Split (1)

train · 11.7k rows

text stringlengths 204 3.13k
1 ≤ Y l t+1 + Bt + (dl 1,t+1 + Bt + (dh 1,t+1 ≤ Y l t+1 + Bt + (dh 1,t+1 + Qh,h,h 1,t+1 + Qh,h,l 1,t+1 + Qh,l,l 1,t+1 + Qh,l,h 1,t+1 + Ql,h,h 1,t+1 + Ql,h,l 1,t+1 ) S1,t + (dh 2,t+1 ) S1,t + (dl 2,t+1 ) S1,t + (dl 2,t+1 ) S1,t + (dh 2,t+1 ) S1,t + (dh 2,t+1 ) S1,t + (dl 2,t+1 ) S2,t + Qh,h,h 2,t+1 + Qh,h,l 2,t+1 + Qh,l,l 2,t+1 + Qh,l,h 2,t+1 + Ql,h,h 2,t+1 + Ql,h,l 2,t+1 ) S2,t ) S2,t ) S2,t ) S2,t ) S2,t (35.26) (35.27) (35.28) (35.29) (35.30) (35.31) C h,h,h t+1 C h,h,l t+1 C h,l,l t+1 C h,l,h t+1 C l,h,h t+1 C l,h,l t+1 C l,l,l t+1 C l,l,h t+1 t+1 t+1 2,t+1 1,t+1 ≤ Y l ≤ Y l ) S2,t + Bt + (dl + Bt + (dl ) S1,t + (dl + Ql,l,l 1,t+1 + Ql,l,h 1,t+1 Regardless of the realization of uncertainty in t + 1, consumption (after imposing the terminal conditions) cannot exceed available resources. Available resources include the exogenous income ﬂow, the payout from holdings
of the riskless one period discount bond, and the uncertain payout from holdings of both stocks (which in principal include both a dividend and capital gain component). + Ql,l,l 2,t+1 + Ql,l,h 2,t+1 ) S1,t + (dh (35.33) (35.32) ) S2,t 1,t+1 2,t+1 The household wishes to maximizes expected lifetime utility. Given the nature of uncer- tainty about t + 1, this can be written: U = u(Ct) + p1u(C h,h,h t+1 ) + p2u(C h,h,l t+1 + p5u(C l,h,h t+1 ) + p3u(C h,l,l t+1 ) + p4u(C h,l,h t+1 ) ) + p6u(C l,h,l t+1 ) + p7u(C l,l,l t+1 ) + p8u(C l,l,h t+1 ) (35.34) The household’s problem is to pick a consumption plan to maximize (35.34) subject to (35.25)-(35.33). It is easiest to characterize the problem by writing it as an unconstrained problem of choosing Bt, SH1,t, and SH2,t in period t. The problem is: 849 max Bt,SH1,t,SH2,t U = u [Yt − P B t Bt − Q1,tSH1,t − Q2,tSH2,t] + 1,t+1 1,t+1 1,t+1 1,t+1 1,t+1 1,t+1 + Bt + (dh + Bt + (dh + Bt + (dl + Bt + (dl + Bt + (dh + Bt + (dh + Bt + (dl p1βu [Y h t+1 p2βu [Y h t+1 p3βu [Y h t+1 p4βu [Y h t+1 p5βu [Y l t+1 p6βu [Y l t+1 p7βu [Y l t+1 p8β
u [Y l t+1 + Qh,h,h 1,t+1 + Qh,h,l 1,t+1 + Qh,l,l 1,t+1 + Qh,l,h 1,t+1 + Ql,h,h 1,t+1 + Ql,h,l 1,t+1 + Ql,l,l 1,t+1 2,t+1 2,t+1 2,t+1 2,t+1 2,t+1 2,t+1 ) SH1,t + (dh ) SH1,t + (dl ) SH1,t + (dl ) SH1,t + (dh ) SH1,t + (dh ) SH1,t + (dl ) SH1,t + (dl + Ql,l,h 1,t+1 1,t+1 1,t+1 + Bt + (dl + Qh,h,h 2,t+1 + Qh,h,l 2,t+1 + Qh,l,l 2,t+1 + Qh,l,h 2,t+1 + Ql,h,h 2,t+1 + Ql,h,l 2,t+1 + Ql,l,l 2,t+1 ) SH2,t] + ) SH2,t] + ) SH2,t] + ) SH2,t] + ) SH2,t] + ) SH2,t] + ) SH2,t] + + Ql,l,h 2,t+1 2,t+1 2,t+1 ) SH1,t + (dh ) SH2,t] (35.35) The ﬁrst order optimality conditions are: ∂U ∂Bt = 0 ⇔ P B t u′(Ct) = β[p1u′(C h,h,h t+1 ) + p2u′(C h,h,l t+1 ) + p3u′(C h,l,l t+1 ) + p4u′(C h,l,h t+1 ) + p5u′(C l,h,h t+1 ) + p6u′(C l,h,l t+1 ) + p7
u′(C l,l,l t+1 ) + p8u′(C l,l,h t+1 )] (35.36) ∂U ∂SH1,t = 0 ⇔ Q1,tu′(Ct) = β[p1(dh 1,t+1 + Qh,h,h 1,t+1 ))u′(C h,h,h t+1 ) + p2(dh 1,t+1 + Qh,h,l 1,t+1 )u′(C h,h,l t+1 ) + p3(dl 1,t+1 + Qh,l,l 1,t+1 ))u′(C h,l,l t+1 ) + p4(dl 1,t+1 + Qh,l,h 1,t+1 ))u′(C h,l,h t+1 ) + p5(dh 1,t+1 + Ql,h,h 1,t+1 ))u′(C l,h,h t+1 ) + p6(dh 1,t+1 + Ql,h,l 1,t+1 ))u′(C l,h,l t+1 ) + p7(dl 1,t+1 + Ql,l,l 1,t+1 ))u′(C l,l,l t+1 ) + p8(dl 1,t+1 + Ql,l,h 1,t+1 ))u′(C l,l,h t+1 )] (35.37) 850 ∂U ∂SH2,t = 0 ⇔ Q2,tu′(Ct) = β[p1(dh 2,t+1 + Qh,h,h 2,t+1 ))u′(C h,h,h t+1 ) + p2(dl 2,t+1 + Qh,h,l 2,t+1 )u′(C h,h,l t+1 ) + p3(dl 2,t+1 + Qh,l,l 2,t+1 ))u′(C h,l,l t+1 ) + p4(dh 2,t+1 + Qh,l,h 2,t+1 ))u′(C h,
l,h t+1 ) + p5(dh 2,t+1 + Ql,h,h 2,t+1 ))u′(C l,h,h t+1 ) + p6(dl 2,t+1 + Ql,h,l 2,t+1 ))u′(C l,h,l t+1 ) + p7(dl 2,t+1 + Ql,l,l 2,t+1 ))u′(C l,l,l t+1 ) + p8(dh 2,t+1 + Ql,l,h 2,t+1 ))u′(C l,l,h t+1 )] (35.38) (35.36)-(35.38) all look somewhat nasty but all have fairly intuitive interpretations. In particular, the terms on the left hand sides are simply the marginal utility costs of purchasing an additional unit of each of the three diﬀerent kinds of assets, while the right hand sides are the expected marginal utility beneﬁts of doing so. The expressions end up looking nasty because there are eight possible states of nature in t + 1. Written more compactly in terms of expectations operators, however, these FOC can be written: t u′(Ct) = β E[u′(Ct+1)] P B Q1,tu′(Ct) = β E[(d1,t+1 + Q1,t+1)u′(Ct+1)] Q2,tu′(Ct) = β E[(d2,t+1 + Q2,t+1)u′(Ct+1)] (35.39) (35.40) (35.41) We again are working in the conﬁnes of an endowment economy in which all assets are in zero net supply. This means that Ct = Yt and Ct+1 = Yt+1 regardless of the realization of t + 1 uncertainty. Because u′(Yt) is known in period t, these can be written in the usual setup wherein the price of the asset is the expected value oﬀ the product of the stochastic discount factor with the t + 1 payout from holding the asset: P B t = E [ βu′(Yt+1) u′(Yt) ] Q1,t = E [ βu′(
Yt+1) u′(Yt) Q2,t = E [ βu′(Yt+1) u′(Yt) (d1,t+1 + Q1,t+1)] (d2,t+1 + Q1,t+1)] (35.42) (35.43) (35.44) We can then write the yields on each asset as the ratio of the expected payout divided by the price, or: 851 1 + rt = 1 P B t = 1 + rs,1,t = E[d1,t+1 + Q1,t+1] Q1,t 1 + rs,2,t = E[d2,t+1 + Q2,t+1] Q2,t = = ] 1 E [ βu′ (Yt+1) u′ (Yt) E[d1,t+1 + Q1,t+1] E [ βu′ (Yt+1) u′ (Yt) E[d2,t+1 + Q2,t+1] (Yt+1) (Yt) E [ βu′ u′ (d1,t+1 + Q1,t+1)] (d2,t+1 + Q2,t+1)] (35.45) (35.46) (35.47) 1,t+1 2,t+1 = 1.1 and Y l t+1 = 1.05 and d2 We construct a numerical example to illustrate how the diﬀerent kinds of stocks might be priced diﬀerently and oﬀer diﬀerent yields. Assume that current income is Yt = 1 and the discount factor is β = 0.95. Let the utility function be the natural log. We assume that = 0.9. Assume that the two values income can take on in the future are Y h 1+1 = 0.95. Assume the ﬁrst stock oﬀers a relatively stable dividend, with dh = 1.5 that the dividend of the second stock is much more volatile. In particular, let dh = 0.5. Let us specify the probabilities of diﬀerent states as follows. Assume that and dl p1 = 0.4 and p4 = 0.
1, with p2 = p3 = 0. This means that there is a 50 percent chance of income being high; conditional on income being high, both stocks pay a high dividend with 80 percent probability (i.e. 0.4/0.5 = 0.8) and pay a low dividend with 20 percent probability (i.e. 0.1/0.5 = 0.2). Similarly, let us assume that p7 = 0.4 and p5 = 0.1, with p6 = p8 = 0. This means that conditional on income being low, both stocks oﬀer a low dividend (state 5) with 80 percent probability and a high dividend with probability 20 percent. For simplicity, we assume that it is never the case that one stock pays a high dividend and the other pays a low dividend. Note that the example has been setup up where all three assets oﬀer an expected payout of 1 in t + 1. 1,t+1 2,t+1 If we work through the numbers, we get the following prices: Pt = 0.959, Q1,t = 0.956, and Q2,t = 0.931. Even though all three assets oﬀer the same expected payout in t + 1, the bond is most valuable, followed by the ﬁrst stock and then the second. This is because the bond’s payout does not covary with the future marginal utility of consumption, whereas the payouts of both stocks are high when future income is high, which means that their payouts covary negatively with the marginal utility of consumption. In terms of yields, the bond yields 0.042, the ﬁrst stock yields 0.045, and the second stock 0.074. The household demands a premium in the form of a higher expected return to hold either stock relative to the bond, and in turn demands a higher premium to be willing to hold the second stock compared to the ﬁrst stock. This is because the dividend payout from the second stock covaries much more negatively with the marginal utility of future consumption than does the ﬁrst stock. Once again, note that the numerical example could be altered in such a way as to change 852 2,t+1 = 3 or dl the prices of the stocks but not the relative yields. For example, suppose that the second = 1. This means that its
expected payout is stock pays a dividend of dh 2, which is double the expected payout on the bond or the ﬁrst stock. It therefore trades at a higher price. Keeping everything the same as in the previous example, we would get P2,t = 1.86, or twice as high as when the expected dividend was 1. But in terms of yields, the expected yield/return on stock 2 is still 0.074. This example yet again underscores the fact that it is most appropriate to compare diﬀerent assets in terms of yields, not prices. 2,t+1 This analysis reveals a potentially important insight when thinking about the cross-section of stock returns. In particular, stocks whose returns are positively correlated with the overall economy (i.e. they are procyclical) ought to command a higher expected return (yield) compared to stocks whose returns are less correlated, or even negatively correlated, with the economy. A household’s objective is to maximize its expected lifetime utility, which entails investing in assets that help it smooth its consumption. An asset that oﬀers low payouts when consumption would otherwise already be low does not help smooth consumption, and hence optimizing households ought to demand a premium in the form of a higher expected return to be willing to hold such a stock. On average, stock returns are procyclical, and this is why the average equity premium over risk-free government debt is positive. But across diﬀerent types of stocks, some are more procyclical than others. Other things being equal, we would expect the most procyclical stocks to trade at the highest expected returns to compensate investors for risk. 35.3 Moving Beyond Two Periods Return to assuming that there is only one kind of equity available for purchase, but instead suppose that the household lives for three periods – t, t + 1, and t + 2. In period t, the household can once again purchase a risk free, one period bond or a risky equity. Its budget constraint is: Ct + P B t Bt + QtSHt ≤ Yt (35.48) In period t + 1, there are four possible states of nature – income could be high or low and the dividend from the stock could be high or low. The household can purchase more one period bonds which pay out in period t + 2, or it could purchase more stock. Resources come from the exogenous income ﬂow, payouts on the one period
riskless bond brought from t to t + 1, and dividend payouts on shares of stock brought from t to t + 1. The ﬂow budget constraint must hold in each state of nature. For period t + 1, this requires: 853 C h,h t+1 C h,l t+1 C l,l t+1 C l,h t+1 + P B,h,h + P B,h,l + P B,l,l t+1 Bh,h t+1 t+1 Bh,l t+1 t+1 Bl,l t+1 t+1 Bl,h t+1 + Qh,h t+1 + Qh,l t+1 + Ql,l t+1 + Ql,h t+1 (SH h,h t+1 (SH h,l t+1 (SH l,l t+1 (SH l,h t+1 + P B,l,h − SHt) ≤ Y h t+1 − SHt) ≤ Y h t+1 + Bt + dh t+1SHt + Bt + dl t+1SHt − SHt) ≤ Y l t+1 − SHt) ≤ Y l t+1 + Bt + dl t+1SHt + Bt + dh t+1SHt (35.49) (35.50) (35.51) (35.52) In (35.45)-(35.52), the ﬁrst superscript refers to the output state, while the second refers to the state of nature for the dividend. There are again four states of nature in t + 2 – income could be high or low, and the dividend could be high or low. But because consumption in these states depends on the realization of uncertainty in t + 1, there are in principle sixteen possible states of nature in t + 2 – four states for each of the four states possible in t + 1 (i.e. 42 = 16). We will denote the realization of the state in t + 2 with a four part superscript. The ﬁrst entry denotes the output state in t + 2, the second the dividend state in t + 2, the third the output state in t + 1, and the fourth the dividend state in t + 1. For example, (h, l, h, l) means high output
in t + 2, low dividend in t + 2, high output in t + 1, and low dividend in t + 1. The t + 2 constraints can be written: C h,h,h,h t+2 + P B,h,h,h,h t+2 Bh,h,h,h t+2 + Qh,h,h,h t+2 (SH h,h,h,h t+2 − SH h,h t+1 ) ≤ Y h t+2 + Bh,h t+1 + dh t+2SH h,h t+1 (35.53) C h,l,h,h t+2 C l,l,h,h t+2 C l,h,h,h t+2 C h,h,h,l t+2 C h,l,h,l t+2 C l,l,h,l t+2 C l,h,h,l t+2 C h,h,l,l t+2 C h,l,l,l t+2 C l,l,l,l t+2 + P B,h,l,h,h Bh,l,h,h t+2 t+2 + P B,l,l,h,h Bl,l,h,h t+2 t+2 + P B,l,h,h,h Bl,l,h,h t+2 t+2 + P B,h,h,h,l Bh,h,h,l t+2 t+2 + P B,h,l,h,l Bh,l,h,l t+2 t+2 + P B,l,l,h,l Bl,l,h,l t+2 t+2 + P B,l,h,h,l Bl,h,h,l t+2 t+2 + P B,h,h,l,l Bh,h,l,l t+2 t+2 + P B,h,l,l,l Bh,l,l,l t+2 t+2 t+2 Bl,l,l,l + P B,l,l,l,l t+2 + Qh,l,h,h t+2 + Ql,l,h,h t+2 + Ql
,h,h,h t+2 + Qh,h,h,l t+2 + Qh,l,h,l t+2 + Ql,l,h,l t+2 + Ql,h,h,l t+2 + Qh,h,l,l t+2 + Qh,l,l,l t+2 + Ql,l,l,l t+2 (SH h,l,h,h t+2 (SH l,l,h,h t+2 (SH l,h,h,h t+2 (SH h,h,h,l t+2 (SH h,l,h,l t+2 (SH l,l,h,l t+2 (SH l,h,h,l t+2 (SH h,h,l,l t+2 (SH h,l,l,l t+2 (SH l,l,l,l t+2 + dl + dl + dh + dh t+2SH h,h t+1 t+2SH h,h t+1 t+2SH h,h t+1 t+2SH h,l t+1 t+2SH h,l t+1 t+2SH h,l t+1 t+2SH h,l t+1 t+2SH l,l t+1 t+2SH l,l t+1 t+2SH l,l t+1 + dh + dh − SH h,h t+1 − SH h,h t+1 − SH h,h t+1 − SH h,l t+1 ) ≤ Y h t+2 ) ≤ Y l t+2 ) ≤ Y l t+2 ) ≤ Y h t+2 + Bh,h t+1 + Bh,h t+1 + Bh,h t+1 + Bh,l t+1 − SH h,l t+1 ) ≤ Y h t+2 + Bh,l t+1 + dl + dl − SH h,l t+1 − SH h,l t+1 − SH l,l t+1 ) ≤ Y l t+2 ) ≤ Y l t+2 ) ≤ Y h t+2 + Bh,l t+1 + Bh,l t+1 + Bl,l t+1 − SH
l,l t+1 ) ≤ Y h t+2 + Bl,l t+1 + dl + dl − SH l,l t+1 ) ≤ Y l t+2 + Bl,l t+1 854 (35.54) (35.55) (35.56) (35.57) (35.58) (35.59) (35.60) (35.61) (35.62) (35.63) C l,h,l,l t+2 C h,h,l,h t+2 C h,l,l,h t+2 C l,l,l,h t+2 C l,h,l,h t+2 + P B,l,h,l,l t+2 + P B,h,h,l,h t+2 + P B,h,l,l,h t+2 + P B,l,l,l,h t+2 + P B,l,h,l,h t+2 Bl,h,l,l t+2 Bh,h,l,h t+2 Bh,l,l,h t+2 Bl,l,l,h t+2 Bl,h,l,h t+2 + Ql,h,l,l t+2 + Qh,h,l,h t+2 + Qh,l,l,h t+2 + Ql,l,l,h t+2 + Ql,h,l,h t+2 (SH l,h,l,l t+2 (SH h,h,l,h t+2 − SH l,l t+1 − SH l,h t+1 ) ≤ Y l t+2 ) ≤ Y h t+2 + dh + Bl,l t+1 + Bl,h t+1 (SH h,l,l,h t+2 (SH l,l,l,h t+2 (SH l,h,l,h t+2 − SH l,h t+1 ) ≤ Y h t+2 + Bl,h t+1 + dl − SH l,h t+1 − SH l,h t+1 ) ≤ Y l t+2 ) ≤ Y l t+2 + dl + Bl,h t+1 + Bl,h t
+1 t+2SH l,l t+1 t+2SH l,h + dh t+1 t+2SH l,h t+1 t+2SH l,h t+1 t+2SH l,h t+1 + dh (35.64) (35.65) (35.66) (35.67) (35.68) Let p1 be the probability that both income and the dividend are high in t + 1, p2 be the probability that income is high and the dividend is low in t + 1; p3 by the probability that both income and the dividend are low in t + 1; and p4 = 1 − p1 − p2 − p3 be the probability that income is low and the dividend is high in t + 1. It must be the case that the sum of these probabilities is one because some state of nature must occur in t + 1. Let q1,... q16, j=1 qj = 1, be the probabilities of each of the sixteen states of nature described in with ∑16 (35.53)-(35.68) occurring. These probabilities must sum to one because some state of nature must materialize in t + 2. Note that a special case of this probability speciﬁcation would be one in which the four states occur in t + 2 independently of what happened in t + 1. If this 1 – i.e. the probability of the (h, h) state in t + 1, p1, were the case, we would have q1 = p2 times the probability of the (h, h) state in t + 2, also p1. In writing it instead the way that we do, we allow for the realizations across time to potentially not be independent. The household’s objective is to maximize the present discounted value of lifetime utility. This can be written: U = u(Ct) + β [p1u(C h,h t+1 )+q2u(C h,l,h,h +β2[q1u(C h,h,h,h ) + p2u(C h,l t+1 )+q3u(C l,l,h,h ) + p3u(C l,l t+1 )+q4u(C l,h,h,h ) + p4u(C l
,h t+1 )] t+2 t+2 + q8u(C l,h,h,l t+2 ) + q9u(C h,h,l,l t+2 )+q5u(C h,h,h,l )+q6u(C h,l,h,l t+2 ) + q10u(C h,l,l,l t+2 t+2 ) + q11u(C l,l,l,l t+2 t+2 t+2 ) + q12u(C l,h,l,l ) + t+2 )+q7u(C l,l,h,l t+2 ) q13u(C h,h,l,h t+2 ) + q14u(C h,l,l,h t+2 ) + q15u(C l,l,l,h t+2 ) + q16u(C l,h,l,h t+2 )] (35.69) We can again invoke terminal conditions to simplify the analysis. In particular, the household will not choose to die with positive stocks of bonds or stock in t + 2. This means that Bt+2 = 0 and SHt+2 = 0 regardless of the realization of uncertainty. It is easiest to think about the problem by substituting in all the constraints into the objective function so as to eliminate the diﬀerent consumption terms. Then the problem amounts to one of picking Bt, SHt, Bt+1 in all four states, and SHt+1 in all four states of nature in t + 1. The ﬁrst order optimality conditions with respect to each of these choices are given below: 855 Bt ∶ P B t u′(Ct) = β [p1u′(C h,h t+1 ) + p2u′(C h,l t+1 ) + p3u′(C l,l t+1 ) + p4u′(C l,h t+1 )] (35.70) SHt ∶ Qtu′(Ct) = β [p1u′(C h,h t+1 )(dh + Qh,h t+1 t+1 p3u′(C l,l t+1 ) + p2u′(
C h,l t+1 + Ql,l )(dl t+1 + Qh,l )(dl t+1 t+1 ) + p4u′(C l,h t+1 t+1 )+ )(dh t+1 + Ql,h t+1 )] (35.71) Bh,h t+1 ∶ p1P h,h t+1 u′(C h,h t+1 ) = β [q1u′(C h,h,h,h t+2 ) + q2u′(C h,l,h,h t+2 ) + q3u′(C l,l,h,h t+2 ) + q4u′(C l,h,h,h t+2 )] Bh,l t+1 Bl,l t+1 ∶ p2P h,l t+1u′(C h,l t+1 t+1u′(C l,l t+1 ∶ p3P l,l ) = β [q5u′(C h,h,h,l ) + q6u′(C h,l,h,l ) + q7u′(C l,l,h,l ) + q8u′(C l,h,h,l t+2 t+2 ) + q10u′(C h,l,l,l t+2 t+2 ) + q11u′(C l,l,l,l t+2 t+2 ) + q12u′(C l,h,l,l t+2 )] ) = β [q9u′(C h,h,l,l t+2 (35.72) )] (35.73) Bl,h t+1 ∶ p4P l,h t+1u′(C l,h t+1 ) = β [q13u′(C h,h,l,h t+2 ) + q14u′(C h,l,l,h t+2 ) + q15u′(C l,l,l,h t+2 ) + q16u′(C l,h,l,h t+2 (35.74) )] (35.75) SH h,h t+1 ∶ p1Q
h,h t+1u′(C h,h t+1 ) = β [q1u′(C h,h,h,h t+2 )(dl t+2 )(dh t+2 + Ql,l,h,h t+2 + Qh,h,h,h t+2 ) + q4u′(C l,h,h,h t+2 ) + q2u′(C h,l,h,h t+2 )(dh t+2 )(dl t+2 + Ql,h,h,h t+2 + Qh,l,h,h t+2 )] )+ (35.76) q3u′(C l,l,h,h t+2 SH h,l t+1 ∶ p2Qh,l t+1u′(C h,l t+1 SH l,l t+1 ∶ p3Ql,l t+1u′(C l,l t+1 SH l,h t+1 ∶ p4Ql,h t+1u′(C l,h t+1 ) = β [q5u′(C h,h,h,l t+2 q7u′(C l,l,h,l )(dl t+2 t+2 )(dh + Qh,h,h,l t+2 ) + q8u′(C l,h,h,l ) + q6u′(C h,l,h,l t+2 )(dh t+2 + Ql,l,h,l t+2 t+2 )(dl + Qh,l,h,l t+2 )] t+2 + Ql,h,h,l t+2 t+2 )+ (35.77) ) = β [q9u′(C h,h,l,l t+2 q11u′(C l,l,l,l )(dl t+2 t+2 )(dh + Qh,h,l,l t+2 ) + q12u′(C l,h,l,l ) + q10u′(C h,l,l,l t+2 )(dh t+2 + Ql,l,l,l t+
2 t+2 t+2 )(dl + Qh,l,l,l t+2 )] t+2 + Ql,h,l,l t+2 )+ (35.78) ) = β [q13u′(C h,h,l,h t+2 )(dl q15u′(C l,l,l,h t+2 t+2 )(dh t+2 + Ql,l,l,h t+2 + Qh,h,l,h t+2 ) + q16u′(C l,h,l,h t+2 ) + q14u′(C h,l,l,h t+2 )(dh t+2 )(dl t+2 + Ql,h,l,h t+2 + Qh,l,l,h t+2 )] )+ (35.79) (35.70)-(35.79) once again look nasty, but have much cleaner interpretations when written using expectations operators. In particular, (35.70)-(35.71) can be written: t u′(Ct) = β E[u′(Ct+1)] P B Qtu′(Ct) = β E[u′(Ct+1)(dt+1 + Qt+1)] (35.80) (35.81) 856 (35.80)-(35.81) are entirely standard and indeed look exactly the same as in the two period model (see, e.g., (35.14)-(35.15)). Because β is constant and Ct is known, these can both be written in the by-now-familiar stochastic discount factor form: P B t = E [βu′(Ct+1) u′(Ct) ] Qt = E [βu′(Ct+1) u′(Ct) (dt+1 + Qt+1)] (35.82) (35.83) Next, let us turn to the other ﬁrst order conditions, which are new since we have added a third period. Note that if one sums (35.72)-(35.75), one gets that the expected value of the product of the bond price in t + 1 with the marginal utility of consumption in period t + 1 (i.e. p1P B,h,h )) equals β times the
