Multiperiod Market Equilibrium

Multiperiod Market Equilibrium Multiperiod Market Equilibrium 1/ 27

Introduction The rst order conditions from an individual s multiperiod consumption and portfolio choice problem can be interpreted as equilibrium conditions for asset pricing. A particular equilibrium asset pricing model where asset supplies are exogenous is the Lucas (1978) endowment economy. We also consider bubbles: nonfundamental asset price dynamics. Multiperiod Market Equilibrium 2/ 27

Asset Pricing in the Multiperiod Model In the multiperiod model, the individual s objective is " T 1 # X max E t U (C s ; s) + B (W T ; T ) C s ;f! is g;8s;i s=t which is solved as a series of single period problems using the Bellman equation: J (W t ; t) = (1) max U (C t; t) + E t [J (W t+1 ; t + 1)] (2) C t ;f! i;tg This led to the rst-order conditions U C (C t ; t) = R f ;t E t [J W (W t+1 ; t + 1)] = J W (W t ; t) (3) E t [R it J W (W t+1 ; t + 1)] = R f ;t E t [J W (W t+1 ; t + 1)] ; i = 1; :::; n (4) Multiperiod Market Equilibrium 3/ 27

Multiperiod Pricing Kernel This model has equilibrium implications even if assumptions about utility, income, and asset return distributions do not lead to explicit formulas for C t and! it. Substituting the envelope condition U C (C t ; t) = J W (W t ; t) at t + 1 into the right-hand side of the rst line of (3), U C (C t ; t) = R f ;t E t [J W (W t+1 ; t + 1)] = R f ;t E t UC C t+1; t + 1 (5) Furthermore, substituting (4) into (3) and, again, using the envelope condition at date t + 1 allows us to write or Multiperiod Market Equilibrium 4/ 27 U C (C t ; t) = E t [R it J W (W t+1 ; t + 1)] = E t Rit U C C t+1; t + 1 (6)

Multiperiod Pricing Kernel cont d 1 = E t [m t;t+1 R it ] = R f ;t E t [m t;t+1 ] (7) where m t;t+1 U C Ct+1 ; t + 1 =U C (Ct ; t) is the SDF (pricing kernel) between dates t and t + 1. The relationship derived in the single-period context holds more generally: Updating equation (6) for risky asset j one period, U C Ct+1 ; t + 1 = E t+1 Rj;t+1 U C Ct+2 ; t + 2, and substituting in the right-hand side of the original (6), one obtains U C (Ct ; t) = E t Rit E t+1 Rj;t+1 U C Ct+2; t + 2 = E t Rit R j;t+1 U C C t+2; t + 2 (8) Multiperiod Market Equilibrium 5/ 27

Multiperiod Pricing Kernel cont d or 1 = E t [R it R j;t+1 m t;t+2 ] (9) where m t;t+2 U C Ct+2 ; t + 2 =U C (Ct ; t) is the marginal rate of substitution, or the SDF, between dates t and t + 2. By repeated substitution, (9) can be generalized to 1 = E t [R t;t+k m t;t+k ] (10) where m t;t+k U C Ct+k ; t + k =U C (Ct ; t) and R t;t+k is the return from any trading strategy involving multiple assets over the period from dates t to t + k. Equation (10) says that equilibrium expected marginal utilities are equal across all time periods and assets. These moment conditions are often tested using a Generalized Method of Moments technique. Multiperiod Market Equilibrium 6/ 27

Lucas Model of Asset Pricing Lucas (1978) derives equilibrium asset prices for an endowment economy where the random process generating the economy s real output is exogenous. Output obtained at a particular date cannot be reinvested and must be consumed immediately. Assets representing ownership claims on this exogenous output are in xed supply. Assuming all individuals are identical (representative), the endowment economy assumptions x the process for individual consumption, making the SDF exogenous. Exogenous output implies the market portfolio s payout (dividend) is exogenous, which makes it easy to solve for the equilibrium price. Multiperiod Market Equilibrium 7/ 27

Including Dividends in Asset Returns Let the return on the i th risky asset, R it, include a dividend payment made at date t + 1, d i;t+1, along with a capital gain, P i;t+1 P it : R it = d i;t+1 + P i;t+1 P it (11) Substituting (11) into (7) and rearranging gives P it = E t " UC C t+1 ; t + 1 U C (C t ; t) (d i;t+1 + P i;t+1 ) # (12) Similar to what was done in equation (8), substitute for P i;t+1 using equation (12) updated one period to solve forward this equation. Multiperiod Market Equilibrium 8/ 27

Including Dividends in Asset Returns cont d " UC Ct+1 P it = E ; t + 1 t U C (Ct ; t) d i;t+1 + U C Ct+2 ; t + 2!# U C Ct+1 ; t + 1 d i;t+2 + P i;t+2 " UC Ct+1 = E ; t + 1 t U C (Ct ; t) d i;t+1 + U C Ct+2 ; t + 2 # U C (Ct ; t) d i;t+2 + P i;t+2 (13) Repeating this type of substitution, that is, solving forward the di erence equation (13), gives us 2 TX U C Ct+j P it = E t 4 ; t + j U C (Ct d i;t+j + U C Ct+T ; t + T ; t) U C (Ct ; t) j=1 (14) where the integer T re ects a large number of future periods. P i;t+t 3 5 Multiperiod Market Equilibrium 9/ 27

