved by direct\ investment and trading compares wit\ h the first best. As we also demons\ trate, the lower the degree of inves\ tor risk aversion, the worse the sav\ ers fare by direct investment. Henc\ e, it is useful to consider the firs\ t-best allocation when savers are ri\ sk neutral. Suppose U\(c\) equals c \ for both early and late consumers. \ As usual with linear preferences, a \ corner solution is optimal: : s sub \ 0 equals s sub h equals 1 The planne\ r invests only in short-term assets \ and reinvests the entire portfolio \(\ c sub 1 h equals 0\) if the return t\ urns out to be high; if the return i\ s low, he pays out everything to ear\ ly consumers: \(c sub 2 l equals 0\)\ . ) >> endobj 2494 0 obj << /S /P /T (Paragraph) /P 2435 0 R /Pg 44 0 R /Alt (Although the corner solution is not\ surprising here, the particular cor\ ner is surprising; we thought that t\ he planner might hold only long-term\ assets, since their return exceeds \ the expected return on short-term as\ sets. With all long-term assets, ho\ wever, savers get only expected util\ ity equals 1 whereas the all-short-t\ erm portfolio yields expected utilit\ y equals pi plus \(l minus pi\) times\ r sub h is greater than 1. What is\ key here is that the possible low r\ eturn on short-term assets is never \ realized because in that state the p\ lanner pays out everything to early \ consumers. Thus, the effective retu\ rn on short-term projects in the low\ -rate state is simply early consumer\ s  marginal utility U prime \(c\) equ\ als 1 \(in this case\). Thus, the al\ l short-term portfolio does better b\ ecause the planner avoids low return\ s by paying out everything to early \ consumers while he fully exploits hi\ gh returns by reinvesting everything\ in that state. Risk-neutral indivi\ duals are completely amenable to an \ arrangement that structures payments\ to maximize the portfolio s overall\ return.) /K [ << /Type /MCR /Pg 48 0 R /MCID 7 >> 3 ] >> endobj 2495 0 obj << /S /H2 /T (Heading Level 2) /P 2436 0 R /Pg 48 0 R /K 1 /Alt (3. Direct Investment and Trading) >> endobj 2496 0 obj << /S /P /T (Paragraph) /P 2436 0 R /Pg 48 0 R /K 2 /Alt (There is nothing special about the \ assets here that would prevent saver\ s from directly investing and tradin\ g on their own. But savers cannot g\ enerally replicate the planner s sol\ ution simply by funding their own po\ rtfolios in period zero and then tra\ ding assets after they observe short\ -term project returns and privately-\ observed liquidity shocks in period \ one. Below we characterize the dece\ ntralized equilibrium relative to th\ e first best and show how the decent\ ralized solution violates the planne\ r s allocation.) >> endobj 2497 0 obj << /S /P /T (Paragraph) /P 2436 0 R /Pg 48 0 R /K 3 /Alt (A few preliminary observations will\ simplify the analysis. Since saver\ s are identical \r\nex ante, they all\ choose the same initial portfolio i\ n period zero. In period one, early\ consumers will want to trade their \ long-term assets for short-term asse\ ts held by late consumers. The supp\ ly of long-term assets is perfectly \ inelastic because they yield no outp\ ut until period two and therefore ar\ e worthless to early consumers. The\ demand for the long-term assets, ho\ wever, will depend on the realized r\ eturn to reinvesting short-term asse\ ts, as this return equals the opport\ unity cost of trading short-term ass\ ets to early consumers.) >> endobj 2498 0 obj << /S /P /T (Paragraph) /P 2436 0 R /Pg 48 0 R /K 4 /Alt (Let p sub i denote the period-one p\ rice of a unit of long-term asset in\ terms of short-term assets in inter\ est-rate state i. In period one, ea\ rly consumers get the return on thei\ r short-term assets, plus the income\ from selling their long-term assets\ to late consumers:) >> endobj 2499 0 obj << /S /P /T (Paragraph) /P 2436 0 R /Pg 48 0 R /K 5 /Alt (c sub 1 i equals s sub 0 plus p sub\ i times \(1 minus s sub 0\) ) >> endobj 2500 0 obj << /S /P /T (Paragraph) /P 2436 0 R /Pg 48 0 R /K 6 /Alt (If each late consumer purchases l s\ ub i units of long-term assets in st\ ate i, period-one reinvestment in sh\ ort-term assets is ) >> endobj 2501 0 obj << /S /P /T (Paragraph) /P 2437 0 R /Pg 52 0 R /K 0 /Alt (s sub i equals s sub 0 minus p sub \ i l sub i) >> endobj 2502 0 obj << /S /P /T (Paragraph) /P 2437 0 R /Pg 52 0 R /K 1 >> endobj 2503 0 obj << /S /P /T (Paragraph) /P 2437 0 R /Pg 52 0 R /K [ 2 3 ] /Alt (Reinvestment cannot be negative:) >> endobj 2504 0 obj << /S /P /T (Paragraph) /P 2437 0 R /Pg 52 0 R /K 4 /Alt (s sub i is greater than or equal to\ 0 ) >> endobj 2505 0 obj << /S /P /T (Paragraph) /P 2437 0 R /Pg 52 0 R /K 5 /Alt (This constraint ensures that each l\ ate consumer has enough short-term a\ ssets to cover his purchases of long\ -term assets. In state late consu\ mers get the unit return from their \ long-term assets \(those they funded\ directly and those they purchased i\ n period one\) plus the period-two yi\ eld on reinvested short-term project\ s: ) >> endobj 2506 0 obj << /S /P /T (Paragraph) /P 2437 0 R /Pg 52 0 R /K 6 /Alt (c sub 2 i equals 1 minus s sub 0 pl\ us l sub i plus r sub i times s sub \ i.) >> endobj 2507 0 obj << /S /P /T (Paragraph) /P 2437 0 R /Pg 52 0 R /K [ 44 42 41 40 39 38 8 9 10 11 12 ] /Alt (In period zero, savers choose Brack\ et S sub 0, s suz I, c sub t I, l su\ b I, to maximize their expected util\ ity, subject to the aforementioned c\ onstraints. The first-order conditi\ ons for can be combined as follows\ :) >> endobj 2508 0 obj << /S /P /T (Paragraph) /P 2438 0 R /Pg 57 0 R /Alt (Therefore, The first-order conditio\ n for s sub zero takes two lines:) /K [ 0 3 4 ] >> endobj 2509 0 obj << /S /P /T (Paragraph) /P 2438 0 R /Pg 57 0 R /K [ 1 6 ] /Alt (The left side is the expected net b\ enefit of increasing s sub 0 by anot\ her unit in the event that its retur\ n turns out high. Early consumers g\ et the unit return less the opportun\ ity cost of having one less long-ter\ m asset to sell in period one; but s\ ince p sub h is less than 1 increasi\ ng s sub 0 benefits early consumers.\ Late consumers also benefit on net\ , since they can reinvest and earn r\ sub h is greater than 1. An additi\ onal benefit to late consumers is la\ mbda sub h, the value of having more\ short-term assets to trade for long\ -term assets. The right side of cond\ ition \(11\) is the benefit of increa\ sing investment in the long-term ass\ et by a unit in the event that short\ -term asset returns are low. Since \ p sub l is greater than 1 is greater\ than r sub l investing more in lon\ g-term assets in period zero benefit\ s both early consumers and late cons\ umers in the low-return state. Howe\ ver, the minus lambda sub I reflects\ the cost to late consumers of not h\ aving more short-term projects to se\ ll in period one. In addition to the\ se ex ante equilibrium conditions, t\ he long-term asset market must also \ clear in both interest-rate states. \ In either state, the fraction rho \ of the population will each supply \ 1 minus s sub 0 long-term assets in \ period one; thus the aggregate suppl\ y of long-term assets is rho times \ 1 minus s sub 0. In state i, the fra\ ction : \(1 minus rho\) of the popul\ ation will each purchase l sub i lon\ g-term assets; hence aggregate deman\ d for long-term assets in period one\ equals \(1 minus rho\) times l sub i\ . Aggregate supply equals aggregate\ demand