expected value of the marginal utility of consumption in t + 2 (i.e. q1u′(C h,h,h,h )). We can do the same thing with (35.76)-(35.79). We get: ) +... q16u′(C l,h,lh t+1 u′(C h,h t+1 t+1 u′(C h,l t+1 t+1 u′(C l,h t+1 t+1 u′(C l,l t+1 ) + p4P B,l,h ) + p2P B,h,l ) + p3P B,l,l t+2 t+2 E [P B t+1u′(Ct+1)] = β E [u′(Ct+2)] E [Qt+1u′(Ct+1)] = β E [u′(Ct+2)(dt+2 + Qt+2)] (35.84) (35.85) (35.84)-(35.85) are the same as (35.82)-(35.83), except led one period into the future and with an expectation operator on both sides of the equality. But we can actually say more than just that the same pricing condition holds in expectation in t + 1. For example, focus on (35.72). Divide both sides of the equality by p1. One gets: t+1 u′(C h,h P B,h,h t+1 ) = β [ q1 p1 u′(C h,h,h,h t+2 ) + q2 p1 u′(C h,l,h,h t+2 ) + q3 p1 u′(C l,l,h,h t+2 ) + q4 p1 u′(C l,h,h,h t+2 )] (35.86) The left hand side of (35.86) is the realized product of the price of the bond and the marginal utility of consumption in the (h, h) state in t + 1. The right hand side is the expected value of the t + 2 marginal utility of consumption conditional on the (h, h) state occurring in t + 1.2 (35.73)-(35.75) can be written in a similar way with a similar interpretation
. Deﬁne 2This is a straightforward application of rules of conditional probability, which states that for two random variables X and Y, the probability of Y given X, P (Y ∣ X), equals the ratio of the probability of both X and Y occurring, P r(Y ∩ X), divided by the probability of X, so P r(Y ∣ X) = P r(Y ∩X). In the way we have laid P r(X) out the probabilities, p1 is the probability of both income and the dividend being high in t + 1, while q1 is the probability of both income and the dividend being high in both t + 1 and t + 2. Hence, q1 is the probability of p1 income and the dividend both being high in t + 2, conditional on both income and the dividend being high in t + 1. Similarly, q2 is the probability of income being high and the dividend being low in t + 2, conditional on p1 income and the dividend both being high in t + 1. And so on. 857 Et+1[⋅] as the expectation conditional on the realization of the state in t + 1. In other words, (35.84) must not only hold in expectation based on what is known in period t, it must hold in in all states of the world in t + 1. Hence: P B t+1 = Et+1 [ βu′(Ct+2) u′(Ct+1) ] (35.87) By similar logic, we can see from (35.76)-(35.79) that (35.85) must not only hold in expectation from the perspective of period t, but also in each realization of uncertainty in t + 1: Qt+1 = Et+1 [ βu′(Ct+2) u′(Ct+1) (dt+2 + Qt+2)] (35.88) (35.88) is particularly useful, because it can be used in conjunction with (35.83) to eliminate Qt+1. In particular, we can write: Qt = E [ βu′(Ct+1) u′(Ct) (dt+1 + Et+1 [βu′(Ct+2) u′(Ct+1) (dt+2 + Qt+2)])] (35.89) Now, if we distribute inside the outer expectations operate, we
get: Qt = E [βu′(Ct+1) u′(Ct) dt+1 + β2 u′(Ct+1) u′(Ct) Et+1 u′(Ct+1)dt+2 + β2 u′(Ct+1) u′(Ct+2) u′(Ct) Et+1 u′(Ct+2) u′(Ct+1)Qt+2] (35.90) (35.90) can be simpliﬁed along two dimensions. First, we can impose a terminal condition on the period t + 2 price of the stock, Qt+2 = 0. This is because the stock is a claim to future dividends, and there is no future from the perspective of t + 2, so in equilibrium the stock should be worthless. Second, an application of the Law of Iterated Expectations says that the unconditional expectation of a conditional expectation is the unconditional expectation. Taken together, we can write: Qt = E [ βu′(Ct+1) u′(Ct) dt+1 + β2u′(Ct+2) u′(Ct) dt+2] (35.91) In other words, the price of the stock ought to equal the present discounted value of future dividends, where discounting is by the stochastic discount factor. For a multi-period framework, the stochastic discount factor can be deﬁned as: mt,t+j = βju′(Ct+j) u′(Ct) 858 (35.92) The stochastic discount factor measures how payouts in t + j ought to be valued from the perspective of period t. Hence, we can more compactly write (35.91) as: Qt = E [ 2 ∑ j=1 mt,t+jdt+j] Now, let us return to (35.91). Note that we can write: Qt = E [ βu′(Ct+1) u′(Ct) dt+1] + E [ βu′(Ct+1) u′(Ct) Et+1 [ βu′(Ct+2) u′(Ct+1) dt+2]] (35.93) (35.94) Now, if there were no uncertainty over the future (or more generally if there were no correlation/covariance between future dividends and
the stochastic discount factor), we could write (35.94) as: But since the bond prices are just inverse of gross yields, (35.95) could be written: Qt = Ptdt+1 + PtPt+1dt+2 (35.95) Qt = dt+1 1 + rt + dt+2 (1 + rt)(1 + rt+1) (35.96) From (35.96), if there were no uncertainty then the stock price would simply be the present discounted value of dividend payouts, where discounting is by the yield on the one period risk-free bond. But in general, (35.96) will not hold because there is uncertainty, and future dividends will be discounted at a potentially higher rate than the gross yield on the risk-free bond. In an endowment economy equilibrium, we must have consumption equal income at all dates and all possible realizations of uncertainty. This means we can replace consumption values with exogenous values of income and use the above-derived expressions to price each asset. As before, we can deﬁne the yields/expected returns on each type of asset as the discount rate which rationalizes the current price of the asset in terms of the expected value of future cash ﬂows. For the bond, the yield is simply: 1 + rt = 1 P B t 1 + rs,t = = 1 E [ βu′ (Yt+1) u′ (Yt) E[dt+1 + Qt+1] Qt ] (35.97) (35.98) (35.98) is the expected return on holding the stock. Note that there are two components to this expected return. It can be written: 859 1 + rs,t = E[dt+1] Qt + E[Qt+1] Qt (35.99) In (35.99), the ﬁrst component is the dividend yield (expected dividend divided by current share prices) and the second component is the capital gain (expected share price divided by current share price). Note that (35.98) can be written: 1 + rs,t = E[dt+1 + Qt+1] (Yt+1) (Yt) (dt+1 + Qt+1)] E [ βu′ u′ The equity premium is simply the ratio of the two yields: 1 + rs,t 1
+ rt = E[dt+1 + Qt+1] E [ βu′ E [ βu′ (Yt+1) u′ (Yt) (Yt+1) u′ (Yt) (dt+1 + Qt+1)] ] (35.100) (35.101) One will note that (35.101) is exactly the same as the expression for the equity risk premium in the two period case, (35.20). The only diﬀerence relative to the two period case is that in the two period case we will have Qt+1 = 0, whereas in the three period case Qt+1 ≠ 0 in general. Provided dt+1 + Qt+1 covaries negatively with u′(Yt+1) (i.e. the payout from the stock is procyclical), investors will demand a higher yield to hold the stock in comparison to the risk-free bond (i.e. the equity premium will be positive). t+1 t+2 t+1 t+2 t+1 t+2 = dl = dh = Y l = Y h t+2 = 1.1 and dl = 1.1 and Y l Let us now do a couple of quantitative experiments with actual numbers. Suppose that β = 0.95 and Yt = 1. Let the utility function be the natural log. Suppose further that income in t + 1 and t + 2 can take on the same two values, Y h = 0.90. t+1 For now, assume that the dividend payout on the stock can take on the same two values = 0.90. It remains to specify the in both future periods: dh probabilities of diﬀerent states materializing, which in a three period context can become somewhat messy. Let us suppose the following probability structure in t + 1: p1 = p3 = 0.4 and p2 = p4 = 0.1. This means that there is a 50 percent chance of the t + 1 income being high (p1 + p2), and a 50 chance of the dividend being high (p1 + p4 = 0.5). Furthermore, we assume that conditional on whatever state materializes in t + 1, there is a 60 percent chance of that same state materializing in t + 2. Conditional on whatever state materializes in t + 1, there is a 20 percent chance of the state “�
��ipping” (i.e. going from e.g. (h, l) to (l, h) or (h, h) to (l, l)). There is a 10 percent chance of going to the other two possible states conditional on the realization of the state in t + 1 (i.e. the probability of going from (h, h) to (h, l) or (l, h) is 0.1 each).3 This means that we are assuming some persistence – whatever state 3Formally, this means we are assuming p1 = p3 = 0.4, p2 = p4 = 0.1, q1 = q11 = 0.24, q2 = q4 = q10 = q12 = 0.04, q3 = q9 = 0.08, q5 = q7 = q13 = q15 = 0.01, q6 = q16 = 0.06, and q8 = q14 = 0.02. 860 materializes in t + 1 is relatively more likely to materialize in t + 2. Table 35.2: Stock and Bond Pricing: Three Periods Pt Qt 1 + rt 1 + rs,t rs,t − rt E[Pt+1] E[Qt+1] E[Yt+1] E[Yt+2] E[Yt+2 ∣ Y h t+1 E[Yt+2 ∣ Y l t+2 E[dt+1] E[dt+2] E[dt+1 ∣ Y h t+1 E[dt+1 ∣ Y 1 t+1 E[dt+2 ∣ Y h t+2 E[dt+2 ∣ Y 1 t+2 ] ] ] ] ] ] (1) (2) (3) (4) 0.96 1.86 1.042 1.049 0.007 0.96 0.95 1 1 1.04 0.96 1 1 1.06 0.94 1.036 0.964 0.95 1.85 1.053 1.053 0.000 0.95 0.95.96 1.87 1.042 1.045 0.003 0.96 0.96 1 1 1.04 0.96 1 1 1 1 1 1 0.96 2.05 1.042 1.049
0.007 0.96 1.05 1 1 1.04 0.96 1 1 1.16 1.04 1.136 1.064 Table 35.2 presents information from numerical experiments under several diﬀerent scenarios described in column headings. Case (1) is the base case as described above. The bond yields about 4.2 percent, and the stock about 4.9 percent. As such, the equity premium is roughly 0.7 percent. The numbers are chosen such that the unconditional expectations of both dividends and income are all the same at 1. But conditional on income being high in t + 1, the expected value of income in t + 2 is higher than the unconditional expectation (1.04). Similarly, conditional on income being low in t + 1, the expectation of income in t + 2 is lower than the unconditional expectation (0.96). This is a feature of the assumed persistence. In both t + 1 and t + 2, the dividend payout is positively correlated with income. Conditional on income being high in t + 1, for example, the expected dividend is 1.06, whereas conditional on income being low, the expected dividend is 0.94. A similar pattern emerges, though not as stark, for t + 2. Note that the expected value of the future stock price is lower than the current value of the stock price. This is an artifact of our assumption of a ﬁnite number of periods – in period t, the stock is a claim on two dividend payments of equal magnitude in expectation, whereas in t + 1 a share of stock is a claim on only one dividend payment. It is therefore natural in this example that Qt > E[Qt+1]. 861 t+2 t++2 Column (2) considers a case in which there is no uncertainty over future income – that = 1. The probabilities are the same and the dividend values are is, Y h t+1 the same. The bond has a higher yield in this case. The reason for this relates back to precautionary saving – if there is no income uncertainty, there is less demand for the bond, and hence it trades at a lower price and higher yield compared to the case with uncertainty. We see that the stock has the same expected return as the bond; there is no equity premium. There is no equity premium because there is no covariance between stock returns and the marginal utility of consumption if there is no uncertainty over income/consumption. t+2
t+1 t+2 t+1 = dl = dl = dh Column (3) considers the case where there is income uncertainty, as in our base case (1), but = 1. assumes that there is no uncertainty over the future dividend, so dh The bond price and yield are identical to the base case, (1). However, even though there is no dividend uncertainty, the stock still commands a higher yield than the bond, with an implied equity premium of about 0.3 percent. If there is no covariance between dividend payments and the marginal utility of consumption, why is there an equity premium? The reason is that there is in eﬀect a maturity diﬀerence between the bond and the stock. The stock has a two period maturity, the bond a one period maturity. Since the stock pays a constant and known dividend, the stock in this example is isomorphic to a two period bond. For the same reasons we encountered in Chapter 34, uncertainty over income means there is a term premium. Hence, the stock trades at a higher yield than the one period risk-free bond. Column (3) reveals an important point. In reality, stocks are securities with no maturity, as corporations can exist in perpetuity. This means that a stock that pays a very consistent dividend (what is sometimes called an “income stock”) ought to trade at a fairly similar yield to long maturity government bonds. The ﬁnal case considered, labeled (4), is one in which the expected value of dividends in both t + 1 and t + 2 is 1.1 instead of 1, but is otherwise identical to case (1). This has the eﬀect of resulting in a higher stock price, but has no eﬀect on the yields of the bond and the stock or the equity premium. The equity premium arises because of covariance between dividends and future income, not the expected level of future income. Moving to more than three periods is reasonably straightforward. As long as there are at least two periods, for any stock the pricing condition can be written as: Qt = E [βu′(Ct+1) u′(Ct) (dt+1 + Qt+1)] (35.102) The price today is simply the expectation of the product of the stochastic discount factor with the cash ﬂows generated by the stock. If time lasts for three or more periods, as in the example above, note that the
expression for Qt+1 will be the same as (35.102), just led 862 forward one period: Qt+1 = Et+1 [ βu′(Ct+2) u′(Ct+1) (dt+2 + Qt+2)] (35.103) (35.103) can be plugged into (35.102) as we did above to give: Qt = E [ βu′(Ct+1) u′(Ct) (dt+1 + Et+1 [βu′(Ct+2) u′(Ct+1) (dt+2 + Qt+2)])] (35.104) This process can be continued by noting that, for any arbitrary number of periods into the future, j ≥ 1, we must have: Qt+j = Et+j [βu′(Ct+j+1) u′(Ct+j) (dt+j+1 + Qt+j+1)] (35.105) Suppose that time continues until period T (i.e. there are T periods subsequent to the present period t). If we successively substitute (35.105) into (35.104), and make use of the Law of Iterated Expectations, we can write: Qt = E [ T ∑ j=1 mt,t+jdt+j] + E [mt,t+T Qt+T ] (35.106) u′ (Ct+j ) (Ct) Where once again mt,t+j = βj u′ is the stochastic discount factor between t and t + j. As long as T is ﬁnite, a terminal condition should be satisﬁed such that Qt+T = 0. This condition simply says that the stock is worthless in the ﬁnal period. Since the stock is a claim on future dividends, if there is no future, the stock should not trade for a positive price in the ﬁnal period. Imposing this terminal condition allows us to write: Qt = E [ T ∑ j=1 mt,t+jdt+j] (35.107) In other words, the stock price should be the expected present discounted value of future dividends. Discounting is by the stochastic discount factor, which, if there is uncertainty, will diﬀer from the yield on the risk-free bond. 35.3.1 The Gordon
Growth Model The Gordon Growth Model is a famous stock pricing equation that is a special case of (35.107). It result in an intuitive and easy to understand pricing equation that provides a number of useful insights. Suppose that the stochastic discount factor varies only with the horizon. In particular, 863 suppose that mt,t+j = ( 1. This would obtain if, for example, the endowment were constant, 1+r so that mt,t+j = βj. Suppose further that dividends grow at a constant rate, g. There is no uncertainty in the dividend process. In particular: )j Iterating (35.108) forward, we get: Dt+1 = (1 + g)Dt Dt+j = (1 + g)jDt (35.108) (35.109) Let the household live forever, so T → ∞. Under these assumptions, the stock price can be solved for using (35.107): Qt = ( =1 Dt Using facts about inﬁnite sums, this can be written: (35.111) can be written: Qt = Dt 1 1 − 1+g 1+r Qt = (1 + g)Dt r − g (35.110) (35.111) (35.112) (35.112) tells us that a stock price depends positively on the level of its dividends, Dt; positively on the growth rate of dividends, g; and negatively on the discount rate, r. One can divide both sides by Dt to write this as a price-dividend ratio (which ought to be reasonably closely-related to a price-earnings ratio): Qt Dt = 1 + g r − g (35.113) While (35.113) only holds under special circumstances, it provides a couple of important insights. Price-dividend ratios can vary (either across time or across stocks) with the growth rate of dividends and the discount rate. Young startup companies may pay little or no dividend at present, but they may have fast expected growth (i.e. high g). We therefore would expect such companies to have a high price-dividend ratio relative to more mature companies which might have high current earnings but weaker long run earnings potential. The riskier is a stock (i.e. the more its payouts covary with the stochastic discount factor), the higher should be the discount rate applied to
the stock, and hence such a stock should 864 trade at a comparatively low price-dividend ratio. Finally, decreases in interest rates in the economy as a whole ought to generally be good for stock prices – as we can see in (35.113), a decline in r results in a higher Qt for a given Dt. 35.4 Bubbles and the Role of the Terminal Condition The term “bubble” is often used in popular parlance to describe the behavior of the stock market (or markets for other assets, such as land or housing). In popular usage, “bubble” seemingly just refers to a situation in which an asset’s price is high (and rising) without observable changes in cash ﬂows from the asset. Economists have a more precise deﬁnition of the term bubble. In particular, in (35.106) the “bubble” component of the price is deﬁned as the expected discounted value of the stock price in the ﬁnal period of time, while the “fundamental” component is the expected presented discounted value of the stream dividends. Concretely: Qt = E [ T ∑ j=1 mt,t+jdt+j] ·„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„‚„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„¶ Fundamental + E [mt,t+T Qt+T ] ·„„„„„„„„„„„„„„„„„„„„„„„„„„„�
�„„„„„„„„„‚„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„¶ Bubble (35.114) Which we can write more compactly as: Qt = QF t + QB t (35.115) When working above, we ruled the bubble term out, setting QB t = 0. What allowed us to do so was the idea that, in the ﬁnal period, any asset should have a zero price, so Qt+T = 0. Again, the intuition for this is that the share of stock is a claim to future dividends; in period t + T, there is no future, and hence no one should be willing to pay a positive price = 0, and the period t price of the asset is simply the for the asset. If Qt+T = 0, then QB fundamental component, which is the presented discounted value of the stream of dividends where discounting is by the stochastic discount factor. A general point which we can conclude from this discussion is that one should not observe bubbles (as economists have deﬁned the term) for assets with a ﬁnite life span / maturity (such as almost all bonds). t But in reality, stocks diﬀer from bonds in that there is no maturity for a stock. In principle, a company exists in perpetuity. This suggests that we should let T → ∞, so that time never ends. If this is the case, then (35.114) may be written: Qt = E [ ∞ ∑ j=1 mt,t+jdt+j] ·„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„‚„„„„„„„„
„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„¶ Fundamental E [mt,t+T Qt+T ] + lim T →∞ ·„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„‚„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„¶ Bubble (35.116) 865 We can still express (35.116) by (35.115), but unlike the case where T is ﬁnite, we cannot necessarily rule the bubble term out. Why is this? If there is no end of time, then there is no reason to think that the price of the stock will ever be zero. But if we can never say that the stock has zero value, we cannot necessarily conclude that QB t = 0. Even if we cannot necessarily rule bubbles out, we can say something about how bubbles must evolve. Recall that the period t price of the stock can be written: Qt = E [mt,t+1(dt+1 + Qt+1)] (35.117) can be written: Qt = E[mt,t+1dt+1] + E[mt,t+1Qt+1] Now, using (35.115), (35.
