Economics 202AProblem 7: Asset Bubbles, Capital In‡ows, and PriceCollapse*First consider a one-good closed economy. There are two periods, periods 1and 2, and there are two assets available, a safe asset o¤ering a predictablegross return of r between periods 1 and 2 and a risky asset (think of o¢ cebuildings or shopping malls) o¤ering a random gross return of R, where R isdistributed on [0, RM AX] with c.d.f H(R) and meanR.Competitive risk-neutral banks have an exogenously determined amountof output B that they lend inelastically to risk-neutral entrepreneurs at thegross interest rate, which equals r in equilibrium. (We will not model thedetermination of B, but one interpretation is suggested in an open-economycontext in the last part of the question.) Banks lend to entrepreneurs becausethey themselves lack the know-how to invest their resources in assets on date1. The total supply of the risky asset on date 1 is …xed and normalizedat 1.1Any entrepreneur who holds x units of the risky asset at the end ofperiod 1 pays a nonpecuniary period 2 cost c(x); where c(0) = c0(0) = 0;c0(x); c00(x) > 0. (Think of this extra cost as the labor or stress involvedin running a risky project; because it is nonpecuniary, it does not have tobe paid out of the entrepreneur’s funds.) If entrepreneurs invest x outputunits in the safe asset, the total gross return is f(x) output units on date2, where the function f() has the usual properties, i.e., it is nonnegative,strictly concave, makes f (0) = 0, and satis…es the Inada conditions. Thedate 1 price of the safe asset is always 1, under the assumption of a costlesstechnology for transforming output into safe assets. In equilibrium, of course,f0(x) = r.Finally, banks cannot observe how entrepreneurs invest borrowed re-sources, and can enter only into debt contracts with entrepreneurs. Thesecontracts are similar to those discussed in David Romer’s macro text: bor-rowers default if the (random) value of their date 2 portfolio is below whatthey owe the bank, but keep any value in excess of what is owed. (We donot, however, explicitly model costs of state veri…cation.)1Imagine that in the background there is an overlapping-generations structure in whichold owners of the risky asset supply it inelastically. You may assume that entrepreneursthemselves have no wealth to invest.1(a) Imagine that a representative entrepreneur buys XRunits of the riskyasset and XSunits of the safe one on date 1. Let P be the date 1 price ofthe risky asset. Show that the date 2 (pecuniary) payo¤ to the entrepreneurunder the contract just described is:(R) = max fRXR rP XR; 0g :Graph this payo¤ function (i.e., graph  against R).(b) What happens to the expected payo¤ E(R) as the variance of Rrises, givenR and XR? What is the intuition?(c) Show why a representative entrepreneur maximizesZRM AXrP(RXR rP XR) dH(R)  c (XR) :Derive his/her …rst-order optimality condition w.r.t. XR.(d) The model’s other equilibrium conditions are: XR= 1, XS+ P = B,and r = f0(XS). Explain each of these. Show that a unique equilibriumexists whenR > c0(1). [Hint: Write the …rst-order condition from part (c)with 1 substituted for XR. Show that in equilibrium, rP > 0 if and only ifR > c0(1). Graph, in the (P; r) plane, the equilibrium …rst-order condition.Finish by graphing the last two equilibrium conditions listed in this partof the question.] Show that in equilibrium, banks earn an expected returnstrictly below r on their loans. (They have no choice but to pay a rent toentrepreneurs.)(e) Prove that the locus de…ning the downward-sloping schedule in thediagram from part (d) can be expressed as:P =1r"RRM AXrPRdH(R)  c0(1)Pr fR  rP g#:(f) We may de…ne the fundamental level of the risky asset’s price as theprice P that would prevail if entrepreneurs …nanced asset purchases entirelyout of their own wealth B (rather than b orrowing the same amount B).(This price will not re‡ect an overvaluation due to entrepreneurs’increased2propensity to gamble on the risky asset and default in low-return states.)Show that in equilibrium,P =1rZRM AX0RdH(R)  c0(1)=1rR  c0(1) Interpret this relationship.(g) Assume that the risk-free interest rate r in the formula of part (f)is the same as the one in the formula of part (e). Show that in that case,P > P . One can think of the di¤erence as a bubble in the asset’s price.(The proof is not completely trivial. Work out an example if you prefer.)(h) Show that in an economy where entrepreneurs …nance investmententirely out of their own wealth B, the equilibrium interest rate r0actuallymust be below the one determined in part (d). Show that, nonetheless, thefundamentals asset price P 0is still below the bubble-ridden price P in part(d).(i) Returning to the diagram in part (d), show how a rise in B (think ofit as an infusion of credit from the banking system) a¤ects r and P . Explainthese e¤ects intuitively.(j) Suppose we have an open economy and that banks are all foreign andwilling to supply loans provided the expected return on the loans equals agiven (world) interest rate rw. Show that for a given capital in‡ow B, thevalues of r and P are determined as in part (d). Show that, however, B isnow endogenous and is determined to equate the expected return on domesticlending (given r and P ) to rw. Prove that r > rw: there is a country premiumin the domestic interest rate. Let the safe technology be given by f (x) = x,where 0 <  < 1. Show how a fall in the world interest rate rwleads to arise in B (higher capital in‡ows), a fall in r, and a rise in P .*Problem inspired by F. Allen an d D. Gale, ”Bubbles and Crises,” EconomicJournal 110 (January 2000):