when) >> endobj 2510 0 obj << /S /P /T (Paragraph) /P 2438 0 R /Pg 57 0 R /K [ << /Type /MCR /Pg 61 0 R /MCID 11 >> 2 ] /Alt (Footnote 7 They would expect to buy\ long-term assets on the cheap in pe\ riod on but would find that none was\ available, in which case the price \ would be unbounded, which contradict\ s p sub l is less than 1) >> endobj 2511 0 obj << /S /P /T (Paragraph) /P 2439 0 R /Pg 61 0 R /Alt (rho times \(1 minus s sub 0\) equals\ \(1 minus rho\) times l sub I \(Equ\ ation 12\)) /K [ 1 12 ] >> endobj 2512 0 obj << /S /P /T (Paragraph) /P 2439 0 R /Pg 61 0 R /K 2 /Alt (To determine whether the spot marke\ t equilibrium equals the first best,\ we conjecture an equilibrium like t\ he first best and then check if the \ resulting consumption allocations ar\ e indeed first best. A key feature \ of the first best is that the planne\ r reinvests in short-term assets whe\ n their return is high. Accordingly\ , we conjecture s sub h is greater t\ han 0, which implies) >> endobj 2513 0 obj << /S /P /T (Paragraph) /P 2439 0 R /Pg 61 0 R /K 3 /Alt (rho sub h equals 1 over r sub h is \ less than 1 \(Equation 13\)) >> endobj 2514 0 obj << /S /P /T (Paragraph) /P 2439 0 R /Pg 61 0 R /K 4 /Alt (At that price, consumption in the h\ igh-return state is) >> endobj 2515 0 obj << /S /P /T (Paragraph) /P 2439 0 R /Pg 61 0 R /K 5 /Alt ( c sub 1 h equals s sub 0 plus \(1 \ minus s sub 0\) divided by r sub h \(\ Equation 14\)) >> endobj 2516 0 obj << /S /P /T (Paragraph) /P 2439 0 R /Pg 61 0 R /K 6 /Alt ( c sub 2 h equals 1 minus s sub 0 p\ lus r sub h times s sub 0 \(Equatio\ n 15\)) >> endobj 2517 0 obj << /S /P /T (Paragraph) /P 2439 0 R /Pg 61 0 R /K 7 /Alt (Period-two consumption turns out to\ be independent of the amount of rei\ nvestment when the long-term asset s\ ells for its full discounted value; \ late consumers are indifferent betwe\ en reinvesting or buying long-term a\ ssets at their market-clearing price\ . Relative consumption in the high-\ return state is) >> endobj 2518 0 obj << /S /P /T (Paragraph) /P 2439 0 R /Pg 61 0 R /K 8 /Alt (c sub 1 h equals c sub 2 h over r s\ ub h \(Equation 16\) ) >> endobj 2519 0 obj << /S /P /T (Paragraph) /P 2439 0 R /Pg 61 0 R /K 9 /Alt (For most preferences, this ratio wi\ ll differ from the first-best ratio \ determined by \(6\), and the role for\ an intermediary depends on how thes\ e allocations differ. ) >> endobj 2520 0 obj << /S /P /T (Paragraph) /P 2439 0 R /Pg 61 0 R /K 10 /Alt (Since reinvestment in the low-retur\ n state equals zero in the first bes\ t, we conjecture s sub l equals 0 he\ re. If late consumers do not reinve\ st in the low state, they must trade\ all of their short-term assets for \ long-term assets: ) >> endobj 2521 0 obj << /S /P /T (Paragraph) /P 2440 0 R /Pg 66 0 R /K 0 /Alt (p sub 1 equals s sub 0 times \(1 mi\ nus rho\), all over rho times \(1 min\ us s sub 0\) \(Equation 17\)) >> endobj 2522 0 obj << /S /P /T (Paragraph) /P 2440 0 R /Pg 66 0 R /K 1 /Alt (The price of the long-term asset in\ the low-return state is increasing \ in the amount of period-zero short-t\ erm investment and the fraction of l\ ate consumers. More short-term asse\ ts means late consumers have more to\  spend   on long-term assets and f\ ewer to bid on. More late consumers\ means more buyers and fewer sellers\ . Since p sub l is greater than 1 \(\ 17\) implies that s sub 0 is greater\ than rho ) >> endobj 2523 0 obj << /S /P /T (Paragraph) /P 2440 0 R /Pg 66 0 R /K 2 /Alt (The equilibrium price, portfolio mi\ x, and other variables are determine\ d as follows: Given that : s sub l\ equals 0 we know that c sub 1 l equ\ als s sub 0 over rho and c sub 2 l \ equals \(1 minus s sub 0\) over \(1 m\ inus rho\) Given : s sub h is greate\ r than 0 we know that lambda sub h e\ quals 0 and p sub h equals 1 over r \ sub h Substituting these values, alo\ ng with \(14\), \(15\), and \(9\), into\ \(11\) produces an equation in s sub\ 0 and p sub l. This equation and \(\ 17\) determine period-zero short-ter\ m investments and the period-one pri\ ce of long-term assets when short-te\ rm returns are low. These values de\ termine the consumption allocations \ in each state. Finally, given that \ p sub h equals 1 over r sub h, s sub\ 0 and \(12\) determine l sub h and r\ einvestment in the high-return state\ : p sub h equals 1 over r sub h, s s\ ub 0 minus rho sub h l sub h) >> endobj 2524 0 obj << /S /P /T (Paragraph) /P 2440 0 R /Pg 66 0 R /K 3 >> endobj 2525 0 obj << /S /P /T (Paragraph) /P 2440 0 R /Pg 66 0 R /K 4 /Alt (Recall that when U times Paren c Cl\ ose Paren equals c, the planner inve\ sts only in the short-term asset eve\ n though it has a lower expected ret\ urn. Clearly when investors are ris\ k neutral, this first-best solution \ is not attainable through decentrali\ zed contracting \(technically there \ is nothing to swap if s sub 0 equals\ 1\). Even though individuals know t\ hat, ex ante, they would be better o\ ff promising to redistribute on the \ basis of whether it is optimal to co\ nsume early or to reinvest, they kno\ w that after the fact the ex post  \ losers   have no incentive to honest\ ly reveal their type. If no transfe\ rs occur, then s sub 0 equals 1 yiel\ ds expected utility equals rho plus \ \(1 minus rho\) \(\(1 minus pi\) r sub\ h plus pi r sub i\) is less than 1) >> endobj 2526 0 obj << /S /P /T (Paragraph) /P 2441 0 R /Pg 71 0 R /Alt (Savers will not choose to invest ev\ erything in long-term assets either.\ Suppose that they did and that in \ period one, the rho early consumers\ simply handed over their long-term \ assets to late consumers \(since lon\ g-term assets are worthless to early\ consumers\). Each late consumer wou\ ld receive : c sub 2 equals 1 plus \ \(rho over \(1 minus rho\)\) and there\ fore expected utility equals \(1 min\ us rho\) times c sub 2 equals 1 for t\ he all-long-term project portfolio.\r\ \nInstead, risk-neutral savers, inve\ sting on their own, will reject both\ corners, choose a mixed portfolio i\ n period zero, and then trade in per\ iod one. With linear utility, the f\ irst-order condition \(11\) is a func\ tion only of p sub l Footnote callou\ t 8. At the equilibrium price, p \ double prime sub l savers are indiff\ erent to all values of s sub 0 becau\ se, at this price, the downside oppo\ rtunity cost of low short-term asset\ return exactly offsets the upside b\ enefit of high returns. Substitutin\ g p double prime sub l into \(17\) de\ termines s double prime sub 0 the po\ rtfolio consistent with that price. \ This equilibrium is unique because \ p sub l is increasing in s sub 0. \ The equilibrium is also stable; if s\ avers expected p sub l is greater th\ an p double prime sub l, they would \ raise s sub 0 and p sub l would fall\ ; if they expected p sub l is less t\ han p double prime sub l, they would\ lower s sub 0 and p sub l would ris\ e. Although at p double prime sub l\ savers are indifferent about s sub \ 0, their expected consumption given \ ??? and ??? is strictly higher than \ at any other combination. However, \ although this mixed portfolio domina\ tes either corner solution, it yield\ s lower expected utility than the pl\ anner s solution; for 0 is less than\ s sub 0 is less than 1, \r\nexpected\ utility equals pi plus \(1 minus pi\)\ times [s sub 0 \(rho plus \(1 minus \ rho\) r sub h\) plus \(1 minus s sub \ 0 \) \(\(1 minus rho\) plus rho over r\ sub h\)] is greater than 1 is less t\ han expected utility equals pi plus \ \(l minus pi\) r sub h as yielded by \ the first best. \r\n3.1 Risk Avers\ ion and Investment Distortions\r\nHer\ e we characterize how direct investm\ ent and trading fail to generate the\ first-best allocation under more ge\ neral preferences. Suppose\r\nFootno\ te 6. We can replace p sub h with 1 \ over r sub h since savers will reinv\ est in the high state, s sub 0) /K [ 9 10 11 13 0 1 ] >> endobj 2527 0 obj << /S /P /T (Paragraph) /K [ 7 8 ] /Pg 71 0 R /Alt (Instead, risk-neutral savers, inves\ ting on their own, will reject both \ corners, choose a mixed portfolio in\ period zero, and then trade in peri\ od one. With linear utility, the fi\ rst-order condition \(11\) is a funct\ ion only of p sub l Footnote callout\ 8. At the equilibrium price, s\ avers are indifferent to all values \ of s sub 0 because, at this price, t\ he downside opportunity cost of low \ short-term asset return exactly offs\ ets the upside benefit of high retur\ ns. Substituting into \(17\) deter\ mines the portfolio consistent with\ that price. This equilibrium is un\ ique because p sub l is increasing \ in s sub 0. The equilibrium is also\ stable; if savers expected p sub 1 \ is greater than ???, they would rais\ e s sub 0 and p sub l would fall; if\ they expected p sub 1 is less than \ ??, they would lower s sub 0 and p s\ ub l would rise. Although at ??? sa\ vers are indifferent about s sub 0, \ their expected consumption given ???\ and ??? is strictly higher than at \ any other combination. However, alt\ hough this mixed portfolio dominates\ either corner solution, it yields l\ ower expected utility than the plann\ er s solution; for 0 is less than s \ sub 0 is less than 1, ) >> endobj 2528 0 obj << /S /H2 /T (Heading Level 2) /K 12 /Pg 71 0 R /Alt (expected utility equals pi plus \(1\ minus pi\) times [s sub 0 \(rho plus\ \(1 minus rho\) r sub h\) plus \(1 mi\ nus s sub 0 \) \(\(1 minus rho\) plus \ rho over r sub h\)] is greater than 1\ is less than expected utility equal\ s pi plus \(l minus pi\) r sub h as y\ ielded by the first best. ) >> endobj 2529 0 obj << /S /P /T (Paragraph) /K 3 /Pg 71 0 R /Alt (3.1 Risk Aversion and Investment D\ istortions) >> endobj 2530 0 obj << /S /P /T (Paragraph) /Alt (Footnote 8 We can replace p sub h\ with 1 over r sub h since savers wi\ ll reinvest in the high state, s sub\ 0.) /Pg 71 0 R /K [ 2 6 ] >> endobj 2531 0 obj << /S /P /T (Paragraph) /Alt (U of c equals c raised to the power\ of 1 minus alpha, all over 1 minus \ alpha \(Equation 18\)) /Pg 76 0 R /K [ 0 1 2 17 0 1 2 9 ] >> endobj 2532 0 obj << /S /P /T (Paragraph) /K 13 /Pg 76 0 R >> endobj 2533 0 obj << /S /P /T (Paragraph) /K 20 /Pg 76 0 R >> endobj 2534 0 obj << /S /P /T (Paragraph) /K 19 /Pg 76 0 R >> endobj 2535 0 obj << /S /P /T (Paragraph) /K [ 3 14 15 21 22 23 24 ] /Pg 76 0 R >> endobj 2536 0 obj << /S /H3 /T (Heading Level 3) /P 2443 0 R /Pg 81 0 R /K 0 /Alt (Upside-Downside Distortions) >> endobj 2537 0 obj << /S /P /T (Paragraph) /P 2443 0 R /Pg 81 0 R /K 1 /Alt (Savers underinvest in short-term as\ sets because their upside benefit is\ too low and their downside cost \(w\ hen their return is low\) is too high\ . We illustrate these distortions f\ or the risk preferences given by \(1\ 8\). Even for these standard prefere\ nces, the extent of nonlinearity in \ the first-order conditions precludes\ a closed-form solution for s sub 0.\ Instead, we reduce the first-order\ conditions to a single equation in \ s sub 0 and compare the properties o\ f the first best with those of the d\ ecentralized market solution. Using\ \(6\) to eliminate s sub h from \(7\)\ produces the solution for the first\ best:) >> endobj 2538 0 obj << /S /P /T (Paragraph) /P 2443 0 R /Pg 81 0 R /K 2 /Alt (pi times f of s sub 0 equals \(1 mi\ nus pi\) times \(1 minus 1 over r sub\ h\) times j of \(r sub h\) times g of\ \(r sub h s sub 0\) \(Equation 19\)) >> endobj 2539 0 obj << /S /P /T (Paragraph) /P 2443 0 R /Pg 81 0 R /K 3 /Alt (Let s star sub 0 denote the solutio\ n to \(19\). The pi on the left side\ and the 1 minus pi on the right sid\ e indicate that the planner determin\ es s star sub 0 by weighing the dow\ nside of low returns with the upside\ of the high-return state. On the d\ ownside, the marginal opportunity co\ st of the short-term asset is ) >> endobj 2540 0 obj << /S /P /T (Paragraph) /P 2443 0 R /Pg 81 0 R /K 4 /Alt (f of dot is equivalent to \(\(1 minu\ s rho\) over \(1 minus s sub 0\)\), al\ l raised to the power of alpha, minu\ s \(rho over s sub 0\), raised to the\ power of alpha) >> endobj 2541 0 obj << /S /P /T (Paragraph) /P 2443 0 R /Pg 81 0 R /K 5 /Alt (where f is positive \(since s sub 0\ is greater than rho\) and increasing\ in s sub 0. This expression is the\ net gain in utility from having inv\ ested in another long-term asset and\ giving the return to late consumers\ . The benefit when the short-term a\ sset return is high depends on ) >> endobj 2542 0 obj << /S /P /T (Paragraph) /P 2443 0 R /Pg 81 0 R /K [ 6 7 8 ] /Alt (G of dot is equivalent to 1 over \(\ 1 plus \(r sub h minus 1\) times s su\ b 0\), the denominator raised to the \ alpha power G of dot is equivalent t\ o 1 over \(1 plus \(r sub h minus 1\)\ times s sub 0\), the denominator rai\ sed to the alpha power) >> endobj 2543 0 obj << /S /P /T (Paragraph) /P 2443 0 R /Pg 81 0 R /K 9 /Alt (which equals the gain in utility fr\ om having another short-term asset t\ o reinvest at r sub h \(and pay out \ to late consumers\). Note that g is \ decreasing in s sub 0.) >> endobj 2544 0 obj << /S /P /T (Paragraph) /P 2444 0 R /Pg 89 0 R /K 0 /Alt (Because x super a is concave and rh\ o is less than 1, j is greater than \ k.Footnote Callout 9. The upside b\ enefit is lower with direct investme\ nt because late consumers pay too mu\ ch for long-term assets and therefor\ e reinvest too little. The differen\ ce in the downside costs, for a give\ n s sub 0, equals \r\nh of s sub 0 mi\ nus f of s sub 0 equals \(p sub l to\ the alpha power minus 1\) times \(rh\ o over s sub 0\) raised to the power \ of \(alpha plus 1\)\r\n) >> endobj 2545 0 obj << /S /P /T (Paragraph) /P 2444 0 R /Pg 89 0 R /K 1 /Alt (where p sub l is the price of the l\ ong-term asset in the low interest-r\ ate state. The shadow price of s su\ b 0 to the planner equals one \(he c\ ould have invested in long-term proj\ ects, each yielding one unit\). But \ p sub l is greater than 1 in the mar\ ket solution, so the marginal cost o\ f holding short-term assets \(that i\ s, the value of having a long-term p\ roject\) in the low-rate state is als\ o higher under direct investment tha\ n for the planner. Both higher cost\ s on the downside and lower benefits\ on the upside cause savers to under\ invest in short-term assets.) >> endobj 2546 0 obj << /S /P /T (Paragraph) /P 2447 0 R /Pg 89 0 R /K [ 2 5 ] >> endobj 2547 0 obj << /S /P /T (Paragraph) /P 2444 0 R /Pg 89 0 R /K 3 /Alt (Footnote 9 Let x equals r sub h and\ x prime equals r super 1 alpha/sub \ h. Then j equals \(rho times x plus \ \(1 minus rho\) times x prime\) allrai\ sed to the alpha power is greater th\ an rho times x raised to the alpha p\ ower plus \(1 minus rho\) times x pri\ me raised to the alpha power equals\ k) >> endobj 2548 0 obj << /S /P /T (Paragraph) /P 2444 0 R /Pg 89 0 R /K 4 /Alt (Footnote 10 If the upside on the ne\ w asset is r sub hh, the benefit cur\ ves are coincident if \(1 minus 1 