118) may be written: (35.117) (35.118) QF t + QB t = E[mt,t+1dt+1] + E[mt,t+1QF t+1 ] + E[mt,t+1QB t+1 ] (35.119) But note that QF t = E[mt,t+1dt+1] + E[mt,t+1QF t+1 ]. Hence: QB t = E[mt,t+1QB t+1 ] (35.120) Assume that the bubble term is uncorrelated with the stochastic discount factor. This means that the expectation of the products on the right hand side of (35.120) is the product of the expectations. This allows us to write: E[QB t+1 ] = (E[mt,t+1])−1 QB t (35.121) The stochastic discount factor will ordinarily be less than one, so its inverse will be greater ≠ 0), it must be expected to grow. than one. (35.121) then tells us that, if a bubble exists (QB t Furthermore, it must be expected to grow at the inverse of the stochastic discount factor (i.e. the growth rate of the bubble term ought to equal the yield on a one period riskless bond). > 0 (note it could also be negative), then one would be paying more than Intuitively, if QB t the stock’s fundamenal value to purchase that stock. One would only be willing to pay than the fundamental value of the stock if one expects that someone else in the future will overpay by even more. Put somewhat more colloquially, you might be willing to behave foolishly if you think there is a greater fool out there. Note that a stock with a bubble component does not oﬀer a higher expected return (adjusting for risk) compared to a stock without a bubble. To see this clearly, let us consider a particularly simple example. Suppose that current and future dividends are ﬁxed at 866 one. Suppose further that the current and future endowments are also ﬁxed at one. This means that mt,t+1 = β and there is no uncertainty over the stochastic discount factor. The fundamental value of the stock is simply the present value of the constant dividend payout, which under these assumptions is
just: QF t = βj ∞ ∑ j=1 (35.122) Using facts about inﬁnite sums, this simply works out to β 1−β. Since the dividend is constant and the endowment is constant as well, the fundamental stock price will be constant throughout time. The expected (gross) return from holding the stock from t to t + 1 without a bubble (over which there is no uncertainty) is simply: Rt+1 = Dt+1 + Qt+1 Qt = 1 + β 1−β β 1−β = 1 β (35.123) In other words, the expected return is just β−1, the inverse of the stochastic discount factor (which would be the same as a yield on a one period bond). Now suppose that the > 0. Suppose that the bubble continues with probability p and stock is in a bubble, with QB t bursts (goes to zero) with probability 1 − p. Since a bubble must grow in expectation, from (35.121), the realized value of the bubble in t + 1 if it continues must satisfy: pQB t+1 + (1 − p)0 = 1 β QB t (35.124) If the bubble continues, the price in t + 1 will be Qt+1 = β 1−β t. If it bursts, the price 1−β. The expected return on the stock (which is no longer certain) is therefore: will be Qt+1 = β βp QB + 1 E[Rt+1] = Dt+1 + E[Qt+1] Qt 1 + p ( β 1−β = Simplifying (35.125), we obtain: + 1 pβ QB ) + (1 − p) β 1−β t + QB t β 1−β E[Rt+1] = 1 1−β β 1−β + 1 β QB t + QB t (35.126) can be written: E[Rt+1] = But (35.127) may be written: 1 + 1−β β QB t β + (1 − β)QB t 867 (35.125) (35.126) (35.127) E[Rt+1] = 1 β (β + (1 − β)QB t β + (1 − β)QB t ) = 1 β (35.128) In other words, for any value of
QB t, the expected gross return on the stock is just 1 β. Hence, the stock being in a bubble does not oﬀer high expected returns. What it does do is increase the volatility of realized returns. If there is no bubble, in this particular example the return is always 1 β. If there is a bubble, with probability p the return will be higher than this and with probability 1 − p the return will be lower. For example, suppose that β = 0.95. If there were no bubble, the return would be constant at 1.0526. Suppose that there is a bubble = 1 and p = 0.8 (i.e. the bubble continues with 80 percent probability). If the bubble of QB t continues, the realized return works out to 1.0658, whereas if the bubble burst the realized (gross) return is 1 (so a net return of 0). The average of these is 0.8 × 1.0658 + 0.2 × 1 = 1.0526, the same as the return would be with no bubble. A stock with a bubble component does not oﬀer a high return in expectation; it oﬀers a high return if the bubble continues and a low return if the bubble bursts. Bubbles increase the volatility of returns, but not the level of returns in an average sense. 35.4.1 A Numerical Example with Bubbles Let us explore a somewhat more complicated numerical example. Doing so will give us some guidance on how one might detect bubbles in the data. Suppose that the representative household is inﬁnitely-lived and has a discount factor of β = 0.95. The utility function is the natural log. Suppose that the household’s endowment is constant at one. This means there is no risk premium for equity over riskless debt. This is not critical but simpliﬁes the computation. Suppose that shares of a stock pay dividends according to the following stochastic process: dt = (1 − ρ) ¯d + ρdt−1 + εd,t, 0 ≤ ρ < 1, εd,t ∼ N (0, s2) (35.129) εd,t is a stochastic shock drawn from a normal distribution with zero mean and standard deviation of s (the variance is s2). The process has an unconditional mean of ¯d. Conditional on dividends observed in
t, the expected value of dividends in the next period is Et dt+1 = (1 − ρ) ¯d + ρdt. ρ is a measure of the persistence in the process for dividends. ρ > 0 means that dt > ¯d is associated with Et dt+1 > ¯d – i.e. if dividends are higher than average today, you will expect them to be higher than average tomorrow as well. If the endowment is constant and consumption must equal income each period, then the stochastic discount factor is mt,t+j = βj. This signiﬁcantly simpliﬁes the analysis. Making use 868 of this, we can write the price of a share of stock as: Qt = E [β (dt+1 + Qt+1)] (35.130) We can again split this into two components – the fundamental component and the bubble component. The fundamental component is the present value of dividends; QF t = ∞ ∑ j=1 βj Et dt+j The expectation of future dt+j given a current dt+j is: Plug (35.132) into (35.131) and re-arrange to get: Et dt+j = ¯d + ρj(dt − ¯d) QF t = ∞ ∑ j=1 βj ¯d + (βρ)j(dt − ¯d) ∞ ∑ j=1 Using facts about inﬁnite summations, this reduces to: QF t = β ¯d 1 − β + βρ(dt − ¯d) 1 − βρ The bubble component must satisfy: The actual price is: E[QB t+1 ] = β−1QB t Qt = QF t + QB t (35.131) (35.132) (35.133) (35.134) (35.135) (35.136) Let us ﬁrst rule out the possibility of bubbles by setting QB t = 0 at all horizons. We can generate a simulation of the stock price by simulating a process for dividends. Suppose that β = 0.95. Suppose further that ¯d = 1, ρ = 0.98, and s = 0.02. We simulate 1000 periods of dividends. At each point, we calculate the fundamental stock price using (35.134). A hypothetical time
series of the stock price across 1000 periods is: 869 Figure 35.3: Simulated Stock Price: No Bubbles Stock prices ﬂuctuate about a mean value of roughly 19. There are periods in the simulation that look a bit like the popular deﬁnition of a bubble – periods in which stock prices at ﬁrst go up markedly and then down quickly (e.g. roughly period 525 to period 600 exhibits a sharp boom and then bust in the stock price). Yet Figure 35.3 is generated with no bubbles at all! This simple plot underscores an important point – it is diﬃcult to identify bubbles even after the fact, much less in real time. Next we re-do the simulation but allow for the possibility of exogenous and stochastic bubbles. Formally, we assume that the bubble term obeys a stochastic Markov chain. In particular, we assume that there are three possible states – negative bubble, no bubble, and positive bubble. The economy begins in the no bubble state. In the subsequent period, there is a 95 percent chance of staying in the no bubble state, a 2.5 percent chance of entering a negative bubble, and a 2.5 percent chance of entering a positive bubble. If the economy moves from no bubble to a positive bubble in period t, then QB 2, and if it moves to a t = − 1 negative bubble, we have QB 2. Once in a bubble state (either positive or negative), t assume that there is a p probability of remaining in the bubble and a 1 − p probability of it “bursting” and returning to zero in the subsequent period. From above, recall that we must have E[QB t. With this setup, the value of the bubble term should it continue t+1 satisﬁes: ] = β1QB = 1 pQB t+1 + (1 − p)0 = β−1QB t (35.137) Or: 870 0200400600800100014151617181920212223 QB t+1 = 1 βp QB t (35.138) In other words, if the bubble continues, it must grow by a suﬃcient amount that in = 0.5 we = 0.6579 should the bubble continue. This is growth of about 31 percent. β is necessary to compensate the household for the expectation it grows at β−1. For example, if
β = 0.95 and p = 0.8, then if QB t must have QB t+1 This additional growth over and above 1 possibility that the bubble bursts. For our numerical simulation, we assume that p = 0.8. This means that, conditional on being in a bubble, there is a 80 percent chance of remaining in the bubble and a 20 percent chance of exiting. Figure 35.4 plots a simulated bubble process under these assumptions. During most periods, the bubble term is zero. There are periods in which it goes positive and negative. If it stays in the bubble, it grows (declines) very rapidly until the bubble bursts, when it immediately jumps back to zero. We observe a very large bubble term emerging around period 250 in the simulation. There are several other smaller bubbles (both negative and positive), though these are not as noticeable. Figure 35.4: Simulated Bubble Process Figure 35.5 plots the simulated actual stock price, Qt = QF t t, along with the fundat. During most periods these are quite similar, though they deviate from one ≠ 0. The most noticeable such period is around period 250, which is a mental price, QF another whenever QB t period of a very large and long-lasting positive bubble, as noted above. + QB 871 01002003004005006007008009001000-100102030405060 Figure 35.5: Actual and Fundamental Stock Price One would be tempted to look at the simulation in Figure 35.5 and conclude that spotting bubbles is fairly straightforward – simply look for periods where stock prices increase or decrease very rapidly. But in practice this may not be so easy and there are some speciﬁcs to this particular simulation. For example, if we were to increase the volatility or persistence of the dividend process (i.e. refer back to Figure 35.3), we could make the stock price series arbitrarily volatile and there might be many episodes which “look like” bubbles but are not. Is there a more robust way to try to detect bubbles in past data? Economist Robert Shiller has popularized a simple measure based on price-earnings ratios (PE ratios) for the stock market as a whole. Empirically, the average PE ratio for the market as a whole is somewhere between 15 and 20. This would be broadly consistent with the simple Gordon Growth model described above if (i) dividends equal earnings, (ii) the discount rate is 7 percent, and
(iii) the growth rate of dividends is 1.5 percent.4 Shiller noted that when the PE ratio is unusually high, this tends to be associated with stocks performing relatively poorly over the ensuing 20 yeas. The reverse is the case when the PE ratio is relatively low. Stock performance here is measured as the cumulative realized return over the ensuing 20 years, assuming all dividends are re-invested into the market. Figure 35.6 below plots a scatter plot of historical PE ratios with subsequent 20 year realized returns. There is a clear negative relationship evident in the data, with a correlation of about -0.35. 4In particular, using the Gordon Growth Model with these numbers, we’d get a ratio of price-dividends (equivalently earnings) of 1.015.07−.015 = 18.45. 872 010020030040050060070080090010001020304050607080 Figure 35.6: PE Ratio and Subsequent 20 Year Realized Return: S&P 500 What is the logic behind why a negative correlation between future realized returns and current PE ratios may be indicative of bubbles? If a stock is experiencing a (positive) bubble, then it ought to trade at a comparatively high PE ratio (the reverse would be true for a negative bubble). While being in a bubble does not have a higher than average return in expectation, the bubble ought to burst at some point in the future, and when it does, the stock will suﬀer a large capital loss (or gain in the case of a negative bubble). For example, with a probability of a bubble continuing of p as described above, the probability of the bubble still being in place 20 periods later is p20. With p = 0.8, this is about 1 percent. Hence, there is a 99 percent chance of the bubble having bursted by a 20 year horizon, which means the owner of the stock will suﬀer a capital loss and hence a low realized return. To see if such a test makes sense, we return to our simulation model described in this subsection. We assume, for the moment, that there are no bubbles. On the 1000 period simulation, for each of the ﬁrst 980 periods, we calculate the realized return on holding the stock over the subsequent 20 periods. The realized return assumes dividends are reinvested.5 Figure 35.7 below plots a scatter plot of current price-dividend ratios in the model with subsequent
realized twenty period (gross) returns. There is clearly not a negative relationship between price-dividend ratios and subsequent returns; in this particular sample of simulated 5To be clear, one could deﬁne the (realized) holding period return as the sum of cash ﬂows over some horizon h relative to the current price of the security: Rt+h = [ ∑ average it will be declining for the horizon h because simply summing dividend payouts ignores compounding eﬀects.. If one computes this, on ] h j=1 Dt+j +Qt+h Qt 1 h 873 00.020.040.060.080.10.120.140.165101520253020 Yr Annualized Return P/E Ratio Correlation = -0.35 data, if anything the correlation looks positive. Figure 35.7: Price-Dividend Ratio and Subsequent Realized Return: Model with No Bubble Next, we re-do the simulation but this time allow for bubbles with the same stochastic process as described above. Figure 35.8 plots the scatter plot between price-dividend ratios and subsequent realized twenty period returns. With bubbles included in the model simulation, we observe a clear negative relationship between price-dividend ratios and subsequent twenty period returns. The correlation in this particular sample is -0.63. Periods of very high price-dividend ratios are associated with lower than average returns over the next twenty periods, while the reverse is true for periods of very low price-dividend ratios. 874 17.51818.51919.5201.041.0451.051.0551.061.065 Figure 35.8: Price-Dividend Ratio and Subsequent Realized Return: Bubbles In the context of the simulation model employed here, it does seem necessary to include bubbles to match the correlation between PE ratios and subsequent realized returns found in the data. In retrospect and at least in an average sense, it seems possible to identify bubbles by periods of unusually high (or low) price-earnings ratios which are followed by unusually poor (or strong) returns. But this test only works after the fact. Detecting bubbles in real time is quite diﬃcult. 35.4.2 Should Monetary Policy Attempt to Prick Bubbles? In the wake of the collapse of the housing market in 2006 and 2007, many have
argued that the Federal Reserve (and central banks more generally) should be tasked with stopping asset price bubbles before they get out of hand. Further, some have argued that central banks have been responsible for bubbles in the ﬁrst place via monetary policies that are too easy. The bursting of bubbles often has detrimental consequences beyond the decline in prices of particular classes of assets aﬀected by a bubble. As we discuss further in Chapter 37, a decline in asset prices could set of a “run” on ﬁnancial institutions when short term liability holders (e.g. depositors) become concerned about the value of the assets held by ﬁnancial institutions. Such runs can be self-fulﬁlling and lead to widespread increases in credit market spreads and large reductions in economic activity. If asset price bubbles lead to runs and ﬁnancial panics with widespread economic consequences, it would be best to try to avoid such bubbles in the ﬁrst place. 875 5101520253011.021.041.061.081.11.121.141.16 A common argument is that central banks ought to adjust interest rates to prevent bubbles from occurring. John Taylor of Stanford university, for example, has argued that the housing market bubble was driven by the Fed keeping its policy rate too low for too long in the early 2000s. It is easy to see why low interest rates can result in high asset prices (see, e.g., the Gordon Growth pricing condition above, (35.112)). It is less obvious why low interest rates could cause bubbles (which are exogenous in the context of the asset pricing model considered above), or why raising interest rates could prick bubbles,6 though it is commonly thought that bubbles are fueled by excessively easy credit (i.e. investors buying stocks on margin, or individuals buying houses with very low interest rate mortgages). Our own view is that conventional monetary policy (by which we mean the adjustment of the monetary base and short term interbank lending rates) ought not to be in the business of trying to prick bubbles or to target asset prices more generally. First, the detection of bubbles in real time is extraordinarily diﬃcult and fraught with hazards. Raising interest rates to prick what appears to be an asset price bubble but which is not might be quite costly. Second, interest rates aﬀect credit decisions of all sorts of actors and sectors, many of which might be completely unrelated
