Economics 202A Lecture Outline, November 15-20, 2007(version 1.1)Maurice ObstfeldThe main point of the model we'll study today is to show how agency costsof investment can be mitigated by a larger decision-maker stake in projects.Thus, more plentiful internal funds can spur investment and, conversely,sharp reductions in decision-maker wealth can cause investment to collapse.Idea of the Bernanke-Gertler (QJE; February 1990) modelTo set the stage, start with a setting much more simple than that ofBernanke-Gertler (BG ). A risk-neutral investor or entrepreneur with totalreal wealth w (observable by outsiders) faces a world capital market in whichthe gross interest rate on loans is (the given constant) r. There are two timeperiods; investment takes place in the rst and consumption in the second.A project requires the input of 1 unit of output on date 1 and has a date2 payo of R with probability p and of 0 with probability 1p. Importantly,p is the entrepreneur's private knowledge. An entrepreneur can undertake atmost one project, and has the option of instead investing his or her wealthat the gross risk-free rate r < R. The cumulative distribution function for pwithin the population of entrepreneurs is H(p).Assume tentatively that an entrepreneur with wealth w can borrow 1 wat the world interest rate r: Lenders can observe the investment outcomeand compel repayment up to the limit of the borrower's resources. For whichvalues of p will entrepreneurs choose to invest in their risky projects? If therewere no nonnegativity constraint on consumption, the cuto value of p wouldbe where the expected returns to risky and riskless investment coincide:p [R r(1 w)] (1 p)r(1 w) = rw:The solution isp fb= r=R;1which, you can conrm, gives an ecient amount of investment. But con-sumption cannot be negative. An entrepreneur whose investment goes sourcan only repay 0 in period 2, so that the problem he or she solves in period1 has a cuto probability given byp [R r(1 w)] = rw;with solutionp =rwR r(1 w)rR:(The last inequality is strict if w < 1.) Notice that unless w = 1 (so the en-trepreneur bears the entire risk of the project), p < p fb. Too many projectswill be undertaken relative to the ecient benchmark. There is a classicproblem of adverse selection, because \bad" borrowers who know they havelow p will borrow and invest. They have a small chance of a big win, butdefault at the lender's expense if the investment fails. Notice that the loweris w, the investor stake, the greater is the incentive to gamble on high-riskprojects (dp =dw > 0).Furthermore, rational lenders, anticipating the behavior above, wouldnever lend at the interest rate r: Instead, they oer a loan contract designedin the expectation that the borrower will default if the project fails. The equi-librium loan contract is simple (and is equivalent to an equity contract in thissimple setting). A borrower undertaking a risky project repays R(1 w) ifthe projec t is successful and 0 otherwise (i.e., there is a default).1Followingour earlier logic, we see that p = p fb= r=R and that the lender's expectedreturn is p fbR = r. The prop osed contract entirely solves the adverse selec-tion problem, delivering the rst-b es t investment level while giving lenderstheir required expected return of r on loans. Thus, entrepreneurial stakesneed not aect aggregate investment in this simple model.To derive contrary results, BG introduce two additional assumptions.First entrepreneurs must pay a xed charge e in order to invest, and pay-ing that cost reveals to them their individual value of p. Second, lenders1A borrower who undertakes the risk-free project repays r(1 w) always. (Lenders canobserve how borrowers use loan proceeds.)2cannot observe whether a borrower who claims to have paid e really has.The assumptions introduce a moral hazard problem, which an optimal loancontract must solve: to induce entrepreneurs to learn p and then to avoidinvesting if they turn out to have very low values of p. As we now see, theresulting contract generally do es not attain the rst-best, and its form makesinvestment sensitive to entrepreneurial wealthThe setupThe economy is closed. The population is a continuum indexed by [0; 1].A fraction consists of risk-neutral entrepreneurs (those with potential in-vestment projects), a fraction 1 of risk-neutral nonentrepreneurs. Anonentrepreneur has wealth wn, an e ntrepreneur wealth we, where wn 1 we, andwav= we+ (1 )wn> ;so that it is feasible (if not optimal in any sense) to fund all investmentprojects.2As in the earlier setting, there is a risk-free technology oering agross rate of return r; call it storage. The last inequality implies that somestorage will occur in equilibrium, so that we can again identify r with thereal rate of interest between periods 1 and 2.Let's look rst at the rst-best (socially ecient) allocation. To that end,dene H(p) (again) as the cumulative distribution function for p within thepopulation of entrepreneurs. For any cuto probability p dene (see BG)^p E (p j p p ) =R1p pdH1 H(p ):(Keep in mind that ^p is a function of p |a fact that would only complicatethe notation were we to continually make it explicit.) A rst simplic ation:since all entrepreneurs are the same ex ante, it is socially optimal either forall or for none to evaluate and learn their projects' success probabilities.2BG let entrepreneurs' wealth vary cross-sectionally, but this is inessential. The mainconsequence is that in their setup, constrained-optimal contracts tailored to dierentwealth levels coexist.3Assume rst that it is socially optimal for all to pay the xed investmentcharge e up front.3Having learned p; it is then socially optimal to invest inthe risky technology if and only if the expected return is not below that onstorage. So we get a cuto probability given byp fbR = r , p fb= r=R:But when is it socially optimal to pay the up-front charge e? It is optimalto pay e only if the expected return on each project from paying e and theninvesting according to the preceding cuto rule exceeds r. Formally thecondition is4h1 H(p fb)ihEpR j p p fb ri e =h1 H(p fb)i(^pfbR r) e > 0:If the last inequality fails to hold, it is optimal for society simply to investall its resources in storage. In what follows we will assume this is not the
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