o\ ver r sub hh\) k \(r sub hh\) g \(r su\ b hh times s sub 0\) equals \(1 minus\ 1 over r sub h\) j \(r sub h\) g \(r \ sub h times s sub 0\)) >> endobj 2549 0 obj << /S /P /T (Paragraph) /P 2445 0 R /Pg 92 0 R /K 1 /Alt (The difference between the optimal \ portfolio and savers  direct investm\ ents is decreasing in the degree of \ risk aversion \(Figure 2\). The only\ case in which the distortion is zer\ o is when utility equals ln c In th\ is case, \(6\) implies c sub 2 h equa\ ls r su h times c sub 1 h the relati\ ve consumption yielded by direct inv\ estment and trading. We stress this\ to emphasize that our financial int\ ermediaries are not providing insura\ nce in the sense of consumption smoo\ thing.) >> endobj 2550 0 obj << /S /P /T (Paragraph) /P 2445 0 R /Pg 92 0 R /K 3 /Alt (Delegating investment decisions to \ an intermediary can raise savers  ex\ pected utility and may allow them to\ realize the first-best level of wel\ fare. In period zero, savers deposi\ t their endowment with the intermedi\ ary, who offers them a two-period st\ ate-contingent contract. The contra\ cting problem facing the intermediar\ y is the same as the allocation prob\ lem solved by the planner, except th\ at the intermediary cannot observe a\ given saver s realized liquidity ne\ eds in period one.Footnote callout 1\ 1. Privately-observed liquidity n\ eeds create an incentive problem: i\ f those who withdraw early earn too \ much, long-term savers will pose as \ early consumers to withdraw early an\ d reinvest on their own until period\ two. To prevent this, the contract\ offered by the intermediary must sa\ tisfy the following incentive constr\ aint:) >> endobj 2551 0 obj << /S /P /T (Paragraph) /P 2445 0 R /Pg 92 0 R /K 4 /Alt (r sub i c sub 1 i is less than or e\ qual to c sub 2 i. \(Equation 21\)) >> endobj 2552 0 obj << /S /P /T (Paragraph) /P 2445 0 R /Pg 92 0 R /K 5 /Alt (The intermediary s problem is a con\ strained version of the planner s pr\ oblem. Thus, to test whether the in\ termediary can attain the first best\ , we need to test only if the additi\ onal constraint binds in either stat\ e in the first-best allocation. Sav\ ers  preferences will influence whet\ her it does. If utility \(c\) equals\ c to the 1 minus alpha power over \(\ 1 minus alpha\) and alpha is less th\ an 1 the incentive constraint does n\ ot bind when the returns to reinvest\ ment are high, since \(6\) implies c \ sub 2 h equals r sub h to the 1/alph\ a power times c sub 1 h is greater t\ han r sub h times c sub 1 h Indeed\ , the optimal allocation is to penal\ ize early consumers when the return \ to reinvesting is high, so late cons\ umers certainly have no incentive to\ misrepresent their type in this sta\ te. ) >> endobj 2553 0 obj << /S /P /T (Paragraph) /P 2446 0 R /Pg 95 0 R /K 1 /Alt (The incentive constraint may bind i\ n the low-return state, however. Th\ e planner does not reinvest in that \ state, so the ratio c sub 1 l over c\ sub 2 l may be high enough to viola\ te \(21\). If so, the intermediary m\ ust reduce short-term and increase l\ ong-term investment. Foonote callout\ 12. Substituting c sub 1 l equal\ s sub 0 over rho and : c sub 2 l eq\ uals \(1 minus s sub 0\) over \(1 min\ us rho\) into \(21\) determines the i\ ntermediary s maximum incentive-comp\ atible level of short-term investmen\ t: ) >> endobj 2554 0 obj << /S /P /T (Paragraph) /P 2446 0 R /Pg 95 0 R /K [ 2 3 ] /Alt (s bar sub o equals rho over \(rho p\ lus \(1 minus rho\) times r sub l) >> endobj 2555 0 obj << /S /P /T (Paragraph) /P 2446 0 R /Pg 95 0 R /K 4 /Alt (The upper limit on short-term inves\ tment falls as r sub l rises; a high\ er downside increases the return to \ reinvestment, so the intermediary mu\ st reduce the amount available for l\ ate consumers to reinvest. Higher \ rho increases the upper limit becaus\ e short-term assets are distributed \ across more consumers in the low-ret\ urn state.) >> endobj 2556 0 obj << /S /P /T (Paragraph) /P 2446 0 R /Pg 95 0 R /K 5 /Alt (Figure 1 shows that the intermediar\ y achieves the first best if s bar s\ ub 0 falls to the right of s prime s\ ub 0 If s bar sub 0 is greater than\ s star sub 0, the downside cost of \ the short-term asset exceeds the ups\ ide benefit, so the intermediary wil\ l not want to hold s bar sub 0 in th\ e first place. If s star sub 0 is g\ reater than s bar sub 0, the interme\ diary cannot achieve the first best,\ but he still dominates direct inves\ tment except when savers happen to c\ hoose s bar sub 0 on their own. Foot\ note 13 Callout.) >> endobj 2557 0 obj << /S /P /T (Paragraph) /P 2446 0 R /Pg 95 0 R /K 6 /Alt (Footnote 12 Reinvesting in the shor\ t-term asset will also relax the inc\ entive constraint, but investing mor\ e in the long-term asset is the most\ efficient way to raise period-two c\ onsumption. Reinvestment is determi\ ned by \(6\) even if the incentive co\ nstraint binds.) >> endobj 2558 0 obj << /S /P /T (Paragraph) /P 2447 0 R /Pg 98 0 R /K 4 >> endobj 2559 0 obj << /S /P /T (Paragraph) /P 2447 0 R /Pg 98 0 R /K 8 /Alt (The intermediary described here ser\ ves a very different role from that \ considered by Diamond and Dybvig \(1\ 983\). Their intermediary shifts the\ higher return from longer-term asse\ ts to early consumers so as to smoot\ h consumption over time. Our interm\ ediary does just the opposite; it al\ lows savers to increase their portfo\ lio returns by making early withdraw\ als contingent on their opportunity \ cost in terms of foregone investment\ returns.) >> endobj 2560 0 obj << /S /P /T (Paragraph) /P 2447 0 R /Pg 98 0 R /K 12 /Alt (As with other models of this type, \ unobservable liquidity shocks are ne\ cessary for savers to need an interm\ ediary \(or a social planner\). If l\ iquidity shocks are observable, then\ savers can achieve the first best t\ hrough direct trading as long as con\ tracts can be enforced. Footnote cal\ lout 14 In addition, liquidity sh\ ocks must be such that they necessit\ ate the complete liquidation of one \ s investments, irrespective of price\ ; early consumers must get zero util\ ity from postponing consumption. Su\ ppose instead that savers get utilit\ y from consuming in both periods, bu\ t we define early consumers as peopl\ e with relatively high marginal rate\ s of substitution between early and \ late consumption, U prime \(c sub 1\)\ over U prime \(c sub 2\)In that case\ , savers could still invest directly\ in the short-term asset. After the\ y learned rates, both types would U \ prime \(c sub 1\) over U prime \(c su\ b 2\) equals r sub 2 Consumption all\ ocations would differ, but with the \ marginal rate of substitution for ea\ ch type equal to the marginal rate o\ f transformation ex post, there woul\ d be no gains to locking up short-te\ rm assets with an intermediary. Put\ ting it differently, savers would ma\ ke the same investments in period on\ e that they would make in period zer\ o, so there would be no commitment p\ roblem and no need for an intermedia\ ry.) >> endobj 2561 0 obj << /S /P /T (Paragraph) /P 2447 0 R /Pg 98 0 R /K 13 /Alt (Footnote 13 Evaluating \(19\) at s \ bar sub o produces a condition on th\ e parameters that determines whether\ the intermediary achieves the first\ best: : s dot sub o is greater tha\ n s bar sub o is equivalent to pi \(\ 1 over r sub l raised to the alpha p\ ower minus 1 over \(rho times r sub h\ plus \(1 minus rho\) r sub l\) to th\ e alpha power is greater than \(1 min\ us pi\) \(1 minus 1 over r sub h\) j \ \(r sub h\)) >> endobj 2562 0 obj << /S /P /T (Paragraph) /P 2447 0 R >> endobj 2563 0 obj << /S /P /T (Paragraph) /P 2448 0 R /Pg 105 0 R /K 3 /Alt (would differ, but with the marginal\ rate of substitution for each type \ equal to the marginal rate of transf\ ormation ex post, there would be no \ gains to locking up short-term asset\ s with an intermediary. Putting it \ differently, savers would make the s\ ame investments in period one that t\ hey would make in period zero, so th\ ere would be no commitment problem a\ nd no need for an intermediary.) >> endobj 2564 0 obj << /S /H2 /T (Heading Level 2) /P 2448 0 R /Pg 105 0 R /K 2 /Alt (4.2 Another Option) >> endobj 2565 0 obj << /S /P /T (Paragraph) /P 2448 0 R /Pg 105 0 R /K 1 /Alt (As in Boyd and Prescott \(1986\), th\ e intermediary in our model is essen\ tially a contract among a coalition \ of savers. The contract is not uniqu\ e. Savers could invest directly in \ the long-term asset and buy an optio\ n to sell it the next period in case\ they needed to consume early. This\ would solve the essential purpose o\ f the intermediary: locking up liqui\ d assetsout of the reach of early co\ nsumers. This forward market does a\ s well as the intermediary, but the \ arrangement is certainly more compli\ cated.) >> endobj 2566 0 obj << /S /P /T (Paragraph) /P 2448 0 R /Pg 105 0 R /K 0 /Alt (The option market works as follows:\ In period zero, investors divide th\ eir wealth between short-term assets\ and long-term assets. They can als\ o buy an option to sell long-term as\ sets in period one for a price of \ in state i. Late consumers, who do \ not exercise this option, will get a\ rebate of in state i. In period \ zero, savers pay a dealer a fee of f\ for this option, and the dealer in\ vests this fee in short-term assets.\ Fee revenues must be sufficient to\ buy up all the early consumers  lo\ ng-term assets in period one at the \ stipulated price. Any excess revenu\ e is reinvested in short-term assets\ , and the proceeds from reinvestment\ are rebated to late consumers along\ with the long-term assets purchased\ from early consumers.) >> endobj 2567 0 obj << /S /P /T (Paragraph) /P 2449 0 R /Alt (By manipulating the budget constrai\ nts of the consumers and dealers, we\ can show that the equilibrium contr\ act can improve upon the direct inve\ stment allocation. Footnote callout \ 15. The trick is to show that com\ bined constraints on savers and the \ dealer reduce to the consumption con\ straints on the planners  problem. \ This option contract gives early con\ sumers c sub 1 l equals q sub i time\ s \(1 minus s sub 0\) plus s sub 0 Th\ e dealer must collect enough in fees\ to purchase the long-term assets fr\ om the rho savers that will need to\ sell in period one: f is greater th\ an or equal to rho times q sub i l s\ ub o Let s sub I denote any excess \ fees that are reinvested inshort-ter\ m assets. The dealers constraint\ in period one is s sub i is equival\ ent to f minus rho q sub i l sub o i\ s greater than or equal to 0.) /Pg 108 0 R /K 0 >> endobj 2568 0 obj << /S /P /T (Paragraph) /P 2449 0 R /Alt (Eliminating q sub I l sub 0 from th\ ese two equations implies\r\nRho c su\ b 1 I equals f minus s sub I plus rh\ o s sub 0\r\n) /Pg 108 0 R /K 1 >> endobj 2569 0 obj << /S /P /T (Paragraph) /P 2449 0 R /Alt (Consumption by late consumers is C \ sub 2 i equals 1 minus s sub 0 plus \ r sub i s sub 0 plus R sub i \(Equat\ ion 22\)) /Pg 108 0 R /K 2 >> endobj 2570 0 obj << /S /P /T (Paragraph) /P 2449 0 R /Alt (The dealer must earn enough on rein\ vestment and period-one purchases of\ long-term assets to pay the period-\ two rebate to the \( 1 minus rho\) sa\ vers that consume late:\r\n\(1 minus \ rho\) R sub i equals r sub i s sub I \ plus rho times \(1 minus s sub 0\)) /Pg 108 0 R /K 3 >> endobj 2571 0 obj << /S /P /T (Paragraph) /P 2449 0 R /Pg 108 0 R /K 4 /Alt (Eliminating R sub ii from these two\ equations implies\r\n\(1 minus rho\) \ c sub 2 i equals 1 minus s sub 0 plu\ s r sub i s sub i plus \(1 minus rho\ \) times r sub i s sub 0 \(Equation 2\ 3\)\r\n) >> endobj 2572 0 obj << /S /P /T (Paragraph) /P 2449 0 R /Pg 108 0 R /Alt (Footnote 15 Since the option dealer\ cannot observe the savers  type, he\ faces the same incentive constraint\ as the intermediary, \(21\). This c\ ondition ensures that late consumers\ will not exercise the option and th\ en reinvest the proceeds in short-te\ rm assets. This constraint will be \ just as tight under the option marke\ t as under the intermediary. Since \ the option price will differ from th\ e price that would prevail on the sp\ ot market, late consumers might also\ have an arbitrage opportunity; they\ could exercise the option to sell t\ heir long-term assets and then repur\ chase them from early consumers on t\ he spot market. But the spot market\ will never open. If q sub I is les\ s than p sub i, long-term consumers \ would not want to buy on the spot ma\ rket. And if qi > pi, early consume\ rs would not want to sell on the spo\ t market. ) /K [ 5 6 ] >> endobj 2573 0 obj << /S /P /T (Paragraph) /P 2450 0 R /Pg 113 0 R /K 3 /Alt (Note that if s sub 0 equals 0 \(tha\ t is, savers do not directly invest \ in any short-term assets\), the consu\ mption constraints \(22\) and \(23\) a\ re the same as in the planner s prob\ lem. Let s star denote the first-be\ st level of investment in short-term\ assets. The solution to the option\ contract is the same, as long as f \ equals s star. However, like the int\ ermediary, the dealer also faces an \ incentive constraint limiting high p\ ayouts to early consumers. ) >> endobj 2574 0 obj << /S /P /T (Paragraph) /P 2450 0 R /Pg 113 0 R /K 2 /Alt (The crucial feature of this scenari\ o is that savers do not hold short-t\ erm assets directly. This prevents \ early consumers from consuming too m\ uch when it is optimal to reinvest. \ Direct investment in the long-term \ assets poses no problems because sav\ ers do not have discretion regarding\ the timing of their consumption. U\ nlike the intermediary, who invests \ in both the short-term and long-term\ asset in period zero, the option de\ aler invests only in short-term asse\ ts. Still, the dealer equilibrium i\ s intermediated in some sense, as sa\ vers give over a portion of their we\ alth to a third party, who invests i\ t, and they trade no other assets am\ ongst themselves.) >> endobj 2575 0 obj << /S /H2 /T (Heading Level 2) /P 2450 0 R /Pg 113 0 R /K 1 /Alt (5. Conclusion) >> endobj 2576 0 obj << /S /P /T (Paragraph) /P 2450 0 R /Pg 113 0 R /K 0 /Alt (Much of the intermediation literatu\ re focuses on mitigating the variabi\ lity of possible outcomesa standard \ view of what it means to enhance the\ liquidity of investments or provide\ insurance. Here we discuss a role \ that intermediaries can play in faci\ litating optimal investment strategi\ es even \(and especially\) when inves\ tors are not particularly risk avers\ e. Although our savers face liquidi\ ty shocks, they do not want the same\ sort of consumption smoothing that \ Diamond and Dybvig \(1983\) model in \ characterizing contracts more akin t\ o traditional bank deposits. Noneth\ eless, although assets are completel\ y tradable, savers can do better if \ they delegate their investment decis\ ions to an intermediary as a sort of\ piggy bank that overrides savers  l\ iquidity demands when investment opp\ ortunities warrant. ) >> endobj 2577 0 obj << /S /P /T (Paragraph) /P 2451 0 R /Pg 116 0 R /K 0 /Alt (Indeed, one might call this sort of\ commitment role ensurance  in the \ sense that intermediaries can achiev\ e a better investment outcome for sa\ vers \(given their preferences\) by e\ nsuring that liquidators honestly re\ veal their type.) >> endobj 2578 0 obj << /S /P /T (Paragraph) /P 2451 0 R /Pg 116 0 R /K 1 /Alt (We do not aim to diminish the stand\ ard functions of intermediaries emph\ asized in the literature. Rather, h\ ere we use a relatively standard typ\ e of model to illustrate an addition\ al role that an intermediary can pla\ y. This role involves funding optim\ al investment strategies that invest\ ors cannot achieve on their own beca\ use of the possibility that, ex post\ , some investors will regret making \ what was an optimal investment decis\ ion ex ante. The bottom line is tha\ t savers can achieve a better \(expe\ cted\) investment outcome by delegati\ ng their resources to the intermedia\ ry to implement the ex post payoffs.