to the asset market exhibiting bubble behavior (e.g. raising interest rates to combat a suspected housing market bubble would aﬀect the ability of corporations to issue commercial paper to cover short term ﬁnancing needs, which is completely unrelated to real estate markets). Third, as hinted at above, it is not clear whether using monetary policy to manage bubbles would even be successful, or if doing so could actually exacerbate the problem of bubbles. A better solution to the problem of asset price bubbles and their bursting seems to be along the lines of so-called macroprudential regulation, which has come to the fore as a policy tool in the wake of the Great Recession. Macroprudential regulation diﬀers from conventional regulation of ﬁnancial institutions which is micro in nature (i.e. microprudential regulation). Macroprudential regulation focuses on the ﬁnancial system as a whole, and seeks to enact policies which limit the extent to which the system as a whole is subject to systemic risk. Rather than trying to prick bubbles or prevent them from happening in the ﬁrst place, macroprudential regulation seeks to make the ﬁnancial system as a whole better able to withstand the bursting of asset price bubbles. Examples of macroprudential regulatory tools include time-varying capital ratios (the ratio of total equity to assets for ﬁnancial institutions). During good times, institutions should have to build up buﬀers in the form of higher capital ratios, which makes them better able to withstand declines in asset values in a bust, and 6Indeed, one could make the argument that raising interest rates could exacerbate the problem of bubbles bursting. As documented above, under conventional theories bubbles must grow at the discount rate. Raising interest rates raises the discount rate on equity, and might cause bubbles (if they exist) to expand at an even faster rate, which would necessitate an even bigger decline in asset prices when the bubble eventually bursts. 876 therefore less likely to suﬀer runs. During bad times, ﬁnancial institutions should face less stringent capital ratios, which ought to reduce their incentive to try to liquidate assets when facing funding diﬃculties. Another type of macroprudential regulatory tool focuses not so much on the level of total debt of a ﬁnancial institution, but rather on its maturity structure. Short maturity debt is much more subject to runs than
long maturity debt. 35.5 Equilibrium Stock Prices with Endogenous Production: the Neoclassical Model In this Chapter we have discussed stock pricing in the context of an endowment economy. This simpliﬁes the analysis and elucidates important insights. We close this chapter by returning to the neoclassical business cycle model, the microfoundations of which are developed in Chapter 12. Though we did not formally think about the value of shares of stock in the representative ﬁrm, we can do so, and will show how to incorporate this setup in this section. For what follows, let us abstract from uncertainty although allowing the future to be uncertain would not fundamentally change any of our results. In Chapter 12, we wrote the ﬂow budget constraints facing the household as (presented below with a slight modiﬁcation of notation to be consistent with recent chapters): Ct + Bt ≤ wtNt + Dt Ct+1 ≤ wt+1Nt+1 + Dt+1 + (1 + rt)Bt (35.139) (35.140) In (35.139)-(35.140), Bt denotes the stock of one period bonds the household takes from t to t + 1; in the terminology of Chapter 34, we have normalized the period t price of these bonds to 1, with the face value given by (1 + rt). wt and wt+1 are the period t and t + 1 real wages, Nt and Nt+1 denote labor supply, and Dt and Dt+1 measure dividend payouts from the household’s ownership in the representative ﬁrm. Dt and Dt+1 are taken as given. In the presentation from Chapter 12 and given above, we do not allow the household to decide whether (or how much) of the ﬁrm’s shares of stock to hold. As such, we cannot study the value of shares of stock in the ﬁrm. But this can easily be modiﬁed in such a way that we can price the ﬁrm’s stock. Even with this modiﬁcation, the other equilibrium prices and allocations will be identical to the setup in which we ignore this decision. In (35.139), the only way for the household to transfer resources intertemporally is through savings bonds, Bt. We can instead modify the problem where
the household can also save through accumulation of shares of stock in the ﬁrm. In particular, we can write the ﬂow 877 budget constraints as: Ct + Bt + Qt(SHt − SHt−1) ≤ wtNt + dtSHt−1 Ct+1 + Qt+1(SHt+1 − SHt) + Bt+1 ≤ wt+1Nt+1 + dt+1SHt + (1 + rt)Bt (35.141) (35.142) In (35.141)-(35.142), SHt−1 denotes the shares of stock the household brings into period t. Because this was chosen in the past, it is exogenous with respect to period t. The household can choose a new value of shares, SHt, to take from t to t + 1, and another value, SHt+1, to take from t + 1 to t + 2. dt and dt+1 denote dividend rates, reﬂecting the dividend payout per share. In terms of the notation from above, we would have Dt = dtSHt−1 and Dt+1 = dt+1SHt. Qt and Qt+1 denote the prices of shares of ownership in the ﬁrm, both of which the household take as given. So as to simplify analysis, suppose that the household simply supplies one unit of labor inelastically in both periods, i.e. Nt = Nt+1 = 1. Furthermore, let us impose the terminal conditions that SHt+1 = 0 (i.e. the household will not choose to die with any ownership stake in the ﬁrm) and Bt+1 = 0 (i.e. the household will not choose to die with a positive stock of savings, and will not be allowed to die in debt). Imposing these terminal conditions as well as the normalizations on labor supply, the ﬂow budget constraints simplify to: Ct + Bt + Qt(SHt − SHt−1) ≤ wt + dtSHt−1 Ct+1 ≤ wt+1 + (dt+1 + Qt+1)SHt + (1 + rt)Bt Lifetime utility for the household is standard: U = u(Ct) + βu(Ct+1) (35.143) (35.144)
(35.145) The houehold’s objective is to pick SHt and Bt so as to maximize (35.145), subject to (35.143)-(35.144). Imposing that the constraints hold with equality, we can write the unconstrained maximization problem as: max Bt,St U = u [wt + dtSHt−1 − Bt − Qt(St − SHt−1)] + βu [wt+1 + (dt+1 + Qt+1)SHt + (1 + rt)Bt] (35.146) The ﬁrst order optimality conditions are: 878 ∂U ∂Bt = 0 ⇔ u′(Ct) = βu′(Ct+1)(1 + rt) ∂U ∂SHt = 0 ⇔ Qtu′(Ct) = βu′(Ct+1)(dt+1 + Qt+1) (35.147) (35.148) (35.147) is the familiar consumption Euler equation. (35.148) implicitly deﬁnes the equilibrium share price of the ﬁrm as: Qt = βu′(Ct+1)(dt+1 + Qt+1) u′(Ct) (35.149) (35.150) is of course exactly the same (minus the fact that we are abstracting from uncertainty) as the general stock pricing expression derived above, (35.15). Imposing the no-bubble terminal condition that Qt+1 = 0, we simply get: Qt = βu′(Ct+1)dt+1 (35.150) t N 1−α t u′(Ct) The ﬁrm is identical to what was presented in earlier chapters. It produces output using Yt = K α, where we have ﬁxed the value of the exogenous productivity variable to one for simplicity. The ﬁrm hires labor on a period-by-period basis at the market real wage. It begins with an initial stock of capital, Kt. It can accumulate new capital through investment, with Kt+1 = It + (1 − δ)Kt. We assume that any new investment must be ﬁnanced with debt (as opposed to the issuance of new shares of stock). The number of existing shares, SHt−1, is taken as
given. Not allowing the ﬁrm to issue new shares requires that SHt = SHt−1.7 The total period t dividend that the ﬁrm returns to the household is simply output less payments to labor: Dt = K α t N 1−α t − wtNt (35.151) The ﬁrm must ﬁnance any investment in period t via borrowing, which must be paid back at (1 + rt) per unit borrowed in t + 1. Any capital left over after production in t + 1 is also returned to the household. The ﬁrm’s total period t + 1 dividend is therefore: Dt+1 = K α t+1N 1−α t+1 − wt+1Nt+1 − (1 + rt)(Kt+1 − (1 − δ)Kt) + (1 − δ)Kt+1 (35.152) The value of the ﬁrm, Vt, is equal to the sum of its current total dividend plus its share 7While it may seem restrictive to assume that the ﬁrm cannot issue new shares, recall from Chapter 12 that the Modigliani-Miller theorem holds in this model. That is, whether the ﬁrm ﬁnances its capital with debt or equity (issuing new shares) is irrelevant for what happens in the model. 879 price, Qt, multiplied by the number of outstanding shares at the end of period t, SHt. The period t dividend represents current income to the household, and Qt represents how much the household values the claim on future dividends. Hence: Using (35.150), we can write (35.153) as: Vt = Dt + QtSHt Vt = Dt + βu′(Ct+1)dt+1 u′(Ct) SHt (35.153) (35.154) But from (35.148), we know that βu′ u′ we can write the total value of the ﬁrm as: (Ct+1) (Ct) = 1 1+rt. Furthermore, dt+1SHt = Dt+1. Hence, Vt = Dt + Dt+1 1 + rt (35.155) In other words, the value of the ﬁrm is simply the present discounted value of dividends. This
is what we assumed in Chapter 12, but here it has been derived formally. The ﬁrm’s objective is to pick It (equivalently Kt+1, since Kt is taken as given), Nt, and Nt+1 to maximize Vt subject to a standard law of motion for capital (i.e. Kt+1 = It + (1 − δ)Kt. The optimality conditions for the ﬁrm are: wt = (1 − α)K α wt+1 = (1 − α)K α rt + δ = αK α−1 t+1 N 1−α t+1 t N −α t t+1N −α t+1 (35.156) (35.157) (35.158) As noted above, we are assuming that the household supplies one unit of labor inelastically in both periods, that the number of initial shares of stock in the ﬁrm is SHt−1 and is given, and that the number of shares outstanding does not change going from t to t + 1. We can therefore assume as a normalization that SHt−1 = SHt = 1 (i.e. there is one share outstanding), which means that there is no distinction between the dividend rate, dt, and the total dividend payout, Dt. Since in equilibrium Bt = It, and It+1 = −(1 − δ)Kt+1 (i.e. the ﬁrm liquidates itself after production in t + 1), the aggregate resource constraints work out in the standard way – Yt = K α = Ct+1 + It+1. Using the optimality condition for period t + 1 t labor demand, the period t + 1 dividend can be written: = Ct + It and Yt+1 = K α t+1 Dt+1 = K α t+1 − (1 − α)K α t+1 + (1 − δ)Kt+1 − (1 + rt)(Kt+1 − (1 − δ)Kt) (35.159) 880 Which can be written: Dt+1 = αK α t+1 + (1 − δ)Kt+1 − (1 + rt)Kt+1 + (1 + rt)(1 − δ)Kt (35.
160) Using (35.158), this can be written: Dt+1 = (rt + δ)Kt+1 + (1 − δ)Kt+1 − (1 + rt)Kt+1 + (1 + rt)(1 − δ)Kt (35.161) But the terms involving (35.161) involving Kt+1 drop out, leaving: But this means that the equilibrium share price of the ﬁrm is: Dt+1 = (1 + rt)(1 − δ)Kt Qt = (1 − δ)Kt (35.162) (35.163) The share price, which represents what the household is willing to pay to own a share of stock in the ﬁrm, is simply equal to the end-of-period capital stock of the ﬁrm, (1 − δ)Kt. If you stop to think about it, it makes a lot of sense that this is the equilibrium share price of the ﬁrm. The ﬁrm is nothing more than a collection of capital. Capital within period t is “stuck” in the ﬁrm. After production in period t takes place, there is (1 − δ)Kt left over. If the ﬁrm liquidated itself at the end of period t (instead of period t + 1), the ﬁrm could return (1 − δ)Kt to the household in period t. It therefore makes sense that this is what the household would be willing to pay for the ﬁrm. If Qt < (1 − δ)Kt, the household would like to buy more shares in the ﬁrm, but the number of shares are ﬁxed. If Qt > (1 − δ)Kt, in contrast, the household would like to sell shares, which would involve liquidating the ﬁrm. Since the number of shares is ﬁxed by assumption, it must be the case that the ﬁrm’s stock price is (1 − δ)Kt for the household to be indiﬀerent between selling or buying shares in the ﬁrm. Stockholder’s equity is deﬁned as the market-value of a ﬁrm times the number of shares outstanding. Since we
have ﬁxed the number of shares outstanding at 1, stockholder’s equity is simply the share price, Qt. This is simply equal to the end-of-period capital stock. This is another manifestation of why the terms equity and capital are often used interchangeably. 35.6 Summary • The ownership of a stock entitles the owner to current and future proﬁts of the company. Because stocks oﬀer volatile dividends over an inﬁnite horizon and stockholders are junior claimants, stocks are riskier than bonds in general. 881 • Because dividend payments and stock prices are likely to be high when consumption is high, equity returns are negatively correlated with the stochastic discount factor. This generates a positive equity premium. • A price of a stock should equal the expected present discounted value of future dividends where the dividends are discounted by the stochastic discount factor. • Under some special circumstances the stock pricing formula reduces to a function of the the growth rate of the dividend, the interest rate, and the dividend. • According to our theory, securities with ﬁnite maturities (e.g. most bonds) do not have bubbles because no one would pay for them after their maturity date. Stocks, in contrast, can have bubbles. • Identifying bubbles is very diﬃcult in practice. That said, our model predicts that the price to earnings ratio might be a good indicator of a potential bubble. • While arguments can be made for using conventional monetary policy to prick bubbles, an attractive alternative is macroprudential regulation which focuses on the ﬁnancial system as a whole. Key Terms 1. Stock 2. Dividends 3. Equity premium 4. Stochastic discount factor 5. Gordon Growth Model 6. Bubble 7. Macroprudential regulation. Questions for Review 1. Why are stocks riskier than bonds in general? 2. According to the Gordon Growth Model, what factors inﬂuence the price to dividend ratio? 3. How does the economist’s deﬁnition of a bubble diﬀer from the deﬁnition in common parlance? 882 4. Explain how bubbles inﬂuence the expected returns of stocks. How do they aﬀect the variance of expected returns? 5. Why might a monetary policy maker want to prick an asset price bubble before it gets out of hand? What are some
barriers to doing this? 6. What are some types of macroprudential regulation? 883 Chapter 36 Financial Factors in a Macro Model Standard macroeconomic models suitable for short and medium run analysis mostly abstract from issues related to ﬁnance and banking. For example, in motivating the underlying decision rules of both the neoclassical and New Keynesian models, we assumed that ﬁrms had to ﬁnance investment via borrowing from a ﬁnancial intermediary, but this intermediary played an extremely passive role. Further, the Modigiliani-Miller theorem held and we could have alternatively dispensed with ﬁnancial intermediation altogether without altering the equilibrium decision rules. Especially in the wake of the Financial Crisis / Great Recession of 2007-2009, macroeconomists (and macroeconomic models) have been heavily criticized for failing to incorporate frictions related to ﬁnancial intermediation in a compelling way. While recent history has certainly proven this criticism valid, it is easy to understand why macroeconomic models typically do not feature particularly sophisticated ﬁnancial structures. For ﬁnancial intermediation to be valuable, there must be non-trivial heterogeneity in place, and it is diﬃcult to model signiﬁcant heterogeneity in a tractable way. Financial intermediation is a form of indirect ﬁnance, as discussed in Chapter 31. Households do not wish to directly ﬁnance the operations of ﬁrms because there are informational frictions (adverse selection and moral hazard). But for there to be informational frictions, there must be diﬀerent types of ﬁrms, so there must be some heterogeneity. While it is conceptually straightforward to think about how such heterogeneity impacts individual decision rules, it is not so straightforward to incorporate it into a general equilibrium description of an economy as a whole. In spite of these challenges, we wish to return to our core model used to study medium and short run ﬂuctuations, yet do so in a way that can speak to ﬁnancial frictions and shocks. To that end, we will incorporate a credit spread variable into the analysis. In particular, we will assume that the (real) interest rate relevant for the representative household is rt, while the real interest rate relevant for the representative ﬁrm is rt + ft, where ft is the spread. Based on our work in Chapter 34, we can alternatively
think of ft as a term or risk premium (or both), though for the most part we will treat it as exogenous. We can think about a ﬁnancial crisis as a situation in which ft rises markedly. In a crisis (see e.g. Chapter 33), there is high demand for short term, liquid securities (like cash). This makes it diﬃcult 884 for ﬁnancial intermediaries to lend to ﬁrms, resulting in increasing interest costs to ﬁrms of obtaining credit. This results in a collapse in investment and aggregate demand, and graphically serves as an additional shock to the IS curve. We can also consider (partially) endogenizing the credit spread variable. We refer to this as the “ﬁnancial accelerator” model after Bernanke, Gertler, and Gilchrist (1999). In this framework, credit spreads endogenously move countercyclically with output. The logic behind this is simple: if default risk is highest when output is low, it stands to reason that credit spreads are high when output is low. This ﬁnancial accelerator mechanism has the eﬀect of amplifying the output eﬀects of both demand and supply shocks. 36.1 Incorporating an Exogenous Credit Spread We return to the model used to study short and medium run ﬂuctuations but add one simple twist. In particular, recall the set of equations underlying the Neoclassical model (Chapter 18) or the New Keynesian model (Chapter 26). The equilibrium consists of decision rules for consumption, labor supply, money demand, and investment demand; an aggregate resource constraint and the Fisher relationship; and an equation characterizing aggregate supply as a function of the nominal price of goods. The eight equations characterizing the equilibrium are shown below in (36.1)-(36.8). Ct = C d(Yt − Gt, Yt+1 − Gt+1, rt) Nt = N s(wt, θt) Pt = ¯Pt + γ(Yt − Y f t ) It = I d(rt + ft, At+1, Kt) Yt = AtF (Kt, Nt) Yt = Ct + It + Gt Mt = PtM d(rt + πe t+1, Yt) rt = it − πe t+
1 (36.1) (36.2) (36.3) (36.4) (36.5) (36.6) (36.7) (36.8) The most general form of these equations is presented, with (36.3) being the aggregate supply (AS) curve. It summarizes a couple of special cases. When γ → ∞, we must have Yt = Y f t and the model collapses to the Neoclassical Model. When γ = 0, in contrast, we have the simple sticky price model. For values of γ lying in between we have the partial sticky price model. The endogenous variables are Yt, Ct, It, Nt, wt, rt, it, and Pt. Exogenous variables 885 are Gt, Gt+1, At, At+1, θt, ¯Pt, and πe it is taken to be exogenous. t+1. The new variable is the credit spread variable, ft, and The only diﬀerence relative to earlier presentations of the model is the inclusion of an additional exogenous variable, ft. It appears in the investment demand equation, (36.4). In particular, we assume that the (real) interest rate relevant for investment decisions is rt + ft. We can think about rt as the risk-free short run interest rate and ft as the premium the ﬁrm pays over this to borrow funds for investment. As in Chapter 34, this premium can be thought of either as a risk or as a term premium, or a risk and term premium rolled together into one. Because of the simpliﬁed nature of the two period model, it is diﬃcult to formally motivate it as such, however. With only two periods, there would be no term premium. And with a representative ﬁrm that never defaults in equilibrium, there would be no risk premium either. To formally motivate a variable like ft, we would need to extend the model beyond two periods and allow for heterogeneity on the ﬁrm side, both of which would involve serious complications. Instead, we will simply treat ft as exogenous. We can come up with an empirical counterpart to ft by measuring the spread between the average yield on risky corporate debt (debt rated by Moody’s as Baa) relative to the yield on a Treasury of similar maturity (10 years). A time series
