Economics 202ALecture Outline #9 (version 1.3)Maurice ObstfeldFinancing the Firm: Modigliani-MillerFirms raise capital by issuing debt as well as equity. Earlier we pricedequity claims to a …rm’s output. If all claims on a …rm are equity claims –claims to dividends –then the value of the …rm clearly is the present valuederived above.Consider a …rm facing complete markets and a riskless real interest rateof r. There are two periods and S states of nature in period 2. The …rm canissue bonds B and equity shares E to …nance its investment in capital K:The payo¤ to this investment tomorrow is A(s)F (K), for s 2 S: Since the…rm may be unable fully to repay its bondholders on date 2, its borrowingrate ~r (the corporate bond rate) will generally exceed r.For simplicity (and without any real loss of generality) assume the depre-ciation rate of capital is  = 1.Let the …rm borrow B on date 1. It will then owe its bondholders (1+ ~r)Bin every state of nature s 2 S;but it declares bankruptcy if it cannot repaythem in full: Let Sbbe the set of bankruptcy states and Snbe the set ofnonbankruptcy states, so that S = Sb[ Sn: By de…nition, in a bankruptcystate, the total output of the …rm is insu¢ cient to cover debt payments:A(s)F (K) < (1 + ~r)B: In this case, bondholders are senior claimants and getwhatever there is, while equity holders get nothing.Thusequity holder payo¤ =A(s)F (K)  (1 + ~r)B for s 2 Sn0 for s 2 Sb;whereasbond holder payo¤ =(1 + ~r)B for s 2 SnA(s)F (K) for s 2 Sb:At the Arrow-Debreu prices, the value of the …rm’s equity, sold on date1, isE =Xs2Snp(s)1 + r[A(s)F (K)  (1 + ~r)B] = K  B: (1)1Lenders, on the other hand, must earn the same return as they would inrisk-free lending. Thus, coporate debt must have the same value (in termsof AD prices) as riskless debt. We can express this condition asXs2Snp(s)1 + r(1 + ~r)B +Xs2Sbp(s)1 + rA(s)F (K) = Band solve for 1+ ~r to obtain1 + ~r =B Ps2Sbp(s)1+rA(s)F (K)Ps2Snp(s)1+rB:Let’s look at the market value of all claims on the …rm. It is the sum ofequity and debt claimsV = E + B=Xs2Snp(s)1 + r[A(s)F (K)  (1 + ~r)B]| {z }+Xs2Snp(s)1 + r(1 + ~r)B +Xs2Sbp(s)1 + rA(s)F (K)| {z }E + B=Xs2Sp(s)1 + rA(s)F (K) = K:This is the basic Modigliani–Miller theorem. The …rm’s market value issimply the value of its ouputs across future states of nature. The division ofclaims between equity and debt is irrelevant.Now consider the implications for investment. The …rm maximizes thevalue of shareholders’ equity E. Suppose the …rm borrows an additionaldollar of debt to invest in capital. Then B goes up by 1, and the …rms’market value V goes up byXs2Sp(s)1 + rA(s)F0(K):IfPs2Sp(s)1+rA(s)F0(K) > 1, then the …rms’ market value V = E + B goes upby more than B does, so clearly equity holders gain. The …rm should do theinvestment. We therefore get the same …rst-order condition for the optimalinvestment levelXs2Sp(s)1 + rA(s)F0(K) = 12as in the case without debt (recall we’ve assumed depreciation  = 1). An-other implication of Modigliani-Miller: the investment rule is una¤ected bythe mode of …nance, debt or equity.Tobin’s q model of investmentImportant early work on investment dynamics was done by Dale Jorgen-son. He was on the Berkeley faculty, and his very smart research assistant(and coauthor) was Robert Hall (of random walk fame), a Berkeley under-grad. The work basically took the investment …rst-order condition from aSolow-type growth model to de…ne a long-run target capital stock, given byAFK(K; L) = r. The ad hoc dynamics I =_K = (K  K) were added in toyield an investment equation.The Tobin’s q model is more sophisticated. It assumes that capital iscostly to install — and if you want to install it more quickly, you pay morein frictional costs. This setup gives rise to intrinsic dynamics of investment,rather than the imposed Jorgenson-Hall dynamics, as well as to endogenouspricing of installed capital relative to its replacement cost (not includinginstallation expense). The Obstfeld-Rogo¤ book do es it in discrete time, buthere I will use continuous time and the maximum principle.The key assumption in the model is that there is a convex installationcost of the form2(I2=K)for installing new capital. We can therefore de…ne a …rm’s present discountedpro…t stream on date t as(t) =Z1ter(st)[A(s)F (K(s); L(s))w(s)L(s)I(s)2(I(s)2=K(s))]ds;which is maximized subject to the constraint_K(s) = I(s)with Ktgiven. The interest rate r is assumed to be constant. (In contrastto before, I assume zero depreciation for ismplicity.)In terms of the language of the Pontryagin Maximum Principle, L andI are the …rm’s control variables, K the state variable for the maximizationproblem.Let us set up the HamiltonianH = AF (K; L)  wL  I 2(I2=K) + qI3where q is the costate variable. We di¤erentiate with repect to the twocontrols, setting the result to zero, to obtainAFL(K; L) = w;IK=q  1:The …rst of these is the standard employment optimality condition, whilethe second states that investment has the same sign as q  1. As suggestedby Tobin (JMCB 1969, on the reading list), investment is positive when thevalue of installed capital exceeds its replacement cost.Next consider the dynamics of the costate. According to the MaximumPrinciple, an optimum is characterized by the di¤erential equation_q  rq = @H@K;that is, by_q  rq = AFK(K; L) 2IK2: (2)To derive a phase diagram showing the implied dynamics, let us assumewe are looking at the aggregate economy and that the labor market clearswith L =L, the full-employment supply of labor. Then the dynamic equa-tions of the model can be written as:_q  rq = AFK(K;L) (q  1)22;_K =q  1K:The steady state of the model occurs where q = 1 and AFK(K;L) = r.(Alternatively, if we wished to work at the …rm level, we could take wages asexogenous to the …rm, then solve for L using AFL(K; L) = w; then substitutethe solution L = (K; A; w) into (2). Nothing much changes, but the mathis slightly more intricate.)Imagine a phase diagram with q on the vertical axis and K on the hor-izontal axis. The locus along which_K = 0 is horizontal at q = 1, with K4rising above it and falling below. On the other hand, consider the slope ofthe schedule along which _q = 0. It is given bydqdK_q=0=AFKKr  (q  1):At q =