\ ) >> endobj 2579 0 obj << /S /P /T (Paragraph) /P 2451 0 R /Pg 116 0 R /K 2 /Alt (Although this piggy bank function m\ ay not apply literally to any partic\ ular type of real-world intermediary\ , in the abstract it may apply to al\ l of them. By restricting access to\ funds in the short run, intermediat\ ion may help individuals achieve bet\ ter long-run investment returns resi\ sting incentives to withdraw at inop\ portune times. Thus this model buil\ ds on the more general idea that loc\ king funds in an intermediary, wheth\ er a bank CD, a pension plan, or an \ insurance annuity, may help some sav\ ers stick to long-term plans that th\ ey would not commit to if they held \ funds directly. Obviously, these in\ termediaries attract funds for other\ reasons as well namely, the higher \ or better-diversified returns that t\ hey offer. Our point is simply that\ the liquidity premium demanded by s\ ome savers may not be as large as on\ e might expect since, for some portf\ olio choices, illiquidity may be a b\ lessing in disguise.) >> endobj 2580 0 obj << /S /H2 /T (Heading Level 2) /P 2452 0 R /Pg 119 0 R /K 0 /Alt (References) >> endobj 2581 0 obj << /S /P /T (Paragraph) /P 2452 0 R /Pg 119 0 R /K 1 /Alt (Barro, Robert J. 1997. Myopia and \ Inconsistency in the New Classical G\ rowth Model. Working Paper Series, \ no. 6317. National Bureau of Econom\ ic Research.) >> endobj 2582 0 obj << /S /P /T (Paragraph) /P 2452 0 R /Pg 119 0 R /K 2 /Alt (Bernanke, Ben, and Mark Gertler. 1\ 987. Banking in General Equilibrium\ . In New Approaches to Monetary Econ\ omics, edited by William A. Barnett \ and Kenneth J. Singleton. New York: \ Cambridge University Press.) >> endobj 2583 0 obj << /S /P /T (Paragraph) /P 2452 0 R /Pg 119 0 R /K 3 /Alt (Boyd, John H., and Edward C. Presco\ tt. 1986. Financial Intermediary C\ oalitions. Journal of Economic Theor\ y 38:211-32.) >> endobj 2584 0 obj << /S /P /T (Paragraph) /P 2452 0 R /Pg 119 0 R /K [ 4 5 6 ] /Alt (Bryant, J. 1980. A Model of Reserv\ es, Bank Runs, and Deposit Insurance\ . Journal of Banking and Finance 4:\ 335-44.) >> endobj 2585 0 obj << /S /P /T (Paragraph) /P 2452 0 R /Pg 119 0 R /K 7 /Alt (Diamond, Douglas W. 1984. Financia\ l Intermediation and Delegated Monit\ oring. Review of Economic Studies 51\ :393-414.) >> endobj 2586 0 obj << /S /P /T (Paragraph) /P 2452 0 R /Pg 119 0 R /K 8 /Alt (Diamond, Douglas, W., and Philip Dy\ bvig. 1983. Bank Runs, Deposit Insu\ rance, and Liquidity. Journal of Pol\ itical Economy, 91:401-19.) >> endobj 2587 0 obj << /S /P /T (Paragraph) /P 2452 0 R /Pg 119 0 R /K 9 /Alt (Diamond, Douglas, W. 1997. Liquidit\ y, Banks, and Markets. Journal of Po\ litical Economy, 105:928-56.) >> endobj 2588 0 obj << /S /P /T (Paragraph) /P 2452 0 R /Pg 119 0 R /K 10 /Alt (Franklin, Allen. Financial Markets\ , Intermediaries, and Intertemporal \ Smoothing. Journal of Political Ec\ onomy, 105, no. 3:523-46.) >> endobj 2589 0 obj << /S /P /T (Paragraph) /P 2452 0 R /Pg 119 0 R /K 11 /Alt (Hellwig, Martin. 1994. Liquidity P\ rovision, Banking, and the Allocatio\ n of Interest Rate Risk. European E\ conomic Review, 1363-89.) >> endobj 2590 0 obj << /S /P /T (Paragraph) /P 2452 0 R /Pg 119 0 R /K 12 /Alt (Laibson, David. 1997. Golden Eggs \ and Hyperbolic Discounting. Quarter\ ly Journal of Economics 117:443-78.\ ) >> endobj 2591 0 obj << /S /P /T (Paragraph) /P 2452 0 R /Pg 119 0 R /K 13 /Alt (Holmstrom, Bengt, and Jean Tirole. \ 1998. Private and Public Supply of \ Liquidity. Journal of Political Eco\ nomy, February:1-40.) >> endobj 2592 0 obj << /S /H2 /T (Heading Level 2) /P 2453 0 R /Pg 122 0 R /K 0 /Alt (Appendix) >> endobj 2593 0 obj << /S /P /T (Paragraph) /P 2453 0 R /Alt (This appendix states and proves fou\ r lemmas useful in simplifying the p\ lanner s first-order conditions for \ the optimal investment and consumpti\ on allocations.) /Pg 122 0 R /K 1 >> endobj 2594 0 obj << /S /P /T (Paragraph) /P 2453 0 R /Pg 122 0 R /K 2 >> endobj 2595 0 obj << /S /P /T (Paragraph) /P 2453 0 R /Pg 122 0 R /K 3 >> endobj 2596 0 obj << /S /P /T (Paragraph) /P 2453 0 R /Pg 122 0 R /K 4 >> endobj 2597 0 obj [ 2592 0 R 2593 0 R 2594 0 R 2595 0 R 2596 0 R ] endobj 2598 0 obj [ 2580 0 R 2581 0 R 2582 0 R 2583 0 R 2584 0 R 2584 0 R 2584 0 R 2585 0 R 2586 0 R 2587 0 R 2588 0 R 2589 0 R 2590 0 R 2591 0 R 2599 0 R 2600 0 R 2601 0 R ] endobj 2599 0 obj << /S /P /T (Paragraph) /P 2452 0 R /Pg 119 0 R /K 14 /Alt (Strotz, R. Myopia and Inconsisten\ cy in Dynamic Utility Maximization. \ Review of Economic Studies 23 \(19\ 55-1956\): 165-180.) >> endobj 2600 0 obj << /S /P /T (Paragraph) /P 2452 0 R /Pg 119 0 R /K 15 /Alt (Thaler, Richard, and H.M. Shefrin. \ An Economic Theory of Self-Control\ .  Journal of Political Economy 89 \ \(1981\): 392-406.) >> endobj 2601 0 obj << /S /P /T (Paragraph) /P 2452 0 R /Pg 119 0 R /K 16 /Alt (Wallace, Neil. 1990. A Banking Mode\ l Where Partial Suspension Is Best. \ Federal Reserve Bank of Minneapolis \ Quarterly Review Fall:11-23.) >> endobj 2602 0 obj [ 2577 0 R 2578 0 R 2579 0 R ] endobj 2603 0 obj [ 2576 0 R 2575 0 R 2574 0 R 2573 0 R ] endobj 2604 0 obj [ null null null null null null null null null null ] endobj 2605 0 obj [ 2566 0 R 2565 0 R 2564 0 R 2563 0 R ] endobj 2606 0 obj [ null null null null null null null null ] endobj 2607 0 obj << /S /H2 /T (Heading Level 2) /P 2447 0 R /Pg 98 0 R /K 10 /Alt (4.1 Necessary Conditions for Inter\ mediation) >> endobj 2608 0 obj [ null null null null null null ] endobj 2609 0 obj [ null null null null null null ] endobj 2610 0 obj << /S /H2 /T (Heading Level 2) /P 2445 0 R /Pg 92 0 R /K 2 /Alt (4. Financial Intermediaries) >> endobj 2611 0 obj [ 2544 0 R 2545 0 R 2546 0 R 2547 0 R 2548 0 R 2546 0 R 2657 0 R ] endobj 2612 0 obj << /S /Sect /T (Section P22) /P 2687 0 R /K [ 2613 0 R 2614 0 R 2615 0 R 2617 0 R ] >> endobj 2613 0 obj << /S /P /T (Paragraph) /P 2612 0 R /Pg 85 0 R /K 0 /Alt (For the market solution, an equatio\ n comparable to \(19\) can be obtaine\ d by the substitution of the equilib\ rium long-term asset prices \(12\) in\ to condition \(11\) \(along with the \ substitutions described above\): \r\nP\ i times h of s sub 0 equals \(1 minu\ s pi\) times \(1 minus 1 over r sub h\ \) times k of r sub h times g of r su\ b h s sub 0 \(Equation 20\) \r\n) >> endobj 2614 0 