of this variable is plotted below in Figure 36.1. We observe that this credit spread appears quite countercyclical (i.e. it rises around times identiﬁed as recessions, most notably during the ﬁnancial crisis / Great Recession of 2007-2009). Whether this increase in credit spreads is a cause or a consequence of cyclical ﬂuctuations is not immediately obvious, and it may well be both. In fact, in what follows we will consider both possibilities. Figure 36.1: Empirical Measure of ft 886 myf.red/g/eTWe1234567198819901992199419961998200020022004200620082010201220142016fred.stlouisfed.orgSource: Federal Reserve Bank of St. LouisMoody's Seasoned Baa Corporate Bond Yield Relative to Yield on 10-Year Treasury Constant Maturity©Percent As long as ft is treated as completely exogenous, its inclusion does not fundamentally alter the graphical depiction of the equilibrium of the model. This is shown below in Figure 36.3 for the partial sticky price model. The inclusion of ft as an exogenous variable simply motivates an additional reason for the IS and hence AD curves to shift, as we shall see below. Figure 36.2: Equilibrium in the Partial Sticky Price Model with Financial Frictions 887 𝑤𝑤𝑡𝑡 𝑃𝑃𝑡𝑡 𝑌𝑌𝑡𝑡 𝑌𝑌𝑡𝑡 𝑌𝑌𝑡𝑡 𝑌𝑌𝑡𝑡 𝑌𝑌𝑡𝑡 𝑁𝑁𝑡𝑡 𝑁𝑁𝑡𝑡 𝐴𝐴𝐴𝐴 𝐼𝐼𝐴𝐴 𝑟𝑟0,𝑡𝑡 𝑌𝑌0,𝑡𝑡=𝑌𝑌0,𝑡𝑡𝑓𝑓 𝑁𝑁0,𝑡𝑡 𝑁𝑁𝑠𝑠(𝑤
𝑤𝑡𝑡,𝜃𝜃𝑡𝑡) 𝐿𝐿𝐿𝐿(𝐿𝐿𝑡𝑡,𝑃𝑃0,𝑡𝑡) 𝐴𝐴𝑡𝑡𝐹𝐹(𝐾𝐾𝑡𝑡,𝑁𝑁𝑡𝑡) 𝑌𝑌𝑡𝑡=𝑌𝑌𝑡𝑡 𝑟𝑟𝑡𝑡 𝐴𝐴𝐴𝐴 𝑃𝑃0,𝑡𝑡=𝑃𝑃�0,𝑡𝑡 𝑤𝑤0,𝑡𝑡 𝑁𝑁𝑑𝑑(𝑤𝑤𝑡𝑡,𝐴𝐴𝑡𝑡,𝐾𝐾𝑡𝑡) 𝐴𝐴𝐴𝐴𝑓𝑓 36.2 Detailed Foundations Although we will think of ft as exogenous for the reasons enumerated above, we can show how to formally include it in a mathematical derivation of the model’s decision rules. This is similar to what is presented in Chapter 12. The model continues to consist of a representative household, a representative ﬁrm, a government, and a ﬁnancial intermediary (or bank). As in Chapter 12, it plays a fairly passive role, but diﬀerently than our earlier presentation, it can earn a proﬁt which is remitted back to the household. In period t, it takes in savings, St, from the household, and lends that savings to the ﬁrm for investment and to the government for debt issuance. In period t + 1, the ﬁrm pays back principal plus interest of rt + ft. The government pays back principal plus interest of rt. The ﬁnancial intermediary returns any proﬁt to the household in the form of a dividend. We do not formally model a decision rule
for the ﬁnancial intermediary, instead simply taking ft as given. In period t, the ﬁnancial intermediary earns nothing – it simply takes in deposits, St, t, and the government in t and has interest expense of from the household and lends them to the ﬁrm in the amount, BI the amount BG rtSt. Its proﬁt is: t. In t + 1, it earns revenues of (rt + ft)BI + rtBG t DI t+1 = (rt + ft)BI t + rtBG t − rtSt (36.9) The government chooses an exogenous stream of spending, Gt and Gt+1. Spending is ﬁnanced via a mix of taxes, Tt, and debt, Bt. The government’s ﬂow budget constraints are: Gt ≤ Tt + BG t Gt + rtBG t ≤ Tt+1 + BG t+1 − BG t (36.10) (36.11) The government borrows from the ﬁnancial intermediary at the same interest rate the household earns on saving, rt. In other words, there is no interest spread facing the government. The household side of the model is identical to what was presented earlier. The household earns income from working, pays lump sum taxes to the government, and also earns dividend income from ownership in the production ﬁrm as well as in the ﬁnancial intermediary. Its ﬂow budget constraints are: Ct + St ≤ wtNt + Dt − Tt Ct+1 + St+1 ≤ wt+1Nt+1 − Tt+1 + Dt+1 + DI t+1 + (1 + rt)St (36.12) (36.13) 888 Its objective is to pick a consumption-saving and labor supply plan to maximize: U = u(Ct, 1 − Nt) + βu(Ct+1, 1 − Nt+1) Optimization gives rise to standard ﬁrst order conditions: uC(Ct, 1 − Nt) = β(1 + rt)uC(Ct+1, 1 − Nt) uL(Ct, 1 − Nt) = wtuC(Ct, 1 − Nt) uL(Ct+1, 1 − Nt+1)
= wt+1uC(Ct+1, 1 − Nt+1) (36.14) (36.15) (36.16) (36.17) Along with the fact that the household only cares about the presented discounted value of government spending, not the timing and sequence of taxes, these can be used to motivate the consumption and labor supply functions presented above, (36.1)-(36.2). The ﬁrm produces output according to the production function Yt = AtF (Kt, Nt), where At is exogenous and the ﬁrm is initially endowed with an exogenous amount of capital, Kt. New capital can be accumulated via investment, Kt+1 = It + (1 − δ)Kt. This investment must t, at the eﬀective real interest rate rt + ft. The ﬁrm’s objective be ﬁnanced via borrowing, BI is to maximize its value, where Vt = Dt + Dt+1. If the price level is ﬂexible, the ﬁrm can choose 1+rt both It and Nt. Otherwise, the ﬁrm chooses It and picks Nt to satisfy demand at its price. For expositional purposes, we write the ﬁrm’s problem where it can choose both Nt and It: max Nt,It Vt = AtF (Kt, Nt) − wtNt+ 1 1 + rt [At+1F (Kt+1, Nt+1) + (1 − δ)Kt+1) − wt+1Nt+1 − (1 + rt + ft)BI t ] (36.18) In (36.18), the (1 − δ)Kt+1 term is the liquidation value of the ﬁrm – any remaining leftover t is the ﬁrm’s = It = Kt+1 − (1 − δ)Kt, we can capital is returned to the household in the period t + 1 dividend. (1 + rt + ft)BI ﬁnancing cost which must be paid back in t + 1. Since BI t re-cast the ﬁrm’s problem as one of choosing Kt+1 instead of It: Vt = AtF (K
t, Nt) − wtNt+ max Nt,Kt+1 1 1 + rt [At+1F (Kt+1, Nt+1) + (1 − δ)Kt+1) − wt+1Nt+1 − (1 + rt + ft)(Kt+1 − (1 − δ)Kt)] (36.19) The ﬁrst order conditions are: 889 ∂Vt ∂Nt = 0 ⇔ AtFN (Kt, Nt) = wt ∂Vt ∂Kt+1 = 0 ⇔ rt + ft + δ = At+1FK(Kt+1, Nt+1) (36.20) (36.21) (36.20) is identical to what we encountered earlier, and (36.21) is the same with the addition of ft on the left hand side. (36.21) implicitly deﬁnes an optimal Kt+1 as a function of rt + ft, where the optimal Kt+1 is decreasing in rt + ft. This means that investment is decreasing in rt + ft, which motivates the decision rule presented above, (36.4). Market-clearing requires that St = It + BG t – i.e. household saving equal ﬁrm investment plus government borrowing, Dt = Yt − wtNt, and BG = Gt − Tt. Combing these together with t (36.12) yields the standard resource constraint of Yt = Ct + It + Gt. In period t + 1 no actors die with positive or negative stocks of assets, so St+1 = BG = 0. Investment is the liquidation t+1 value of any remaining capital, It+1 = (1 − δ)Kt+1. (36.13) then implies Yt+1 = Ct+1 + It+1 + Gt+1. Referencing back to our work in Chapter 15, note that even in the absence of price stickiness (i.e. γ → ∞), the equilibrium allocations will in general not coincide with what a benevolent social planner would choose unless ft = 0. To see this, not that one of the ﬁrst order conditions for a hypothetical planner’s problem (see Chapter 15 for details)
would be: uC(Ct, 1 − Nt) = βuC(Ct+1, 1 − Nt+1)(At+1FK(Kt+1, Nt+1 + (1 − δ)) (36.22) Combining (36.21) with (36.15), we would obtain: uC(Ct, 1 − Nt) = βuC(Ct+1, 1 − Nt+1)(At+1FK(Kt+1, Nt+1 + (1 − δ) − ft) (36.23) (36.22) and (36.23) are only the same in the event that ft = 0. If ft > 0, then the equilibrium allocation will in general feature consumption that is too high, and investment that is too low, relative to what the planner would prefer.1 Given our assumptions on labor supply, however, the equilibrium level of output will coincide with the eﬃcient level. For the purposes of thinking about optimal monetary policy when prices are sticky, we will ignore the fact that ft ≠ 0 means that the neoclassical equilibrium is not eﬃcient. Formally, we can motivate this assumption by thinking that the average level of ft is small (which it is in the data, see Figure 36.3), and hence we can safely ignore the eﬀect of ft ≠ 0 on thinking about the eﬃcient level of output. 1To see this, not that the bigger ft is, the smaller uC(Ct, 1 − Nt) must be for (36.23) to hold, so Ct will be bigger. 890 36.3 Equilibrium Eﬀects of an Increase in the Credit Spread We are now in a position to analyze the economic eﬀects of a change in the exogenous credit spread variable, ft. Suppose that this increases. The eﬀects of this are shown in Figure 36.3. This has the eﬀect of shifting the IS curve in to the left – for a given rt, the cost of investment is higher, and hence there will be a reduction in investment. This results in the AD curve shifting in to the left. To the extent to which the AS curve is non-horizontal, the price level falls. The fall in the price level causes the LM curve to shift out to the
right somewhat. In equilibrium, rt is lower and Yt is lower. Lower Yt must be met by lower Nt given no change in At or Kt. This necessitates a reduction in the real wage. 891 Figure 36.3: Increase in Credit Spread What happens to consumption and investment in equilibrium? For investment, on the one hand rt is lower but on the other hand ft is higher. We can nevertheless conclude that It must be lower in equilibrium. Suppose that γ → ∞, so that the AS curve is vertical. In this case, rt would fall, but Yt would be unchanged. rt being lower with Yt unaﬀected would mean Ct is higher, but higher Ct with no change in Yt necessitates a lower It. Hence, even if 892 𝑤𝑤𝑡𝑡 𝑃𝑃𝑡𝑡 𝑌𝑌𝑡𝑡 𝑌𝑌𝑡𝑡 𝑌𝑌𝑡𝑡 𝑌𝑌𝑡𝑡 𝑌𝑌𝑡𝑡 𝑁𝑁𝑡𝑡 𝑁𝑁𝑡𝑡 𝐴𝐴𝐴𝐴 𝐼𝐼𝐴𝐴 𝑟𝑟0,𝑡𝑡 𝑌𝑌0,𝑡𝑡=𝑌𝑌0,𝑡𝑡𝑓𝑓 𝑁𝑁0,𝑡𝑡 𝑁𝑁𝑠𝑠(𝑤𝑤𝑡𝑡,𝜃𝜃𝑡𝑡) 𝐿𝐿𝐿𝐿(𝐿𝐿𝑡𝑡,𝑃𝑃0,𝑡𝑡) 𝐴𝐴𝑡𝑡𝐹𝐹(𝐾𝐾𝑡𝑡,𝑁𝑁𝑡𝑡) 𝑌�
��𝑡𝑡=𝑌𝑌𝑡𝑡 𝑟𝑟𝑡𝑡 𝐴𝐴𝐴𝐴 𝑃𝑃0,𝑡𝑡=𝑃𝑃�0,𝑡𝑡 𝑤𝑤0,𝑡𝑡 𝑁𝑁𝑑𝑑(𝑤𝑤𝑡𝑡,𝐴𝐴𝑡𝑡,𝐾𝐾𝑡𝑡) 𝐴𝐴𝐴𝐴𝑓𝑓 𝐼𝐼𝐴𝐴′ 𝐴𝐴𝐴𝐴′ 𝐿𝐿𝐿𝐿(𝐿𝐿𝑡𝑡,𝑃𝑃1,𝑡𝑡) 𝑟𝑟1,𝑡𝑡 𝑃𝑃1,𝑡𝑡 𝑤𝑤1,𝑡𝑡 𝑁𝑁1,𝑡𝑡 𝑌𝑌1,𝑡𝑡 Original equilibrium Direct effect of increase in credit spread Indirect effect of price change Hypothetical flexible price 0 subscript: original equilibrium 1 subscript: new equilibrium f superscript: hypothetical flexible price prices were ﬂexible, It would fall in equilibrium. For the more general case in which prices are sticky (either partially or completely), rt will fall by less than it would in the ﬂexible price case. Hence, we can conclude that It must fall whenever ft increases. It is not possible to deﬁnitively sign the eﬀect of an increase in ft on Ct. On the one hand, rt is lower, which would work to raise Ct. On the other hand, Yt is also lower, which would work to lower Ct. Which eﬀect dominates is unclear. In the case where prices are ﬂexible (see immediately above), Yt is unaﬀected so we can be sure
that Ct rises when ft increases. When prices are sticky and Yt falls, however, we cannot be sure. It is plausible that Ct is far more sensitive to Yt than to rt, so it seems a safe bet that if prices are at all sticky Ct would decline when ft increases. Large increases in credit spreads seem to be a recurrent feature of ﬁnancial crises, as we discuss further in Chapter 37. In particular, we can think about ﬁnancial crises as “runs” in which there is a ﬂight from perceived risky debt (i.e. loans to the ﬁrm in the context of this model) into safer forms of debt (reserves for banks, checking accounts or currency for households). This ﬂight to safety triggers an increase in the credit spread, which leads to a reduction in investment and economic activity more generally, as is shown in Figure 36.3. 36.4 The Financial Accelerator Figure 36.1 shows that credit spreads (diﬀerences in yields between risky and riskless debt of the same maturity) are quite countercyclical. Immediately above we considered exogenous increases in credit spreads as a source of ﬂuctuations, and argued that large increases in spreads, such as that observed during the Great Recession, are likely to emerge during periods of ﬁnancial crisis. There are, however, smaller ﬂuctuations in credit spreads that also appear countercyclical. For example, in Figure 36.1 credit spreads rose near the end of both the 1990 and 2001 recessions, albeit not be nearly as much as spreads rose during the Great Recession. It seems reasonable to think that there may be an endogenous component to credit spreads that accounts for these smaller cyclical patterns. This is reasonable because it seems likely that ﬁrm default risk is highest when the economy is weakest; to compensate for heightened default risk, ﬁnancial intermediaries require a higher spread to lend to production ﬁrms. This endogenous component to credit spreads can amplify cyclical ﬂuctuations. It is called the “ﬁnancial accelerator” in, for example, Bernanke, Gertler, and Gilchrist (1999). A worsening economy (lower Yt) results in higher credit spreads (higher ft), which further worsens the economy. Formally, suppose that the credit spread variable has both an endogenous and an exogenous 893 component as follows:
ft = ¯ft − aYt (36.24) In (36.24), ¯ft is the exogenous component of the credit spread variable. Changes in it have eﬀects like those summarized in Figure 36.3. −aYt is the endogenous component, with a ≥ 0. The parameter a measures the strength of the ﬁnancial accelerator mechanism. The bigger it is, the more sensitive the credit spread variable is to overall economic conditions. We can think about the full description of the model (showing the partial sticky price version, which is the most general version) as the same as (36.1)-(36.8) above, with an additional endogenous variable, ft, and an additional equation, (36.24). The inclusion of a ﬁnancial accelerator mechanism will have two eﬀects on the model. The ﬁrst will be to make the IS and hence the AD curve both ﬂatter. The second will be that the IS and hence the AD curves will shift more in response to exogenous shocks than they would without the ﬁnancial accelerator. To see this formally, begin by deﬁning total desired expenditure, Y d t, as the sum of the consumption function, (36.1); the investment demand function, (36.4); and government spending: Y d t = C d(Yt − Gt, Yt+1 − Gt+1, rt) + I d(rt + ¯ft − aYt, At+1, Kt) + Gt (36.25) Autonomous expenditure is (36.25) evaluated when Yt = 0: E0 = C d(−Gt, Yt+1 − Gt+1, rt) + I d(rt + ¯ft, At+1, Kt) + Gt We once again assume that autonomous expenditure is positive, E0 > 0. Total desired expenditure is a function of total income. Diﬀerentiating (36.25) with respect to Yt, we get: (36.26) ∂Y d t ∂Yt = ∂C d(⋅) ∂Yt − a ∂I d(⋅) ∂rt (36.27) (⋅) In (36.27), ∂Cd is the derivative of the consumption
function with respect to its ﬁrst ∂Yt arugment (net income), while ∂I d is the derivative of the investment demand function with ∂rt respect to its ﬁrst argument (the interest rate relevant for the ﬁrm, so rt + ft = rt + ¯ft − aYt). The −a term shows up multiplying this derivative because −a is the derivative of rt + ¯ft − aYt with respect to Yt. If we once again call the partial of the consumption function with respect to current net income the MPC and treat it as a constant, we have: (⋅) ∂Y d t ∂Yt = M P C − a ∂I d(⋅) ∂rt 894 (36.28) (⋅) Note that since ∂I d ∂rt < 0, the derivative of desired expenditure with respect to current income in (36.28) is bigger than if a = 0 (i.e. no ﬁnancial accelerator mechanism). We can graphically see how this impacts the derivation of the IS curve using the “Keynesian Cross” diagram shown below in Figure 36.4. To make things as clear as possible, we draw in two expenditure lines which cross the 45 degree at the same level of Yt for an initial interest rate of r0,t. The one without a ﬁnancial accelerator is shown in black, whereas the one with the accelerator is steeper and shown in red. We then consider a reduction in the real interest rate from r0,t to r1,t. This causes both expenditure lines to shift up in an equal amount (assuming the same interest sensitivity of investment demand), shown in blue and orange, respectively. When the expenditure line is steeper, the resulting change in Yt for a given rt is bigger. Hence, the presence of a ﬁnancial accelerator mechanism makes the IS curve ﬂatter. Figure 36.4: The IS Curve with the Financial Accelerator We can also see this algebraically. The IS curve is implicitly deﬁned by taking (36.25) and equating Yt = Y d t : Yt = C d(Yt − Gt, Yt+1 − Gt+1, rt) + I d(rt + ¯ft − aYt,
At+1, Kt) + Gt (36.29) Total diﬀerentiate (36.29) about a point: 895 𝑌𝑌𝑡𝑡𝑑𝑑=𝐶𝐶𝑑𝑑�𝑌𝑌𝑡𝑡−𝐺𝐺𝑡𝑡,𝑌𝑌𝑡𝑡+1−𝐺𝐺𝑡𝑡+1,𝑟𝑟0,𝑡𝑡�+𝐼𝐼𝑑𝑑�𝑟𝑟0,𝑡𝑡+𝑓𝑓̅𝑡𝑡,𝐴𝐴𝑡𝑡+1,𝐾𝐾𝑡𝑡�+𝐺𝐺𝑡𝑡 𝑌𝑌𝑡𝑡𝑑𝑑=𝑌𝑌𝑡𝑡 𝑌𝑌𝑡𝑡𝑑𝑑 𝑌𝑌𝑡𝑡 𝑌𝑌𝑡𝑡 𝑟𝑟𝑡𝑡 𝑟𝑟0,𝑡𝑡 𝑟𝑟1,𝑡𝑡 𝐼𝐼𝐼𝐼 𝑌𝑌𝑡𝑡𝑑𝑑=𝐶𝐶𝑑𝑑�𝑌𝑌𝑡𝑡−𝐺𝐺𝑡𝑡,𝑌𝑌𝑡𝑡+1−𝐺𝐺𝑡𝑡+1,𝑟𝑟0,𝑡𝑡�+𝐼𝐼𝑑𝑑�𝑟𝑟0,𝑡𝑡+𝑓𝑓̅𝑡𝑡−𝑎𝑎�
�𝑌𝑡𝑡,𝐴𝐴𝑡𝑡+1,𝐾𝐾𝑡𝑡�+𝐺𝐺𝑡𝑡 𝑌𝑌𝑡𝑡𝑑𝑑=𝐶𝐶𝑑𝑑�𝑌𝑌𝑡𝑡−𝐺𝐺𝑡𝑡,𝑌𝑌𝑡𝑡+1−𝐺𝐺𝑡𝑡+1,𝑟𝑟0,𝑡𝑡�+𝐼𝐼𝑑𝑑�𝑟𝑟1,𝑡𝑡+𝑓𝑓̅𝑡𝑡,𝐴𝐴𝑡𝑡+1,𝐾𝐾𝑡𝑡�+𝐺𝐺𝑡𝑡 𝐼𝐼𝐼𝐼 Standard model Financial accelerator 𝑌𝑌𝑡𝑡𝑑𝑑=𝐶𝐶𝑑𝑑�𝑌𝑌𝑡𝑡−𝐺𝐺𝑡𝑡,𝑌𝑌𝑡𝑡+1−𝐺𝐺𝑡𝑡+1,𝑟𝑟0,𝑡𝑡�+𝐼𝐼𝑑𝑑�𝑟𝑟1,𝑡𝑡+𝑓𝑓̅𝑡𝑡−𝑎𝑎𝑌𝑌𝑡𝑡,𝐴𝐴𝑡𝑡+1,𝐾𝐾𝑡𝑡�+𝐺𝐺𝑡𝑡 dYt = ∂C d(⋅) ∂Yt (dYt − dGt) + ∂C d(⋅) ∂Yt+1 (dY
t+1 − dGt+1) + ∂C d(⋅) ∂rt ∂I d ∂At+1 drt + ∂I d ∂rt dAt+1 + ∂I d ∂Kt (drt + d ¯ft − adYt)+ Treating all exogenous variables as ﬁxed, (36.30) reduces to: dYt = ∂C d(⋅) ∂Yt Solving for dYt/drt, we obtain: dYt + ∂C d(⋅) ∂rt drt + ∂I d ∂rt (drt − adYt) dYt drt = (⋅) (⋅) ∂Cd ∂rt 1 − ∂Cd(⋅) ∂Yt + ∂I d ∂rt + a ∂I d(⋅) ∂rt dKt + dGt (36.30) (36.31) (36.32) Since both partial derivatives in the numerator are negative, the overall slope is negative – i.e. the IS curve is downward-sloping. Since ∂I d < 0, if a > 0 then the denominator is ∂rt smaller the bigger is a, so the overall slope is more negative, and, in a graph with Yt on the horizontal axis, consequently ﬂatter.2 (⋅) A ﬂatter IS curve results in a ﬂatter AD curve as well. We can see this graphically in Figure 36.5 below. A given shift of the LM curve due to a change in the price level results in a larger change in Yt (and a smaller change in rt) the ﬂatter is the IS curve. 2Note that we are making an additional assumption. This is that a ∂I d ∂rt (⋅) denominator in (36.32) negative, which would make the IS curve upward-sloping. is not so negative to make the 896 Figure 36.5: The AD Curve with the Financial Accelerator Not only does the presence of a ﬁnancial accelerator mechanism aﬀect the shape of the IS and AD curves, it also impacts how they shift in response to an exogenous shock. We can see
this graphically in Figure 36.6. This ﬁgure considers an exogenous increase in Gt which shifts the expenditure line up by a given amount regardless of whether there is a ﬁnancial accelerator mechanism or not. Holding the real interest rate ﬁxed, a given upward-shift in the expenditure line results in a larger horizontal shift to the right in the IS curve (and hence also in the AD curve). We draw the ﬁgure where we think of this shift as resulting from an increase in Gt, but qualitatively we would get the same picture if we considered a reduction in Gt+1, an increase in At+1, or a reduction in ¯ft. 897 𝑌𝑌𝑡𝑡 𝑟𝑟𝑡𝑡 𝑌𝑌𝑡𝑡 𝑃𝑃𝑡𝑡 𝐼𝐼𝐼𝐼 𝐿𝐿𝐿𝐿(𝑃𝑃0,𝑡𝑡) 𝐿𝐿𝐿𝐿(𝑃𝑃1,𝑡𝑡) 𝐿𝐿𝐿𝐿(𝑃𝑃2,𝑡𝑡) 𝑃𝑃0,𝑡𝑡 𝑃𝑃1,𝑡𝑡 𝑃𝑃2,𝑡𝑡 𝐴𝐴𝐴𝐴 𝐼𝐼𝐼𝐼 𝐴𝐴𝐴𝐴 No accelerator Financial accelerator Figure 36.6: Shift of the IS Curve with the Financial Accelerator We can use the familiar IS-LM-AD-AS curves to summarize the equilibrium of the model. This is done so in Figure 36.7. We show two IS curves and two AD curves, one corresponding to the case of the ﬁnancial accelerator mechanism being present (i.e. a > 0, shown in orange), the other not (shown in black). We draw in the curves where the equilibrium levels of output, the price level, and the real interest rate are nevertheless the same initially.