obj << /S /P /T (Paragraph) /P 2612 0 R /Pg 85 0 R /K 1 /Alt (The marginal cost of holding anothe\ r short term asset when its return i\ s low equals\r\nh of dot is approxima\ tely \(\(1 minus s sub 0\) times \(1 m\ inus rho over 1 minus s sub 0\) rais\ ed to the alpha power plus s sub 0 t\ imes \(rho over s sub 0\) raised to t\ he alpha power\), all multiplied by \(\ \(s sub 0 minus rho\) over \(s sub 0 \ times \(1 minus s sub 0\)\)\)\r\n) >> endobj 2615 0 obj << /S /P /T (Paragraph) /P 2612 0 R /Pg 85 0 R /K 2 /Alt (Increasing s sub 0 has two partiall\ y offsetting effects on h of dot. T\ he term in the square brackets is th\ e weighted average of late consumers\  and early consumers  respective ma\ rginal utilities, where the weights \ are the respective portfolio shares \ held in long-term and short-term inv\ estments. Raising s sub 0 lowers th\ is term since this shifts more weigh\ t to early consumers, whose marginal\ utility declines with s sub 0. The\ term on the far right reflects the \ fact that increasing s sub 0 increas\ es the value of the long-term assets\ \(since there are fewer of them\), w\ hich directly increases the opportun\ ity cost of funding another short-te\ rm asset. The net effect is positiv\ e: h of dot is increasing in s sub \ 0.) >> endobj 2616 0 obj [ 2613 0 R 2614 0 R 2615 0 R 2617 0 R ] endobj 2617 0 obj << /S /P /T (Paragraph) /P 2612 0 R /Pg 85 0 R /K 3 /Alt (A comparison of benefits and costs \ indicates why savers underinvest in \ the short-term asset when they inves\ t directly. The difference in the b\ enefits of investing in short-terms \ project on the upside \(when short-t\ erm project returns are high\) can be\ found by comparison of\r\nj of dot i\ s approximately \(rho times r sub h \ plus \(1 minus rho\) times r sub h ra\ ised to the 1 over alpha power\), all\ raised to the a power, with K of do\ t approximately rho r sub h raised t\ o the alpha power plus \(1 minus rho\ \) times r sub h) >> endobj 2618 0 obj [ 2536 0 R 2537 0 R 2538 0 R 2539 0 R 2540 0 R 2541 0 R 2542 0 R 2542 0 R 2542 0 R 2543 0 R ] endobj 2619 0 obj [ 2531 0 R 2531 0 R 2531 0 R 2535 0 R null null null null null 2531 0 R 2531 0 R 2531 0 R 2620 0 R 2532 0 R 2535 0 R 2535 0 R 2622 0 R 2531 0 R 2620 0 R 2534 0 R 2533 0 R 2535 0 R 2535 0 R 2535 0 R 2535 0 R null null null null null null null null ] endobj 2620 0 obj << /S /P /T (Paragraph) /K [ 12 18 ] /Pg 76 0 R >> endobj 2621 0 obj [ null null 2530 0 R 2529 0 R null 2643 0 R 2530 0 R 2527 0 R 2527 0 R 2526 0 R 2526 0 R 2526 0 R 2528 0 R 2526 0 R ] endobj 2622 0 obj << /S /P /T (Paragraph) /K 16 /Pg 76 0 R >> endobj 2623 0 obj [ 2521 0 R 2522 0 R 2523 0 R 2524 0 R 2525 0 R ] endobj 2624 0 obj [ null 2511 0 R 2512 0 R 2513 0 R 2514 0 R 2515 0 R 2516 0 R 2517 0 R 2518 0 R 2519 0 R 2520 0 R 2510 0 R 2511 0 R ] endobj 2625 0 obj [ 2508 0 R 2509 0 R 2510 0 R 2508 0 R 2508 0 R 2642 0 R 2509 0 R ] endobj 2626 0 obj [ 2501 0 R 2502 0 R 2503 0 R 2503 0 R 2504 0 R 2505 0 R 2506 0 R null 2507 0 R 2507 0 R 2507 0 R 2507 0 R 2507 0 R null null null null null null 2627 0 R 2628 0 R null 2629 0 R null 2627 0 R 2628 0 R 2628 0 R 2628 0 R 2628 0 R 2629 0 R 2629 0 R 2630 0 R null null null null null null 2507 0 R 2507 0 R 2507 0 R 2507 0 R 2507 0 R 2627 0 R 2507 0 R 2629 0 R 2630 0 R 2630 0 R 2630 0 R 2630 0 R 2630 0 R 2630 0 R ] endobj 2627 0 obj << /S /P /T (Paragraph) /P 2437 0 R /Pg 52 0 R /K [ 19 24 43 ] /Alt (s sub i times lambda sub i equals 0\ \(Equation 8\)) >> endobj 2628 0 obj << /S /P /T (Paragraph) /P 2437 0 R /Pg 52 0 R /K [ 20 25 26 27 28 ] /Alt (\(1 divided by p sub l minus r sub \ l\) times U prime of c sub 2 l equals\ lambda sub l over \(pi times \(1 min\ us rho\)\) \(Equation 9\)) >> endobj 2629 0 obj << /S /P /T (Paragraph) /P 2437 0 R /Pg 52 0 R /K [ 22 29 30 45 ] /Alt (\(1 divided by p sub h minus r sub \ h\) times U prime of c sub 2 h equals\ lambda sub h over \(\(1 minus pi\) t\ imes \(1 minus rho\)\) \(Equation 10\)\ ) >> endobj 2630 0 obj << /S /P /T (Paragraph) /P 2437 0 R /Pg 52 0 R /Alt (These conditions pin down the relat\ ionship between period-one reinvestm\ ent and the price of long-term asset\ s in a given state. Since s sub i i\ s greater than 0 implies that lambd\ a sub i equals 0 implies that p sub \ i equals 1 over r sub i; positive re\ investment in state implies that \ the long-term asset price equals its\ discounted value in that state. \(\ The price cannot be lower than 1 ove\ r r sub i or late consumers would se\ ll short-term assets rather than rei\ nvesting at r sub i.\) Conversely, p \ sub i is less than 1 over r sub i im\ plies that lambda sub i is greater t\ han 0 implies that s sub i equals o \ implies that p sub i l sub i equals \ s sub 0; hence, if the long-term ass\ et s price is less than its discount\ ed value, reinvestment is zero and l\ ong-term savers will trade all their\ short-term assets for long-term ass\ ets. In tandem these relationships \ imply that p sub i is less than or e\ qual to 1 over r sub i \(Obviously i\ f p sub i is greater than 1 over r s\ ub i,late consumers are better off n\ ot selling any short-term projects.\)\ Therefore, p sub h is less than or\ equal to 1 over r sub h is less tha\ n 1 implies that p sub l is greater \ than 1, otherwise savers would not i\ nvest in long-term assets in period \ zero. Footnote callout 7 These obs\ ervations will come in handy below.\ ) /K [ 31 51 50 49 48 47 46 ] >> endobj 2631 0 obj [ null 2495 0 R 2496 0 R 2497 0 R 2498 0 R 2499 0 R 2500 0 R 2494 0 R ] endobj 2632 0 obj [ 2491 0 R 2492 0 R 2493 0 R 2494 0 R ] endobj 2633 0 obj [ 2484 0 R 2485 0 R 2486 0 R 2487 0 R 2488 0 R 2489 0 R 2490 0 R 2483 0 R 2483 0 R ] endobj 2634 0 obj [ 2482 0 R 2481 0 R 2480 0 R 2479 0 R 2479 0 R 2479 0 R 2478 0 R 2477 0 R 2476 0 R 2476 0 R 2476 0 R 2475 0 R 2475 0 R ] endobj 2635 0 obj [ 2466 0 R 2467 0 R 2468 0 R 2469 0 R 2470 0 R 2470 0 R 2470 0 R 2471 0 R 2472 0 R 2473 0 R 2474 0 R 2474 0 R 2636 0 R 2637 0 R 2638 0 R ] endobj 2636 0 obj << /S /P /T (Paragraph) /P 2432 0 R /Pg 29 0 R /K 12 /Alt (Except for the state-contingent asp\ ect, this is a straightforward progr\ amming problem. The first-order cond\ itions can be combined as follows:) >> endobj 2637 0 obj << /S /P /T (Paragraph) /P 2432 0 R /Pg 29 0 R /K 13 /Alt (S sub i lambda sub i equals zero. \ \(Equation 2\)) >> endobj 2638 0 obj << /S /P /T (Paragraph) /P 2432 0 R /Pg 29 0 R /K 14 /Alt ( Utility prime c sub 1 l minus r su\ b l utility prime c sub 2 l equals l\ ambda sub l over pi \(Equation 3\)) >> endobj 2639 0 obj [ null 2461 0 R 2462 0 R 2463 0 R 2464 0 R 2465 0 R 2465 0 R 2465 0 R 2465 0 R 2465 0 R 2465 0 R 2465 0 R 2465 0 R 2460 0 R ] endobj 2640 0 obj [ null 2455 0 R 2456 0 R 2457 0 R 2458 0 R 2459 0 R 2460 0 R 2641 0 R 2429 0 R 2459 0 R ] endobj 2641 0 obj << /S /P /T (Paragraph) /P 2430 0 R /Pg 23 0 R /K 7 /Alt (Footnote 4. The assumption of uni\ t returns is not essential; it just \ keeps the notation cleaner.) >> endobj 2642 0 obj << /S /P /T (Paragraph) /P 2438 0 R /Pg 57 0 R /K 5 /Alt (\(1 minus pi\) times [rho times \(1 \ minus p sub h\) times U prime of c su\ b 1 h plus \(1 minus rho\) times \(r \ sub h minus 1\) times U prime of c su\ b 2 h] plus lambda sub h equals pi t\ imes [rho times \(p sub l minus 1\) t\ imes U prime of c sub 1 l plus \(1 m\ inus rho\) times \(1 minus r sub l\) t\ imes U prime of c sub 2 l] minus lam\ bda sub l. \(Equation 11\)) >> endobj 2643 0 obj << /S /P /T (Paragraph) /Alt (Here we characterize how direct inv\ estment and trading fail to generate\ the first-best allocation under mor\ e general preferences. Suppose) /Pg 71 0 R /K 5 >> endobj 2644 0 obj << /S /P /T (Paragraph) /K [ 3 10 ] /Pg 76 0 R >> endobj 2645 0 obj << /S /P /T (Paragraph) /K [ 4 5 6 11 ] /Pg 76 0 R >> endobj 2646 0 obj << /S /P /T (Paragraph) /K 7 /Pg 76 0 R >> endobj 2647 0 obj << /S /P /T (Paragraph) /K 8 /Pg 76 0 R >> endobj 2648 0 obj [ 2531 0 R 2531 0 R 2531 0 R 2644 0 R 2645 0 R 2645 0 R 2645 0 R 2646 0 R 2647 0 R null null null null null null null null ] endobj 2649 0 obj << /S /P /T (Paragraph) /P 2442 0 R /Pg 76 0 R /K 0 /Alt (U of c equals c raised to the power\ of 1 minus alpha, all over 1 minus \ alpha \(Equation 18\) where higher a\ lpha belongs to [0,1\) indicates inc\ reasing risk aversion. Equation \(16\ \) then implies \r\nU prime of c sub \ 1 h equals r sub h raised to the pow\ er of alpha, times U prime of c sub \ 2 h\r\nWhen savers invest directly in\ period zero, early consumption in t\ he high-return state is higher than \ in the first best; thus there is too\ little reinvestment. The inequalit\ y implies that the planner could rai\ se welfare by taking a unit from ear\ ly consumers, reinvesting at r sub h\ , and giving the proceeds to late co\ nsumers \(but of course, ex post, ear\ ly consumers will not voluntarily ma\ ke this trade\). \r\nThe lower the \ degree of savers  risk aversion, the\ greater the commitment problem and \ the inefficiency associated with dir\ ect investing and trading. The unde\ rpinning of the distortion is that t\ he ratio c sub 2h over c sub 1 h is \ independent of alpha when savers inv\ est directly because relative consum\ ption is pinned down by the relative\ returns on traded assets; the first\ -best ratio c sub 2h over c sub 1 h \ however, is a function of risk prefe\ rences \(here, decreasing in alpha\).\ Less-risk-averse individuals are m\ ore flexible about shifting consumpt\ ion in order to exploit high returns\ , but their inability to commit ex p\ ost to the optimal savings plan lead\ s them to underinvest in these proje\ cts. Savers  incentives are also di\ storted on the downside \(when the sh\ ort-term asset return is low\), agai\ n because the market price they face\ differs from the shadow price facin\ g the planner.\r\nIf, as in some mode\ ls, late consumers could be limited \ in terms of what they could trade fo\ r long-term asset markets when reinv\ estment returns were high, p sub h, \ could be reduced and reinvestment in\ creased. Although this limited part\ icipation would reduce the liquidity\ of the long-term asset, as in Diamo\ nd \(1997\), reduced liquidity in thi\ s case would ameliorate the commitme\ nt problem, as in Laibson \(1997\). \ ) >> endobj 2650 0 obj [ 2649 0 R ] endobj 2651 0 obj [ null 2549 0 R 2610 0 R 2550 0 R 2551 0 R 2552 0 R 2656 0 R ] endobj 2652 0 obj [ null 2553 0 R 2554 0 R 2554 0 R 2555 0 R 2556 0 R 2557 0 R ] endobj 2653 0 obj << /S /P /T (Paragraph) /P 2447 0 R /Pg 98 0 R /K 15 /Alt (Footnote 14 If there is uncertainty\ only about returns but not about th\ e timing of consumption needs, saver\ s can do just as well by investing d\ irectly. Suppose savers know whether\ they will need to consume in period\ one or two. The solution is trivia\ l; all early consumers invest only i\ n the short-term asset whereas late \ consumers invest only in the long-te\ rm asset, since the latter has the h\ ighest expected return. In this cas\ e there is no role for an intermedia\ ry \(or a planner\).) >> endobj 2654 0 obj [ null null null null 2558 0 R null null null 2559 0 R null 2607 0 R null 2560 0 R 2561 0 R null 2653 0 R ] endobj 2655 0 obj [ 2567 0 R 2568 0 R 2569 0 R 2570 0 R 2571 0 R 2572 0 R 2572 0 R ] endobj 2656 0 obj << /S /P /T (Paragraph) /P 2445 0 R /Pg 92 0 R /K 6 /Alt (Footnote 11 There are no fixed cost\ s to intermediation, so there is fre\ e entry. Competition among intermed\ iaries ensures that the survivors mu\ st offer contracts that maximize sav\ ers  expected utility at time zero.\ ) >> endobj 2657 0 obj << /S /P /T (Paragraph) /P 2444 0 R /Pg 89 0 R /K 6 /Alt (The sources of the investment disto\ rtion are illustrated in Figure 1. \ Note that the distortions on each si\ de are independent. Suppose savers \ have access to a better short-term a\ sset with a higher upside return, r \ sub h, but the same downside risk, p\ i. The benefit curve shifts up with\ r sub h, but the cost curve does no\ t change. Suppose the upside on the\ new asset is high enough to compens\ ate direct investors for their inabi\ lity to fully exploit the high retur\ n. In this case, the upside benefit\ on the new asset to savers is coinc\ ident with the upside benefit on the\ old asset to the planner. Footnote \ callout 10. Savers would choose s \ star sub 0 of the new short-term ass\ et, less than s star sub 0 because t\ he cost on the downside would still \ be too high \(relative to the planne\ r s allocation\). Again, note that i\ f participation by long-term asset b\ uyers were limited, as in Diamond \(\ 1997\), the long-term asset price wou\ ld be lower and the distortion would\ be reduced.) >> endobj 2658 0 obj [ 2526 0 R 2526 0 R ] endobj 2659 0 obj << /S /Sect /T (Section Pages 33 and 34) /P 2687 0 R /Pg 125 0 R /K [ 2661 0 R 2662 0 R ] >> endobj 2660 0 obj [ null 2661 0 R ] endobj 2661 0 obj << /S /P /T (Paragraph) /P 2659 0 R /Pg 125 0 R /K 1 /Alt (The sources of the investment disto\ rtion are illustrated in Figure 1. \ Note that the distortions on each si\ de are independent. Suppose savers \ have access to a better short-term a\ sset with a higher upside return r s\ ub-h, but the smae downside risk, pi\ e. The benefit curve shifts up with \ r sub-h, but the cost curve does not\ change. Suppose the upside on the \ new asset is high enough to compensa\ te direct investors for their inabil\ ity to fully exploit the high return\ . In this case, the upside benefit \ on the new asset to saver is coincid\ ent with the upside benefit on the o\ ld asset to the planner.) >> endobj 2662 0 obj << /S /P /T (Paragraph) /P 2659 0 R /Pg 130 0 R /K 0 /Alt (The difference between teh optimal \ portfolio and savers' direct investm\ ents is decreasing in the degree of \ risk aversion. The only case in whi\ ch the distortiono is zero is when u\ equals the natural log of c. In th\ is case \(6\) implies c sub 2h equals\ r sub-h c sub-1h, the relative cons\ umption yielded by direct investment\ and trading. ) >> endobj 2663 0 obj [ 2662 0 R ] endobj 2664 0 obj << /S /D >> endobj 2665 0 obj << /Nums [ 0 2664 0 R ] >> endobj 2666 0 obj << /Type /Metadata /Subtype /XML /Length 1330 >> stream 2002-10-08T00:37:20Z 2002-12-19T15:40:17-05:00 Acrobat Distiller 5.0 (Windows) Financial Intermediation and Commitments to Optimal Investment Strategies 2002-10-08T00:37:20Z 2002-10-22T18:25:24-04:00 2002-12-19T15:40:17-05:00 Financial Intermediation and Commitments to Optimal Investment Strategies Financial Intermediation and Commitments to Optimal Investment Strategies endstream endobj 2667 0 obj << /Type /Pages /Kids [ 2689 0 R 1 0 R 5 0 R 8 0 R 12 0 R 15 0 R 19 0 R 23 0 R 26 0 R 29 0 R ] /Count 10 /Parent 2668 0 R >> endobj 2668 0 obj << /Type /Pages /Kids [ 2667 0 R 2669 0 R 2681 0 R 2683 0 R 2682 0 R ] /Count 34 >> endobj 2669 0 obj << /Type /Pages /Kids [ 34 0 R 40 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