We consider the partial sticky price model, with the neoclassical model (vertical AS curve) and simple sticky price model (horizontal AS curve) being special cases. 898 𝑌𝑌𝑡𝑡𝑑𝑑=𝐶𝐶𝑑𝑑�𝑌𝑌𝑡𝑡−𝐺𝐺0,𝑡𝑡,𝑌𝑌𝑡𝑡+1−𝐺𝐺𝑡𝑡+1,𝑟𝑟0,𝑡𝑡�+𝐼𝐼𝑑𝑑�𝑟𝑟0,𝑡𝑡+𝑓𝑓̅𝑡𝑡,𝐴𝐴𝑡𝑡+1,𝐾𝐾𝑡𝑡�+𝐺𝐺𝑡𝑡 𝑌𝑌𝑡𝑡𝑑𝑑=𝑌𝑌𝑡𝑡 𝑌𝑌𝑡𝑡𝑑𝑑 𝑌𝑌𝑡𝑡 𝑌𝑌𝑡𝑡 𝑟𝑟𝑡𝑡 𝑟𝑟0,𝑡𝑡 𝐼𝐼𝐼𝐼 𝑌𝑌𝑡𝑡𝑑𝑑=𝐶𝐶𝑑𝑑�𝑌𝑌𝑡𝑡−𝐺𝐺0,𝑡𝑡,𝑌𝑌𝑡𝑡+1−𝐺𝐺𝑡𝑡+1,𝑟𝑟0,𝑡𝑡�+𝐼𝐼𝑑𝑑�𝑟𝑟0,𝑡𝑡+𝑓𝑓̅𝑡𝑡−𝑎𝑎𝑌𝑌𝑡
𝑡,𝐴𝐴𝑡𝑡+1,𝐾𝐾𝑡𝑡�+𝐺𝐺𝑡𝑡 𝑌𝑌𝑡𝑡𝑑𝑑=𝐶𝐶𝑑𝑑�𝑌𝑌𝑡𝑡−𝐺𝐺1,𝑡𝑡,𝑌𝑌𝑡𝑡+1−𝐺𝐺𝑡𝑡+1,𝑟𝑟0,𝑡𝑡�+𝐼𝐼𝑑𝑑�𝑟𝑟0,𝑡𝑡+𝑓𝑓̅𝑡𝑡,𝐴𝐴𝑡𝑡+1,𝐾𝐾𝑡𝑡�+𝐺𝐺𝑡𝑡 𝐼𝐼𝐼𝐼 Standard model Financial accelerator 𝑌𝑌𝑡𝑡𝑑𝑑=𝐶𝐶𝑑𝑑�𝑌𝑌𝑡𝑡−𝐺𝐺1,𝑡𝑡,𝑌𝑌𝑡𝑡+1−𝐺𝐺𝑡𝑡+1,𝑟𝑟0,𝑡𝑡�+𝐼𝐼𝑑𝑑�𝑟𝑟0,𝑡𝑡+𝑓𝑓̅𝑡𝑡−𝑎𝑎𝑌𝑌𝑡𝑡,𝐴𝐴𝑡𝑡+1,𝐾𝐾𝑡𝑡�+𝐺𝐺𝑡𝑡 𝐼𝐼𝐼𝐼′ 𝐼𝐼𝐼𝐼′ Figure 36.7: IS-LM-AD-AS Curves with Financial Accelerator We are
now in a position to analyze how the ﬁnancial accelerator impacts the eﬀects of shocks. Consider ﬁrst a positive supply shock (e.g. an increase in At or a reduction in θt). This causes the AS curve to shift right. The ﬂatter is the AD curve, the more output reacts (and the less the price level and the real interest rate react). This can be seen below. We can conclude that the ﬁnancial accelerator ampliﬁes the output eﬀects of supply shocks. 899 𝑌𝑌𝑡𝑡 𝑟𝑟𝑡𝑡 𝑌𝑌𝑡𝑡 𝑃𝑃𝑡𝑡 𝐼𝐼𝐼𝐼 𝐿𝐿𝐿𝐿(𝑃𝑃0,𝑡𝑡) 𝑃𝑃0,𝑡𝑡 𝐴𝐴𝐴𝐴 𝐼𝐼𝐼𝐼 𝐴𝐴𝐴𝐴 No accelerator Financial accelerator 𝑟𝑟0,𝑡𝑡 𝑌𝑌0,𝑡𝑡 𝐴𝐴𝐼𝐼 Figure 36.8: Supply Shock with Financial Accelerator What about demand shocks? Consider a shock which causes the IS curve to shift to the right (e.g. an increase in Gt). To make the graph a bit more readable, consider the simple sticky price model, so that the AS curve is perfectly horizontal and we need not worry about secondary eﬀects on the position of the LM curve arising due to price level changes. As noted above, the presence of a ﬁnancial accelerator mechanism results in the IS and AD curves shifting horizontally by more whenever there is an exogenous shock to the IS curve. This results in bigger movements in equilibrium output compared to a situation in which there is no ﬁnancial accelerator mechanism. 900 𝑌𝑌𝑡𝑡 𝑟𝑟𝑡𝑡 𝑌𝑌�
�𝑡 𝑃𝑃𝑡𝑡 𝐼𝐼𝐼𝐼 𝐿𝐿𝐿𝐿(𝑃𝑃0,𝑡𝑡) 𝑃𝑃0,𝑡𝑡 𝐴𝐴𝐴𝐴 𝐼𝐼𝐼𝐼 𝐴𝐴𝐴𝐴 No accelerator Financial accelerator 𝑟𝑟0,𝑡𝑡 𝑌𝑌0,𝑡𝑡 𝐴𝐴𝐼𝐼 𝐴𝐴𝐼𝐼′ 𝐿𝐿𝐿𝐿(𝑃𝑃1,𝑡𝑡) 𝑌𝑌1,𝑡𝑡 𝑃𝑃1,𝑡𝑡 𝑟𝑟1,𝑡𝑡 𝐿𝐿𝐿𝐿(𝑃𝑃1,𝑡𝑡′) 𝑟𝑟1,𝑡𝑡′ 𝑃𝑃1,𝑡𝑡′ 𝑌𝑌1,𝑡𝑡′ 0 subscript: original equilibrium 1 subscript: new equilibrium ‘ superscript: new equilibrium w/ accelerator Figure 36.9: Demand Shock with Financial Accelerator, Simple Sticky Price Model In summary, then, the ﬁnancial accelerator mechanism ampliﬁes the output responses to both demand and supply shocks. In addition to helping account for the countercyclical behavior of empirical measures of credit spreads in the data, it can also help generate more output volatility in general, which is something quantitative models generally have diﬃculty in doing. 36.5 Summary • For ﬁnancial intermediation to be valuable there must be some friction or market incompleteness. Otherwise a ﬁrms ﬁnancial structure is irrelevant. • We add ﬁnancial frictions in the model by assuming that the rate ﬁrms borrow at exceeds the rate
at which households save. This is a credit spread and, while exogenous, can be motivated by some sort of risk or term premium. • If the credit spread is positive, the equilibrium allocations will not be Pareto eﬃcient. This is true even if prices are completely ﬂexible. • An increase in the credit spread shifts the IS curve to the left so output falls in equilibrium. Given that empirical measures of the credit spread seem to rise during 901 𝑌𝑌𝑡𝑡 𝑟𝑟𝑡𝑡 𝑌𝑌𝑡𝑡 𝑃𝑃𝑡𝑡 𝐼𝐼𝐼𝐼 𝐿𝐿𝐿𝐿(𝑃𝑃0,𝑡𝑡) 𝑃𝑃0,𝑡𝑡 𝐴𝐴𝐴𝐴 𝐼𝐼𝐼𝐼 𝐴𝐴𝐴𝐴 No accelerator Financial accelerator 𝑟𝑟0,𝑡𝑡 𝑌𝑌0,𝑡𝑡 𝐴𝐴𝐼𝐼 𝐼𝐼𝐼𝐼′ 𝐼𝐼𝐼𝐼′ 𝐴𝐴𝐴𝐴′ 𝐴𝐴𝐴𝐴′ 𝑌𝑌1,𝑡𝑡 𝑌𝑌1,𝑡𝑡′ 𝑟𝑟1,𝑡𝑡 𝑟𝑟1,𝑡𝑡′ 0 subscript: original equilibrium 1 subscript: new equilibrium ‘ superscript: new equilibrium w/ accelerator Blue line: post-shock Purple line: post-shock w/ acelerator recessions (and, in particular, ﬁnancial crises), ﬂuctuations in the credit spread are likely contributors to recessions. • Default is most likely when the economy is the weakest. Creditors realize this and charge higher interest rates. Those high interest rates in turn make default
more likely and extends a recession. This is the basic idea of the ﬁnancial accelerator. • The inclusion of the ﬁnancial accelerator ﬂattens the IS and AD curves and ampliﬁes both supply and demand shocks. Key Terms 1. Credit spread 2. Financial accelerator Questions for Review 1. Explain the ﬁnancial accelerator mechanism. 2. Will an increase in ft have a bigger eﬀect on output when γ is big or small? 3. Would you expect an increase in ft to have bigger eﬀects at or away from the ZLB? Exercises 1. Suppose there is an increase in ft. (a) Graphically depict this change in ft. (b) What is optimal monetary policy in this case? Show this graphically. How do all the endogenous variables change relative to Part 1a? (c) Suppose there is no change in monetary policy. How can the government change Gt to achieve the same level of output as in Part 1b? adjust such that there is no change in output? Graphically depict this. Hos do all of the other endogenous variables change? (d) Does the ﬁscal policy in Part 1c implement the Pareto optimal alloca- tions? Explain. 902 Chapter 37 Financial Crises and The Great Recession Financial crises and the ensuing economic recessions associated with them are a recurrent theme in modern, developed economies. In the last century, there have been two major ﬁnancial crises followed by deep recessions in the United States – the Great Depression from 1929-1933 and the Great Recession from 2007-2009. Many economists thought that such crises were a thing of the past prior to the Great Recession. The Great Recession proved these economists wrong. In this chapter, we brieﬂy outline the typical structure of ﬁnancial crises and stress the fundamental similarity between the Great Recession and the earlier Great Depression. We use the short run New Keynesian model as a laboratory to think about the Great Recession as well as the myriad non-standard policy interventions in its wake. The material in this chapter builds oﬀ of Chapters 26-29 and Chapters 31, 33, and 36, although much of it should be self-contained. Many books have and will be written about ﬁnancial crises more generally and about the recent Great Recession in particular. Our objective is not to provide a full, detailed account
of the Great Recession, nor is it to develop a full-ﬂedged critique of modern macroeconomics in light of the recession. Rather, we wish to give a brief overview of ﬁnancial crises more generally, a brief overview of the facts surrounding the Great Recession, and then we want to use the tools and models developed elsewhere in this book to think about the events surrounding the recession as well as the unconventional policy interventions in its wake. For the interested reader, a good book-length treatment of ﬁnancial crises with a focus on the Great Recession is found in Gorton (2012). Much of the material in this chapter follows this book. Some of the material is also closely related to Mishkin (2016). 37.1 Financial Crises: The Great Depression and Great Recession Financial crises occur with some regularity in market economies. Financial crises were common in the US throughout the 19th and early part of the 20th century, prior to the founding of the Federal Reserve and the centralization of the monetary and banking systems. The US has experienced two major ﬁnancial crises, both followed by deep recessions, in the last one hundred years. The Great Depression lasted from 1929-1933, with another milder 903 recession later in the 1930s. The US economy did not fully recover from this event until after World War II. The Great Recession happened in the US from 2007-2009. While economic growth resumed in the US in the latter half of 2009, the economy’s performance, unlike many other recent US recessions, was not so robust as to make up for the losses during the recession. Both the Great Depression and Great Recession were global in nature. Financial crises are associated with major disruptions between the ﬂow of funds from savers to borrowers. The tell-tale sign of a ﬁnancial crisis is an increase in credit spreads (i.e. the diﬀerence between the interest rates faced by ﬁrms for long term, illiquid projects and the interest rates on short term, liquid savings instruments like checking accounts and government bonds). In Chapter 36 we formally show how to incorporate an exogenous credit spread into an otherwise conventional macro model. An increase in credit spreads leads to a sharp reduction in demand and a potentially deep recession. The deep recession is often made deeper by monetary policy being hampered by the zero lower bound on interest rates. Further, as documented in Chapter 29, the economy’s inability to
correct itself to the medium run neoclassical equilibrium can be hampered by the ZLB, which makes the output contraction both deeper and more prolonged than it otherwise might be. In the paragraph above we focus on the macroeconomic nature of recessions due to ﬁnancial crises – increases in credit spreads lead to a contraction in aggregate demand which can be exacerbated by the ZLB. But what causes the increase in credit spreads? Typically, ﬁnancial crises are preceded by asset price booms and busts. Although popular in the ﬁnancial press, we hesitate to associate the term “bubble” with such boom and bust episodes, because what a “bubble” is in from an economists’ perspective is slightly diﬀerent than what the ﬁnancial press typically means, which is a period of excessive asset price appreciation followed by a large asset price decline. See the discussion on bubbles in Chapter 35. Figure 37.1 plots the real S&P 500 stock market index in the years leading up to and immediately after the Great Depression. The data for this and Figure 37.2 can be obtained from Robert Shiller’s website. The asset price boom and bust associated with the Great Depression involved the general stock market. In the three years leading up to the onset of the Great Depression, the S&P 500 index more than doubled. This is more than a 25 percent average appreciation per year, which is astounding given that the typical price appreciation over more than one hundred years of data is about 7 percent. Things came crashing down abruptly on “Black Tuesday,” October 29, 1929. There was a massive selloﬀ which can be seen in Figure 37.1, leading to depressed share prices. The decline in share prices continued for several more years. At the trough in 1932, the market as a whole was valued at less than 25 percent of its peak value in the fall of 1929. 904 Figure 37.1: S&P 500 Stock Market Index Around Great Depression The asset price boom and bust preceding the Great Recession was not the general stock market but rather the housing market. Figure 37.2 plots the behavior of a nationwide price index for houses in the United States in the years surrounding the Great Recession. In the years between roughly 2000 and the end of 2006, home prices in the US more than doubled. Home prices nationwide began to ﬂatline in 2006 and started to decline at the end of 2006
and through 2007, well before the economic contraction began. Home prices continued to decline long after the recession was oﬃcially over. Figure 37.2: Real Home Price Index Around Great Recession Our objective is not to completely understand why asset prices boomed and then busted in these two historical events, but it is useful to brieﬂy mention the leading explanations. During 905 501001502002503003504004502526272829303132333435Real S&P 500 Stock Market Index120130140150160170180190200200420052006200720082009201020112012Real Home Price Index the 1920s Federal Reserve credit was easy (i.e. interest rates were low), general euphoria about the stock market was high, and lending practices (e.g. buying stocks on margin, i.e. buying stocks with borrowed funds) were not well-regulated. During the 2000s, a variety of factors combined to depress interest rates (making the ﬁnancing of homes cheaper) and lending standards declined (i.e. mortgages were extended to borrowers who were previously thought to not be credit-worthy). Like the 1920s, a general sense of euphoria also set in where people believed that real estate was a path to sure riches. These factors combined to push up the demand for housing, and with supply limited house prices rose substantially throughout the country. Signiﬁcant declines in asset prices can aﬀect the economy through two primary channels. The ﬁrst is a wealth eﬀect channel. To the extent to which households own assets (e.g. stocks or houses), then declines in the values of these assets can reduce the present discounted value of lifetime income and reduce the demand for consumption. This eﬀect is undoubtedly present in the data (perhaps most especially for homes, where individuals can use equity accumulated in one’s home as collateral for loans to ﬁnance expenditure), although probably not particularly strong. The potentially bigger problem is that declines in asset prices can trigger banking panics and full-ﬂedged ﬁnancial crises. As discussed in Chapters 31 and 33, banks (or more generally bank-like institutions who engage in credit intermediation) fund longer term, illiquid investment projects with shorter term, liquid debt. For the Great Depression, the shorter term, liquid debt was made up mostly of demand deposits. When asset prices declined,
individual depositors became concerned about the potential solvency of their banks. This led them to rush, en masse, on their banks, demanding cash in exchange for their deposits. The banking system as a whole never has suﬃcient cash to meet withdrawal demands en masse, nor should it, in a sense, as we discuss in Chapter 33. The bank runs that plagued the economy in the Great Depression are memorialized in the bank run scene from the famous holiday movie It’s a Wonderful Life. To come up with cash, banks were forced to sell other assets. This led to kind of a spiraling eﬀect where depressed asset prices raised concerns about bank solvency, which forced more sales of assets in what is sometimes called a “ﬁre sale,” which further depressed asset prices. The end result is that banks and other institutions intermediating credit were forced into a situation of not making loans but instead trying to sell loans and other related assets. This led to a sharp reduction in available credit and a large increase in the interest rate spreads over safer US government debt. 906 Figure 37.3: Interest Rate Spread Around Great Depression Figure 37.3 plots the monthly time series of the average yield on Baa rated corporate debt relative to the yield on the ten year US Treasury Notes. This is an empirical measure of the interest rate spread series, ft, which we consider in the model to follow. As we can see in the ﬁgure, credit spreads rose from just above 2 percent prior to the Depression to more than 7 percent at its height. It is of interest to note that the observed increase in credit spreads did not occur until some two years after the onset of the Depression. This period coincided with the greatest wave of bank failures. The ﬁrst major bank runs did not begin until the fall of 1930, which is when we observe credit spreads ﬁrst increasing. The most vicious period of bank runs occurred between the second half of 1932 and into the ﬁrst part of 1933, which coincides with the period in which credit spreads were the highest. During the Great Recession, the decline in home prices led to a more broad-based ﬁnancial panic. We provide more detail in Section 37.2. Because of concerns about housing related mortgage backed securities, which many ﬁnancial institutions were heavily exposed to, interbank lending markets dried up. There was, in a sense, a classic banking panic
, but it wasn’t a run of depositors on commercial banks. Rather, during the Great Recession there was a run of institutions on other institutions. Short term funding dried up. Struggling to come up with liquidity, ﬁnancial institutions were forced to sell assets unrelated to the housing market. The prices of all ﬁnancial assets declined and interest rate spreads increased massively. The increase in interest rate spreads is documented in Figure 37.4, which is similar to Figure 37.3, but for the more recent period. The Baa-10 Yr Treasury spread went from less than 2 percent to more than 3 percent throughout 2007. It then skyrocketed to roughly 6 percent in the fall of 2008, when Lehman Brothers, a famous investment bank, failed. 907 123456782526272829303132333435Baa Spread over 10 Yr Treasury Figure 37.4: Interest Rate Spread Around Great Recession High credit spreads reduce aggregate demand. In terms of our model, which we will provide more detail on below, the increase in ft shifts the IS curve in to the left and results in the AD curve shifting in as well. In the short run output should fall, and may fall by more than it otherwise would if the ZLB binds. During both the Great Depression and Great Recession, economic activity contracted signiﬁcantly. In Figures 37.5 and 37.6, we plot the behavior of the Industrial Production Index in windows around both events. The IP index is available monthly and goes back further in time than does the GDP series we typically focus on, so it is good to use when studying the Depression. For point of comparison, we also plot the IP series during the Great Recession. In both episodes IP falls precipitously. It is worth nothing that the scales are diﬀerent in the two ﬁgures – the IP index is normalized to be 100 in 2012. Hence, the decline from roughly 104 to about 86 during the Great Recession represents a little more than a 15 percent decline in IP (which is similar to measures of GDP relative to a trend). During the Great Depression, the decline in the IP index from roughly 8 to 4 represents a 50 percent decline in economic activity. 908 1234567200420052006200720082009201020112012Baa Spread over 10 Yr Treasury Figure 37.5: Industrial Production Index Around Great Depression Figure 37.6: Industrial Production Index Around Great Recession In terms of lost
output, the Great Depression was signiﬁcantly worse than the Great Recession. By some measures, the unemployment rate during the Great Depression topped 25 percent, whereas it only maxed out at 10 percent in the most recent crisis. A consensus among economists is that the Great Depression was as bad as it was because of poor policies. Friedman and Schwartz (1971) argue forcefully that mistakes by the Federal Reserve (which was only in its second decade of existence when the Depression hit) signiﬁcantly worsened the Depression. The Fed did not, they argue, fully understand its role as lender of last resort. It allowed many banks to fail and the money supply to consequently contract with dire economic consequences. In contrast, during the more recent crisis, the Federal Reserve went 909 34567892526272829303132333435Industrial Production Index84889296100104108200420052006200720082009201020112012Industrial Production Index out of its way to try to provide stimulus to the economy. While it is still a matter of some debate, many economists (ourselves included) believe that the Great Recession was not as bad as it might have been because of the extraordinary policy actions taken by the Fed. We will discuss some of these extraordinary policy actions in more depth below. 37.2 The Great Recession: Some More Speciﬁcs on the Run In this section we provide a somewhat more detailed overview of the events leading up to the Financial Crisis of 2007-2009. This section is meant as a complement to the rest of this chapter. Our analysis follows the narrative from Mishkin (2011) and Gorton (2010b) pretty closely. The interested reader is referred to these works for more detail. Our central thesis follows from these authors and is that the ﬁnancial crisis was a “run” on the shadow banking system. This run resulted in widening credit spreads which had adverse macroeconomic consequences, particularly when combined with a binding zero lower bound on interest rates. The Financial Crisis had its origins in the housing market. As shown in Figure 37.2, home prices started to level oﬀ towards the end of 2005 and into 2006 and soon thereafter began to decline. Any successful narrative of the crisis must account for why declines in housing prices led to widespread ﬁnancial panic and runs on ﬁnancial institutions. While large, on its own the amount of outstanding mortgage related debt was not large enough to bring down
the entire ﬁnancial system. How the collapse in housing prices led to a widespread ﬁnancial panic relates back to several changes in the ﬁnancial system that had been ongoing for several years. Some of these are detailed in Section 31.4 of Chapter 31. A signiﬁcant fraction of credit intermediation had moved out of the traditional, regulated banking sector into the so-called shadow banking sector. Institutions like the now defunct investment banks (e.g. Bear Stearns and Lehman Brothers) essentially provided the funding for mortgage loans by buying mortgage backed securities. These institutions in turn funded themselves with short term loans from large institutional investors (like pension funds). These short term loans were often in the form of repurchase agreements (repos). In a repo, one party lends another money (often overnight, but never for a very long maturity) at a pre-negotiated interest rate. What makes this loan safe is the posting of collateral by the borrower – in the event that the borrower does not make good on its promised repayment, the lender gets to keep the collateral. The rise of large, institutional investors created a demand for “deposit-like” assets. These large institutional investors might have several hundred million dollars on which they would like to earn a safe interest rate before deciding what to do with it. Because deposit insurance will not cover deposits over $250,000, traditional bank deposits are not safe ways to earn 910 interest on vast sums of money. Short term repurchase agreements emerged to ﬁll this void – one party lends to another short term, and what makes the loan safe is the collateral that the borrower posts. This created a demand for relatively safe assets to serve as a collateral – the US government does not provide enough Treasury securities to meet the demand for relatively safe collateral that currently exists. Securitized mortgages came to be seen as a good form of collateral to make the short term funding from things like repo agreements viable. In other words, the demand for “deposit-like” assets was met with securitization of mortgages to serve as collateral to make these short term assets safe. For this reason, Gorton (2010a) refers to the shadow banking system as a system of “securitized banking.” As was done in Chapter 31, it will be helpful to illustrate these concepts through an example. Suppose that there is an investment bank, call it Bear Stearns,
which holds mortgage backed securities (MBS). It ﬁnances the purchase of these securities with short term funding, for example in the form of repurchase agreements. Suppose that the counterparty providing Bear with Repo is a large institutional investor, call it Fidelity. The balance sheet of Bear might look like: Table 37.1: T-Account for Bear Stearns Assets MBS $500 Other Securities: $100 Cash: $100 Liabilities plus Equity Repo: $500 Equity $200 In this example, which is slightly diﬀerent than what appears in Chapter 31, Bear holds $700 million in assets and has $500 million in liabilities, with $200 million in equity. $500 million of its assets are MBS, while another $100 million are held in other ﬁnancial securities (say, AAA rated corporate bonds) and $100 million is in cash. Bear ﬁnances itself with $500 million in repo, for which the $500 million in MBS serve as collateral. Bear is in essence borrowing (at the repo rate) to ﬁnance its holdings of MBS. If the MBSs and other securities yield more than Bear has to pay for the Repo, then Bear turns a proﬁt from this transaction – it is in essence borrowing low (at the repo rate) and lending high (at whatever the yield on the MBSs is). This is also a good deal for Bear’s counterparty, in this example Fidelity. Fidelity gets to earn some interest while it parks $500 million in cash. This “deposit” is safe so long as Bear does not fail and the underlying collateral does not lose value. What triggered the panic, and why did it extend beyond just mortgage-related debt products? How did a decline in house prices bring the entire ﬁnancial system to its knees? Mortgage backed securities are based on underlying actual mortgages. Some of these mortgages were to so-called “subprime” borrowers – borrowers either with poor credit histories, little 911 money down (i.e. little or no equity in their home), low incomes, or a combination of all these. The structure of many of these so-called subprime loans made the cash ﬂows from the underlying mortgages particularly sensitive to house prices. In essence, borrowers could get a loan at essentially no money down with a low interest rate, often called a “teaser rate
.” For example, suppose that you purchase a home for $500,000, and that the monthly mortgage payment on a conventional loan (e.g. a thirty year ﬁxed rate mortgage with a 20 percent downpayment) would be $2500. You can aﬀord up to a $2500 monthly payment, but you do not have the resources to make a downpayment. You would therefore not be able to purchase this home with a traditional mortgage. Would buying this house with a non-traditional mortgage make any sense? Suppose that you can get a no money down loan for a period of two years where the monthly payment is only $2500 a month, which you can aﬀord. The interest rate is scheduled to “reset” in two years which would raise the monthly payment to $3000, which you cannot aﬀord. If the house appreciates in value, to say $650,000 within the next two years, you can reﬁnance the loan. In a reﬁnance, you take out a new loan to pay oﬀ an existing loan. Your existing loan is valued at $500,000, so you need a loan for this amount. But if the home is now worth $650,000, the loan you are taking out is 75 percent of the value of the asset under consideration. This allows you to reﬁnance at an interest rate you can aﬀord, and you can keep your house payment at $2500 a month. In essence, you can use the accumulated equity in your home as a down payment on the reﬁnance deal. This all works out as planned so long as the home appreciates in value. But what if it doesn’t, or worse yet declines in value? Then you are in trouble. If the home does not appreciate in value, you cannot reﬁnance the loan at more favorable terms after two years – you have no equity built up in the home. This means that your monthly payment resets to $3000 a month, which you cannot aﬀord. You have no choice but to default on the home – i.e. quit making payments, at which point a bank seizes the property. If you quit making payments, whoever owns the mortgage loan will experience lower cash ﬂows than anticipated. In a traditional setting, the bank issuing the loan would have suﬀered this loss
. But with securitized banking, the mortgage issuer did not have a claim to the cash ﬂows from your mortgage, which had instead been pooled together with other mortgages into mortgage backed securities. This meant that whoever owned the MBS would suﬀer a loss. In the hypothetical example above, the owner of the MBS is the investment bank Bear Sterns. Bear Sterns and other ﬁnancial institutions were heavily exposed to MBS, the cash ﬂows of which were in turn quite sensitive to declines in home prices. The decline in home prices triggered a run because large institutional investors became worried that the MBS which were serving as collateral for repo agreements were not worth what they thought. This led to a drying up of short term funding for institutions like Bear Sterns, in a way conceptually 912 isomorphic to a mass withdrawal of deposits from a traditional bank. It did not matter that subprime mortgages were a small fraction of outstanding mortgage debt, or that only a small minority of mortgages ever actually went into delinquency (late on payments) or outright default (failure to make promised repayments). Large institutional investors knew that there were some “bad” mortgage loans out there, but were not sure where. As a consequence, they did not want to accept MBS as collateral, and short term funding for institutions like Bear dried up. Gorton (2010a) has likened this to an e-coli scare – even if you know that most beef does not have e-coli, because you are not sure where the e-coli is, you decide to stop purchasing all beef. In a similar way, even though investors knew that most mortgage-related debt was sound, they knew there was some bad debt out there, and decided to not accept any mortgage debt as collateral. As discussed in Chapter 31, an important feature of repurchase agreements is the haircut, which is deﬁned as the percentage diﬀerence between the amount of a loan and the required collateral. Fearing that their counterparties were at risk and that the underlying collateral was not valuable, short term funders (like Fidelity in the example) began demanding haircuts on repurchase agreements. Prior to the crisis, haircuts were zero. At the height of the crisis, haircuts rose to more than 40 percent. A 40 percent haircut would mean, for example, that Fidelity would only lend Bear $300 million in exchange for $500 million in
40 percent haircut). In Table 37.1, Bear only initially has $100 million in cash. Suppose that it can sell its other securities at their market value to raise the other $100 million. Its new balance sheet would look like: Table 37.2: T-Account for Bear Stearns Assets MBS $500 Other Securities: $0 Cash: $0 Liabilities plus Equity Repo: $300 Equity $200 In this situation, Bear is close to being in trouble. Any more withdrawal of funds by refusing to roll over short term funding, or only doing so with an even higher haircut, would lead Bear into failure. This is, in fact, what happened. As Bear (and other institutions) tried to come up with cash to meet short term funding shortfalls, they sold assets unrelated to mortgage backed securities (like AAA rated corporate debt). When many ﬁnancial institutions try to sell assets at the same time, the price of these assets declines. This creates a feedback eﬀect where declines in the price of these assets make these institutions look ever more vulnerable, which could lead to further pressures on short term funding and even more asset sales. As documented by Gorton (2010a), this resulted in an apparently perverse outcome – yields on AAA rated corporate debt were higher (prices were lower) than yields on AA rated corporate debt, as can be seen in Figure 37.8. This occurred because institutions, facing liquidity pressures, naturally tried to ﬁrst sell their “best” assets to raise cash. But as many institutions tried to do the same thing all at once, the price of these assets declined (and associated yields went up, as bond prices and yields move opposite one another, as discuessed in Chapter 34). 914 Figure 37.8: AA-AAA Spread In summary, things spread from housing related debt to a more general ﬁnancial crisis because of concerns about the backing collateral in short term debt agreements between ﬁnancial institutions. This led to a drying up of short term funding, which forced some institutions to try to sell assets to raise cash. This selling of assets further depressed asset prices (and increased yields) and put more liquidity pressures on the institutions. Financial institutions quit lending – both to one another as well as to consumers and businesses with legitimate needs for credit. As a consequence, credit spreads throughout the economy rose. Figure 37.9 plots the daily spread between the 3 month LIBOR (London Interbank O
declined and funding dried up. Financial institutions of all stripes were trying to sell assets and draw back on credit extension. As a result, the interest rates relevant for consumer and business loans increased markedly. Things began to stabilize in ﬁnancial markets as the calendar moved to 2009, and by the summer of that year the TED spread was lower than it had been prior to the crisis. 916 012345AprilJulyOctoberJanuaryAprilJulyOctoberJanuaryAprilJulyOctober200720082009TED SpreadBNP Paribashalts redemptionBear StearnsBailoutLehman FailureAIG RescueRun on Reserve Primary Fund To a large extent, we feel that this moderation in spreads was due to the extraordinary rescue interventions by the Fed and other government agencies. We discuss this below after employing our New Keynesian AD-AS model to think about the macroeconomic consequences of the ﬁnancial crisis. 37.3 Thinking About the Great Recession in the AD-AS Model Having provided some background detail, we now proceed to employ the New Keynesian IS-LM-AD-AS model, augmented to accounted for an exogenous credit spread as in Chapter 36, as a lens through which to think about the Great Recession. We do so with the obvious caveat that the model is an abstraction of a very complicated reality. Nevertheless, we feel that the model does a good job at making sense of what happened, and, given that, the model can be used to think about the unconventional policies tried by the Fed in the aftermath of the crisis. Roughly speaking, the Great Recession can be divided into three stages. The ﬁrst stage was the housing bust which began in late 2006 and continued throughout 2007. The second stage was the ﬁnancial crisis, which began in late 2007 and intensiﬁed throughout 2008. The third and most virulent stage is the further intensiﬁcation of the ﬁnancial crisis at the end of 2008 and into the ﬁrst half of 2009, exacerbated by the fact that the zero lower bound on interest rates had become binding by the end of 2008. As shown above in Figure 37.2, home prices in the US began to decline in 2006 and continued the decline throughout 2007. The direct macroeconomic consequences of the home price decline were not large. As discussed in Chapter 9, wealth more generally, and housing in particular, can be an argument in a household’s consumption function. A decline in house prices represents a reduction
in wealth, which, other factors being equal, ought to reduce consumption demand. In terms of our graphical model, this would result an inward shift of the IS curve and a resulting inward shift of the AD curve. This is documented in Figure 37.10 below. The ﬁgure focuses just on the IS-LM and AD-AS diagrams, and abstracts from diagrams related to the labor market. The inward shift of the AD curve resulted in a lower level of interest rates and a mild slowdown in output, although the slowdown in output was barely perceptible. 917 Figure 37.10: 2007 Decline in House Prices If the decline in house prices were all that happened, there may not have even been a recession and it certainly would not have been described as “Great” after the fact. The Great Recession entered a more pernicious stage in late 2007 and throughout 2008. As noted in the section above, declines in housing prices nationwide raised concerns of solvency risks for counterparties in interbank lending markets. This precipitated what amounted to a conventional bank run, although it was a run of institutions on other institutions and did not revolve around deposits. As a consequence of the liquidity crisis, ﬁnancial institutions were forced to sell assets to raise cash. Loan supply was signiﬁcantly reduced, and credit spreads increased. The increase in credit spreads, measured by the variable ft in our model, would result in a further inward shift of the IS and AD curves. This is documented in Figure 37.11 below. 918 𝐴𝐴𝐴𝐴 𝐴𝐴𝐴𝐴06 𝐴𝐴𝐴𝐴07 𝐼𝐼𝐴𝐴06 𝐼𝐼𝐴𝐴07 𝐿𝐿𝐿𝐿(𝐿𝐿06) 𝑟𝑟06 𝑟𝑟07 −𝜋𝜋𝑡𝑡+1𝑒𝑒 𝑟𝑟𝑡𝑡 𝑌𝑌𝑡𝑡 𝑌𝑌𝑡𝑡 𝑃𝑃𝑡𝑡 2006 equilibrium 2007 IS shock due to decline in house prices
𝑌𝑌06 𝑌𝑌07 𝑃𝑃06 𝐿𝐿𝐿𝐿(𝐿𝐿07,↓𝑃𝑃) 𝑃𝑃07 Figure 37.11: 2007-2008 Financial Crisis The increase in credit spreads further shifted the IS curve in to the left, with a resulting inward shift of the AD curve. Output began to contract, albeit not particularly signiﬁcantly yet. What is important is that, by the fall of 2008, interest rates had been moved close to zero. In other words,, by the end of 2008 the ZLB was binding, which means that the economy’s equilibrium was at the vertical portion of the AD curve (and the ﬂat portion of the LM curve). Figure 37.12 below plots the time series of the Federal Funds Rate. We can see that the rate was moved essentially to zero by the later part of 2008 and remained there until the end of 2015 (which is not shown in the ﬁgure). 919 𝐴𝐴𝐴𝐴 𝐴𝐴𝐴𝐴06 𝐴𝐴𝐴𝐴07 𝐼𝐼𝐴𝐴06 𝐼𝐼𝐴𝐴07 𝐿𝐿𝐿𝐿(𝐿𝐿06) 𝑟𝑟06 𝑟𝑟07 𝑟𝑟08=−𝜋𝜋𝑡𝑡+1𝑒𝑒 𝑟𝑟𝑡𝑡 𝑌𝑌𝑡𝑡 𝑌𝑌𝑡𝑡 𝑃𝑃𝑡𝑡 2006 equilibrium 2007 IS shock due to decline in house prices 2008 IS shock due to increase in credit spreads 𝑌𝑌06 𝑌𝑌07 𝑃𝑃06 𝐿𝐿𝐿𝐿(𝐿𝐿07,↓𝑃𝑃) 𝑃𝑃07 �
��𝑃08 𝐼𝐼𝐴𝐴08 𝐴𝐴𝐴𝐴08 𝑌𝑌08 Figure 37.12: Federal Funds Rate Around Great Recession Things got worse in the fall of 2008 – this is detailed in the section above with the failure of Lehman Brothers, the rescue of AIG, and the run on the Reserve Primary Fund in September of that year. Things were also exacerbated by the uncertainty surrounding the government’s planned rescued packages (to be discussed more in the section below). From Figure 37.4 above, one can see that while credit spreads increased over 2007 and the ﬁrst half of 2008, the biggest increase in credit spreads was at the end of 2008 and persisted into 2009. The intensiﬁcation of the ﬁnancial crisis can be mapped into the model with a further increase in ft moving from 2008 into the ﬁrst half of 2009. This is shown below in Figure 37.13. 920 0123456200420052006200720082009201020112012Effective Federal Funds Rate Figure 37.13: The ZLB and the Intensiﬁcation of the Financial Crisis in 2008-2009 An important point to be emphasized is that the eﬀects of the ﬁnancial crisis, as manifested in higher credit spreads in late 2008 and into the ﬁrst half of 2009, were exacerbated by the fact that the ZLB was binding by the end of 2008. In Chapter 29 we showed how the economy is particularly susceptible to negative IS shocks at the ZLB. In normal times, negative IS shocks (as for example would happen with an increase in ft) are partially oﬀset by lower interest rates. But at the ZLB this is not possible. Hence, the AD curve shifts in signiﬁcantly more after a negative IS shock when the ZLB binds in comparison to normal times. This is documented in Figure 37.14. This ﬁgure shows the eﬀects of an increase in ft both with the ZLB binding (as characterized the US economy at the end of 2008) compared to a situation in which the ZLB does not bind (as indicated by dashed lines in the ﬁgure). Our model would predict that output would have reacted much less to the increase in credit spreads had the ZLB not been
binding. 921 𝐴𝐴𝐴𝐴 𝐴𝐴𝐴𝐴06 𝐴𝐴𝐴𝐴07 𝐼𝐼𝐴𝐴06 𝐼𝐼𝐴𝐴07 𝐿𝐿𝐿𝐿(𝐿𝐿06) 𝑟𝑟06 𝑟𝑟07 𝑟𝑟08=𝑟𝑟09=−𝜋𝜋𝑡𝑡+1𝑒𝑒 𝑟𝑟𝑡𝑡 𝑌𝑌𝑡𝑡 𝑌𝑌𝑡𝑡 𝑃𝑃𝑡𝑡 2006 equilibrium 2007 IS shock due to decline in house prices 2008 IS shock due to increase in credit spreads 𝑌𝑌06 𝑌𝑌07 𝑃𝑃06 𝐿𝐿𝐿𝐿(𝐿𝐿07,↓𝑃𝑃) 𝑃𝑃07 𝑃𝑃08 𝐼𝐼𝐴𝐴08 𝐴𝐴𝐴𝐴08 𝑌𝑌08 𝐼𝐼𝐴𝐴09 𝐴𝐴𝐴𝐴09 𝑃𝑃09 𝑌𝑌09 2008-2009 IS shock due to intensification of credit spread increases Figure 37.14: Intensiﬁcation of the Financial Crisis in 2008-2009 with and without the ZLB Overall, the model, in spite of its simplicity, can provide a fairly good account of what actually happened during the Great Recession. Output declined throughout 2008, but most precipitously toward the end of 2008 and early 2009 when the ZLB was binding. This is consistent with the predictions of the model. What about the behavior of prices? Figure 37.15 plots the Personal Consumption Expenditure price deﬂator (solid dark line, left scale) and the annualized rate of change in
this price index (blue line, right scale). As our simple model would predict, the price level declined fairly signiﬁcantly in the second half of 2008 and into the early part of 2009. Outright declines in the price level are rare in post-Depression US business cycles. The inﬂation rate went from mildly positive before and during the early stages of the recession to sharply negative at the end of 2008. 922 𝑟𝑟𝑡𝑡 𝑃𝑃𝑡𝑡 𝑌𝑌𝑡𝑡 𝑌𝑌𝑡𝑡 𝐼𝐼𝐼𝐼08 𝐼𝐼𝐼𝐼08 𝐿𝐿𝐿𝐿08 𝐿𝐿𝐿𝐿09(↓𝑃𝑃) 𝑟𝑟08=𝑟𝑟09=−𝜋𝜋𝑡𝑡+1𝑒𝑒 𝑟𝑟09′ 𝑃𝑃08 𝑃𝑃09 𝑃𝑃09′ 𝑌𝑌08 𝑌𝑌09 𝑌𝑌09′ 2008 equilibrium 2008-2009 IS shock due to increase in credit spreads Hypothetical AD effect with no ZLB Hypothetical indirect LM effect with no ZLB 𝐴𝐴𝐴𝐴08 𝐴𝐴𝐴𝐴09 𝐴𝐴𝐴𝐴09′ Figure 37.15: Price Level and Inﬂation Rate Around Great Recession The model is of course not a perfect description of reality. While there were concerns to this eﬀect, the economy never went into a deﬂationary spiral as described in Chapter 29. Indeed, survey measures of inﬂation expectations remained high during and in the immediate wake of the Great Recession, much higher than most models would have predicted (Coibion and Gorodnichenko 2015). Second, even though the ZLB continued to bind through the end of 2015, the economy began to recover in the second half of
2009. This is not necessarily consistent with our basic analysis laid out in Chapter 29, wherein an economy will not recover on its own after a series of negative demand shocks when the ZLB binds. The fact that the economy did start to recover could be evidence against the validity of the model, or it could be evidence supporting a conclusion that the unconventional policy actions taken (to be discussed below) had some beneﬁcial eﬀects. While output did begin to recover in the second half of 2009, the recovery was not robust. In Figure 37.16, we plot the natural log of real GDP in and around the Great Recession. We also show, in a dashed line, a hypothetical trend level of GDP beginning with the onset of the Great Recession. This trend line is computed by taking the average growth rate of GDP from 2004-2007 and assuming that it would have continued after 2007. Relative to trend, GDP fell by roughly 10 percent by the middle part of 2009. While real GDP did begin to grow at that point, it did not growth fast enough to “catch up” to the hypothetical trend line. Several years after the recession (and indeed continuing to the present day), GDP remained 10 or more percent below a hypothetical pre-Recession trend line. 923 889296100104108-6-4-20246200420052006200720082009201020112012PCE IndexPCE InflationPrice LevelInflation Rate Figure 37.16: Real GDP and Hypothetical Trend 37.4 Unconventional Policy Actions The ﬁnancial crisis and ensuing Great Recession spurred several policy actions. Some of these were fairly standard (e.g. the Fed cutting interest rates), but the causes, severity, and ZLB problems associated with the Great Recession dictated more than conventional policy actions. The conventional policy action was monetary accommodation and the cutting of the Fed’s key policy interest rate, the Fed Funds Rate. As documented in Figure 37.12, the Fed lowered the Fed Funds Rate from about 5 percent prior to the start of the crisis to zero by the end of 2008. We can divide the “unconventional” policy actions into roughly three groups. The ﬁrst involved the Fed’s extraordinary rescue actions and attempts to provide liquidity to interbank lending markets. These actions were warranted given that the cause of the crisis was essentially a wholesale run by some ﬁnancial institutions on other institutions. The Fed was merely acting as a
lender of last resort. The second involved ﬁscal stimulus. Most economists would agree that monetary policy should be the ﬁrst line of defense against a recession. During a time where monetary policy is constrained by the ZLB, ﬁscal policy may not only be the only available tool but might also be more eﬀective in comparison to normal times. Finally, the third set of policies involved unconventional monetary policies – principally forward guidance and quantitative easing. These were attempts to lower longer term and risky interest rates without the usual channel of lowering short term and riskless interest rates like the Fed Funds Rate (which was an action not available to policy makers because of the ZLB). Quantitative easing and forward guidance are also discussed in some detail in Chapter 34. 924 9.509.559.609.659.709.759.80200420052006200720082009201020112012Real GDPPre-Recession Trend We will discuss each of these three broad areas of unconventional policy in the subsections below. There is a common theme to the diﬀerent policy interventions, however. In particular, they were all designed to stimulate demand through shifting the IS curve. This makes some sense given that the conventional story about the cause of the recession was an adverse shock to the IS curve (i.e. an increase in credit spreads, ft). What is unconventional about these policies is that the typical policy response to a recession centers on using monetary policy to stimulate demand through shifting the LM curve. This policy option was not on the table due to the zero lower bound problem. 37.4.1 Federal Reserve Lending The ﬁnancial crisis was essentially a liquidity crunch. Fearing losses related to mortgagerelated securities, short term funding for ﬁnancial institutions dried up and these institutions were forced to sell assets to raise cash. It was essentially a classic bank run, only it was a run of institutions on other institutions, as opposed to a run of depositors on conventional banks (as in the Great Depression, for example). There was a run on the shadow banking system, and not on commercial demand deposits, because there is nothing akin to deposit insurance for short term funding markets like repurchase agreements and commercial paper. As discussed at the end of Chapter 33, outside of deposit insurance and suspension of convertibility, the classic way to deal with a run is for the central bank to step in as the lender of last resort. The basic mechanics of this are for the central
bank to step in and lend funds to ﬁnancial institutions facing funding shortfalls. This lending would allow these banks to come up with the cash to meet their short term funding shortfalls without having to sell assets. The hope is that by encouraging institutions to not sell assets, asset prices will remain high, yields low, and credit will continue to ﬂow. The Fed engaged in several extraordinary lending programs at the height of the ﬁnancial crisis. Our objective is not to provide great detail on all of these lending programs, but rather to give a broad overview and highlight a few important ones. The traditional means of lending as a last resort is through the Fed’s discount window. Banks can go to the discount window and get loans at the discount rate, allowing them to deal with temporary liquidity shortfalls. For a couple of reasons, the discount window itself was not particularly important during the ﬁnancial crisis and ensuing Great Recession, although emergency lending and liquidity provision was. Banks have always been weary of a “stigma” attached to going to the discount window, fearing that other market participants gaining knowledge of this would lead the market to perceive an institution as weak. Second, because the “run” during the Great Recession happened outside of the conventional banking sector, the institutions facing 925 short term funding shortfalls were not designated as commercial banks and as such were not eligible for conventional discount loans. The Fed got around these issues and extended credit to the ﬁnancial system more generally in several diﬀerent ways. Relatively early in the crisis, in December of 2007, the Fed established the Term Auction Facility (TAF). The TAF distributed loans to banks through a competitive auction process, and did so in a way that the anonymity of ﬁrms receiving loans was maintained (in contrast to traditional discount window lending). The TAF was quite successful – at its height, more than $400 million in credit was extended through this facility. Other non-traditional lending facilities were designed to open Fed lending to more than just commercial banks. For example, the Primary Dealer Credit Facility (PDCF) was a program designed to lend to institutions which did not have access to the discount window. In the solid black line in Figure 37.17, we plot the total credit outstanding from the Fed to ﬁnancial institutions. This went from under $100 billion prior to the crisis to about $1.5 trillion at the height of the crisis at
the end of 2008. This series returned to more normal levels by the end of 2009. These data are available for download from the Federal Reserve Bank of Cleveland. Figure 37.17: Emergency Fed Lending In addition to traditional outright lending, the Fed also engaged in similar programs to extend liquidity to key markets. For example, the Asset Backed Commercial Paper Money Market Lending Facility (AMLF) lent money to institutions where the lent money was designed to purchase asset-backed commercial paper from mutual funds. This was in response to large scale withdrawals from mutual funds, and was designed to stem the run on these funds. The Term Asset Backed Securities Lending Facility (TALF) was a facility designed to improve consumer credit. The idea was to purchase asset backed securities (for example 926 0200,000400,000600,000800,0001,000,0001,200,0001,400,0001,600,000200720082009201020112012201320142015Lending to Financial InstitutionsLiquidity Provision to Key MarketsMillions based on credit cards and student loans). The liquidity from these purchases would hopefully spur lending to households and businesses. The Commercial Paper Funding Facility (CPFF) was similar in that it purchased commercial paper, with the idea being that this would increase issuance of commercial paper (short term unsecured corporate debt). Finally, the Fed created several special purpose vehicles by the moniker Maiden Lane to extend credit to failing ﬁnancial institutions like Bear Stearns and AIG. Total liquidity provision exceeded $400 billion at the height of the crisis. It is plotted in the solid blue line in Figure 37.17. All told, the Fed directly lent or injected close to $2 trillion into private ﬁnancial markets through its extraordinary lending facilities. This lending peaked at the height of the crisis (end of 2008 and into early 2009), and was essentially back to normal levels by 2012. The objective of all this new lending was to restore calm and liquidity to ﬁnancial markets and to get ﬁnancial institutions lending again. In terms of the AD-AS model, we can think about Fed lending and rescue operations as in essence trying to reverse the increases in credit spreads (ft in our notation) that characterized the height of the recession. Especially to the extent to which the crisis was triggered by an increase in credit spreads, this policy makes a lot of sense. Figure 37.18 plots in an AD-AS diagram the desired eﬀect
s of the Fed’s lending activities taking the 2009 equilibrium as a starting point. We can think of the extraordinary lending activities as essentially attempts to reduce ft, which would work to undo the inward shift of the IS and AD curves due to the ﬁnancial crisis. 927 Figure 37.18: AD-AS Eﬀects of Federal Reserve Lending Programs Were these lending activities successful at stimulating the economy and preventing the recession from being much worse? The data are seemingly consistent with this hypothesis. As can be seen in Figures 37.4 and 37.9, credit spreads were back to normal levels by the end of 2009 and early 2010, which coincides with the period in which the Fed pulled back on its extensive lending to the ﬁnancial markets. Furthermore, output and labor market variables began to stabilize in the middle of 2009, shortly after many of the Fed’s new lending facilities had been put into place. 37.4.2 Fiscal Stimulus Most economists prefer monetary policy as the principal tool to ﬁght recessions. But in extreme circumstances, using stimulative ﬁscal policy (some combination of increasing 928 𝐴𝐴𝐴𝐴 𝐿𝐿𝐿𝐿 𝑟𝑟09=𝑟𝑟09′=−𝜋𝜋𝑡𝑡+1𝑒𝑒 𝑟𝑟𝑡𝑡 𝑌𝑌𝑡𝑡 𝑌𝑌𝑡𝑡 𝑃𝑃𝑡𝑡 2009 equilibrium Desired effects of lending and liquidity provision to ↓𝑓𝑓𝑡𝑡 𝑃𝑃09′ 𝐼𝐼𝐴𝐴09′ 𝐴𝐴𝐴𝐴09′ 𝑌𝑌09′ 𝐼𝐼𝐴𝐴09 𝐴𝐴𝐴𝐴09 𝑃𝑃09 𝑌𝑌09 government spending and reducing taxes) might make sense and even could be quite desirable. These circumstances include a situation in which monetary policy is constrained by the zero lower bound, which characterized the
US economy from the end of 2008 through 2015. The American Recovery and Reinvestment Act (ARRA) was the “stimulus package” passed by Congress and signed into law by then President Obama in early 2009. This bill was in response to the ﬁnancial crisis and Great Recession, which was in its most virulent phase at the time of the bill’s passing. The Recovery Act was designed to inject roughly $800 billion in stimulus into the economy over a ten year period starting in 2009. A little more than half of this was designed to be federal spending, particularly on infrastructure, while the remainder was split between tax credits, tax cuts, and federal subsidies for state and local spending. In the context of our AD-AS model, where we presume Ricardian Equivalence holds, we have to focus on the spending-based features of the Recovery Act. We can simply think of the Act as engineering an increase in Gt. In ordinary times, this would cause the IS curve to shift to the right and the AD curve to shift to the right as well, although the rightward shift of the AD curve would be smaller than the shift of the IS curve because of “crowding out” associated with increases in the real interest rate. At the ZLB, in contrast, the vertical AD curve ought to shift out horizontally to the right by the same amount as the shift of the IS curve because there is no crowding out if the interest rate is ﬁxed at its lower bound (provided the shift of the IS curve is not so large as to make the ZLB no longer binding). In essence, this is just the reverse of what is documented in Figure 37.14 above – IS shocks (whether emerging from changes in ft or Gt) have bigger eﬀects when the ZLB binds than when it does not. This suggests that ﬁscal stimulus might be particularly eﬀective at inﬂuencing output when the economy is at the ZLB (see, e.g. Christiano, Eichenbaum, and Rebelo 2011). 929 Figure 37.19: AD-AS Eﬀects of Fiscal Stimulus Figure 37.19 shows the desired eﬀects of the ﬁscal stimulus in the AD-AS model. The increase in government spending (or more broadly the combined eﬀects of the increase in spending and
tax cuts to the extent to which Ricardian Equivalence does not hold) cause the IS curve to shift to the right. The stimulative eﬀect on the AD curve is larger than it would be if the ZLB does not bind. The empirical evidence on the economic consequences of ﬁscal policy in general, and the ARRA in particular, is quite mixed, with no clear answers emerging. It is thus somewhat diﬃcult to say whether or not the ARRA worked as intended. For example, Conley and Dupor (2013) argue that the ARRA did little more than create government jobs at the expense of private jobs. Chodorow-Riech, Feivson, Liscow, and Woolston (2012) and Wilson (2012) oﬀer more positive takes on the stimulative eﬀects of the ARRA. A survey of economists from the University of Chicago’s IGM Forum indicates than more than 80 percent of surveyed economists felt that the economy was better at the end of 930 𝐴𝐴𝐴𝐴 𝐿𝐿𝐿𝐿 𝑟𝑟09=𝑟𝑟10′=−𝜋𝜋𝑡𝑡+1𝑒𝑒 𝑟𝑟𝑡𝑡 𝑌𝑌𝑡𝑡 𝑌𝑌𝑡𝑡 𝑃𝑃𝑡𝑡 2009 equilibrium Desired effects of fiscal stimulus from ARRA 𝑃𝑃10′ 𝐼𝐼𝐴𝐴10′ 𝐴𝐴𝐴𝐴10′ 𝑌𝑌10′ 𝐼𝐼𝐴𝐴09 𝐴𝐴𝐴𝐴09 𝑃𝑃09 𝑌𝑌09 2010 than it would have been without the stimulus package. Another aspect of the government’s ﬁscal intervention in the wake of the Great Recession was the Troubled Asset Relief Program (TARP). Although the TARP shared some similarities with extraordinary Federal Reserve lending and it was conducted in consultation with the Fed, it was
a program by the Treasury and is thus best classiﬁed as a ﬁscal program. The basic idea of TARP was for the Treasury to purchase “troubled assets” (like mortgage backed securities and collateralized debt obligations) from the nation’s biggest banks. The basic idea of this program is fairly simple. The crisis occurred because concerns about the valuation of assets on ﬁnancial institutions balance sheets led to a run in which short term funding for these ﬁrms dried up. This run required institutions to try to sell these assets, which further depressed the price of these assets and intensiﬁed the run. The idea was that the government could come in and buy the questionable assets. This would give these institutions needed liquidity and would hopefully help to stem the run. In exchange, the Treasury received equity shares in the banks receiving funds. The economic consequences of TARP remain hotly debated and few concrete observations have emerged concerning its eﬀectiveness (or lack thereof). 37.4.3 Unconventional Monetary Policy Conventional monetary policy involves adjusting the money supply and short term interest rates so as to stabilize the economy about its potential. This was of course the ﬁrst line of defense in response to the Great Recession – the Fed aggressively lowered its key policy rate all the way to zero by the end of 2008. Even with this, along with the extraordinary lending activities undertaken by the Fed which are documented above, the Fed felt that this was not enough and that the economy needed more stimulus. Once the Federal Funds rate had hit its zero lower bound in late 2008, the Fed began to experiment with several diﬀerent forms of unconventional monetary policies. To implement conventional monetary policy, the Fed buys and sells primarily short term government debt (i.e. Treasury bills). By aﬀecting the amount of this debt in circulation the Fed can manipulate very short term interbank lending rates. Interbank lending rates (e.g. the Federal Funds rate) are not directly relevant for households or ﬁrms in making spending and investment decisions. The credit households and ﬁrms receive is typically risk adjusted (i.e. Baa corporate bond rates have a higher perceived default risk than government bonds) and longer term (i.e. a 10 year maturity as opposed to overnight). But interest rates relevant for these actors are nevertheless aﬀected by short term interbank lending rates because
debt instruments with diﬀerent risk and maturity characteristics are nevertheless to some degree substitutes. This 931 means that if the Fed Funds rate goes up or down by a certain amount, we would expect other interest rates in the economy to move in a similar direction. See also the more detailed discussion in Chapter 34. In the context of our model, we would think about conventional monetary policy as aﬀecting rt but not ft (the credit spread) – hence the rate relevant for investment decisions, rt + ft, moves in the same way as rt for a ﬁxed ft. With the zero lower bound binding, it was not possible to use conventional monetary policy to stimulate the economy – i.e. it, and hence rt, could not be lowered any further, so this means was not available to impact rt + ft. Purchases of short term Treasuries would do nothing to impact interbank lending rates given that the banking system was generally in a state of hoarding cash. The Fed instead had to think outside of the box. The Fed resorted to unconventional policies, which we group into two parts: quantitative easing and forward guidance. Quantitative easing involves purchases of longer term and/or riskier debt, with the objective to push up the price of these assets and the associated yields (interest rates) down. Forward guidance involves communicating to the public the expected future path of short term interest rates. To discuss quantitative easing and forward guidance intelligently we must take a brief step back and think a bit about the risk and term structure of interest rates. More detail is provided in Chapter 34. In reality there are many diﬀerent interest rates in an economy – e.g. the Fed Funds rate, the 3 month Treasury Bill rate, the 10 year Treasury note, a 30 year Treasury bond rate, a 15 year mortgage rate, a 30 year mortgage rate, the Aaa corporate bond rate, the Baa corporate bond rate, etc. These rates diﬀer both according to perceived risk (i.e. a Aaa rated corporate bond is perceived as being less risky than Baa rated bonds, and interest rates are typically higher the greater is perceived risk of default) and time to maturity (i.e. the 10 year Treasury versus a 3 month Treasury Bill; for the most part, interest rates are higher the longer the time to maturity, i.e. yield curves slope up). Most macroeconomic models, including those models presented in this book
, abstract from this complexity and focus on one or at most two interest rates. In terms of the models studied in this book, the short term, riskless real interest rate is rt, while the (real) interest rate relevant for investment decisions is rt + ft. We can think about the ﬁrst interest rate as being something the Fed can manipulate, while the second is a stand-in for the risk and term structure of interest rates. For debt instruments, bond price and yield (an alternative name for interest rate) move opposite one another. This is easiest to see for a so-called discount bond. A discount bond promises the holder of the bond some pre-speciﬁed payment at some point in the future (say, for simplicity, one period into the future) called the face value. The bond sells at a price lower than the face value. The percentage diﬀerence between the bond price and the face value is the implied interest rate (or yield). In particular, we could think of P B F V as t = 1 1+iB,t 932 relating the price of the bond, its interest rate, and its face value. For example, if I buy a $100 face value bond for $90, and the bond matures (i.e. pays the face value) in one year, then the implicit interest rate on the bond is 11 percent. Although payment details diﬀer for diﬀerent kinds of bonds, the basic idea that the interest rate (yield) and price move opposite one another always holds. What is the connection between all the diﬀerent kinds of interest rates in the world? There are two polar extreme theories, with reality likely lying somewhere in between. The ﬁrst theory is called segmented markets, and is based on the idea that debt instruments with diﬀerent risk and/or maturity characteristics are not substitutable at all. Under segmented markets, we can think about there being separate demand-supply diagrams for bonds with diﬀerent characteristics. Demand and supply determine price and interest rates for each type of bond, and there are no spillover eﬀects between markets for diﬀerent types of debt. Figure 37.20 plots a hypothetical demand-supply diagram for a particular kind of bond. Demand for the bond is downward-sloping. Demand comes from savers who want to hold the bond
to earn interest – the lower the bond price, the higher the interest rate, and the more of the bond savers would want to hold. Supply is upward-sloping. Bond supply comes from debtors who want to borrow funds. The higher is the price, the lower is the interest rate, and hence the more funds these debtors would like to borrow. Demand and supply intersect to determine an equilibrium quantity and price. Figure 37.20: Demand and Supply for a Bond, Segmented Markets The basic idea behind the Fed’s various rounds of quantitative easing was for the Fed to purchase large quantities of mortgage related debt securities and/or longer maturity Treasury 933 𝑃𝑃𝑡𝑡𝐵𝐵 𝐵𝐵𝑡𝑡 𝐷𝐷 𝑆𝑆 𝑃𝑃0,𝑡𝑡𝐵𝐵 𝐵𝐵0,𝑡𝑡 securities.1 The Fed would create base money by creating reserves to purchase these securities from private banks. Quantitative easing (or QE, for short) proceeded in three distinct waves. The ﬁrst wave, or what was called QE1 after the fact, was announced in November of 2008. At ﬁrst this involved only purchases of mortgage backed securities, but as the program continued the Fed also purchased longer maturity Treasury securities. This program continued through the summer of 2010. At its peak the Fed held some $2 trillion dollars in mortgage backed securities and longer term Treasuries. Figure 37.21 plots the evolution of the Fed’s holdings of mortgage backed securities and longer term Treasury securities.2 Prior to the crisis, the Fed held none of these securities. Figure 37.21: Unconventional Asset Holdings by Fed The second round of quantitative easing, or QE2, was announced in November of 2010. QE2 focused on purchasing longer maturity Treasury securities. The uptick in the Fed’s holding of these securities can be clearly seen in Figure 37.21. The third round of quantitative easing, or QE3, began in November of 2012 and focused mostly on purchasing more mortgage backed securities. The eﬀects of this program on the Fed’s holdings of these securities can clearly be seen in the Figure. Quantitative easing programs formally ceased (i.e
. the Fed ceased buying new securities) towards the end of 2014. By the end, the Fed had purchased close to $4 trillion in longer maturity Treasury and mortgage backed securities. The objective of the quantitative easing programs was to increase the market prices of the debt instruments the Fed was buying. By doing so, this would result in lower interest 1The Fed’s purchases of mortgage backed securities was restricted to securities issued by the government sponsored enterprises Fannie Mae, Freddie Mac, and Ginnie Mae. For this reason, one sometimes sees the term agency backed securities (ABS) when referring to the Fed’s asset purchases. 2These data are available for download from the Federal Reserve Bank of Cleveland. 934 0400,000800,0001,200,0001,600,0002,000,0002,400,000200720082009201020112012201320142015MBS HoldingsLong Term Treasury HoldingsMillionsQE1QE2QE3 rates, which would hopefully be passed on to consumers and businesses in the form of lower interest rates on mortgage loans, business loans, and the like. This is easy to see using the demand-supply analysis from Figure 37.20. The QE programs involved a large increase in the demand for these types of debt. The increase in demand ought to increase the price of this debt and lower the interest rate, as shown in Figure 37.22. Figure 37.22: Desired Eﬀects of Quantitative Easing Programs Former Fed chairman Ben Bernanke once famously quipped that “quantitative easing works in practice but not in theory.” In essence he was referring to the alternative theory of the term structure of interest rates called the expectations hypothesis. In this theory, bonds of diﬀerent maturities are perfect substitutes. This ends up meaning that the interest rate on a long maturity bond is approximately equal to the average of expected short term interest rates. Formally, if a bond has a N period maturity, then the interest rate on that bond should equal the average of expected interest rates on one periods maturity bonds, plus a term premium to account for the fact that longer maturity bonds carry more risk in the event they need to be sold before maturity: iN,t = 1 N (i1,t + ie 1,t+1 +... ie 1,t+N −1 ) + tpt (37.1) In (37.1), iN,t is the interest rate

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