Including Time Preference If utility is of the form U (C t ; t) = t u (C t ), where = 1 1+ < 1, then (14) becomes 2 TX P it = E t 4 u j C Ct+j u C (Ct ) d i;t+j + T u C Ct+T 3 u C (Ct ) P 5 i;t+t j=1 Assuming individuals that have in nite lives or a bequest motive and lim T!1 E t T u C (Ct+T ) = 0 (no u C (C t ) P i;t+t (15) speculative price bubbles ), then 2 1X P it = E t 4 u 3 j C Ct+j u C (Ct ) d i;t+j5 (16) j=1 Multiperiod Market Equilibrium 10/ 27

Dividends and Consumption In terms of the SDF m t; t+j j u C Ct+j =u C (Ct ): 2 3 1X P it = E t 4 m t; t+j d i;t+j 5 (17) j=1 Lucas (1978) makes (17) into a general equilibrium model by assuming an in nitely-lived representative individual and where risky asset i pays a real dividend of d it at date t. The dividend is nonstorable and non-reinvestable. With no wage income, aggregate consumption equals the total dividends paid by all of the n assets at that date: nx Ct = d it (18) i=1 Multiperiod Market Equilibrium 11/ 27

Examples If the representative individual is risk-neutral, so that u(c) = C and u C is a constant (1), then (17) becomes 2 3 1X = E t 4 j d i;t+j 5 (19) P it j=1 If utility is logarithmic (u (C t ) = ln C t ) and aggregate dividend d t = P n i=1 d it, the price of risky asset i is given by 2 3 1X P it = E t 4 j Ct 5 j=1 2 1X = E t 4 j=1 j Ct+j d i;t+j d t 3 d i;t+j 5 (20) d t+j Multiperiod Market Equilibrium 12/ 27

Examples cont d Under logarithmic utility, we can price the market portfoio without making assumptions regarding the distribution of d it in (20). If P t is the value of aggregate dividends, then from (20): 2 3 1X P t = E t 4 j d t d t+j 5 d t+j j=1 = d t 1 Note that higher expected future dividends, d t+j, are exactly o set by a lower expected marginal utility of consumption, m t; t+j = j d t =d t+j, leaving the value of a claim on this output process unchanged. (21) Multiperiod Market Equilibrium 13/ 27

Examples cont d For more general power utility, u (C t ) = Ct =, we have 2 3 1X P t = E t 4 j dt+j 5 j=1 = dt 1 E t d t 1 d t+j 2 3 1X 4 j d 5 t+j (22) which does depend on the distribution of future dividends. The value of a hypothetical riskless asset that pays a one-period dividend of $1 is P ft = 1 R ft = E t j=1 " dt+1 d t 1 # (23) Multiperiod Market Equilibrium 14/ 27

Examples cont d We can view the Mehra and Prescott (1985) nding in its true multiperiod context: they used equations such as (22) and (23) with d t = C t to see if a reasonable value of produces a risk premium (excess average return over a risk-free return) that matches that of market portfolio of U.S. stocks historical average excess returns. Reasonable values of could not match the historical risk premium of 6 %, a result they described as the equity premium puzzle. As mentioned earlier, for reasonable levels of risk aversion, aggregate consumption appears to vary too little to justify the high Sharpe ratio for the market portfolio of stocks. The moment conditions in (22) and (23) require a highly negative value of to t the data. Multiperiod Market Equilibrium 15/ 27

Labor Income We can add labor income to the market endowment (Cecchetti, Lam and Mark, 1993). Human capital pays a wage payment of y t at date t, also non-storable. Hence, equilibrium aggregate consumption equals C t = d t + y t (24) so that equilibrium consumption no longer equals dividends. The value of the market portfolio is: 2 P t = E t 4 P u 3 C C 1 t+j j=1 j u C (Ct ) d t+j5 = E t " P1 j=1 j C t+j C t 1 d t+j# (25) Multiperiod Market Equilibrium 16/ 27

Labor Income cont d Now specify separate lognormal processes for dividends and consumption: ln Ct+1=C t = c + c t+1 (26) ln (d t+1 =d t ) = d + d " t+1 where the error terms are serially uncorrelated and distributed as t 0 1 ~N ; (27) 0 1 " t Now what is the equilibrium price of the market portfolio? Multiperiod Market Equilibrium 17/ 27

Labor Income cont d When e < 1, the expectation in (25) equals where d (1 ) c + 1 2 e P t = d t 1 e (28) h i (1 ) 2 2 c + 2 d (1 ) c d (29) (See Exercise 6.3.) Equation (28) equals (21) when = 0, d = c, c = d, and = 1, which is the special case of log utility and no labor income. With no labor income ( d = c, c = d, = 1) but 6= 0, we have = c + 1 2 2 2 c, which is increasing in the growth rate of dividends (and consumption) when 1 > > 0. Multiperiod Market Equilibrium 18/ 27

Labor Income cont d When > 0, greater dividend growth leads individuals to desire increased savings due to high intertemporal elasticity (" = 1= (1 ) > 1). Market clearing requires the value of the market portfolio to rise, raising income or wealth to make desired consumption rise to equal the xed supply. The reverse occurs when < 0, as the income or wealth e ect will exceed the substitution e ect. For the general case of labor income where is given by equation (29), a lower correlation between consumption and dividends (decline in ) increases. Since @P t =@ > 0, lower correlation raises the value of the market portfolio because it is a better hedge against uncertain labor income. Multiperiod Market Equilibrium 19/ 27

Rational Asset Price Bubbles De ne p t P it u C (C t ), the product of the asset price and the marginal utility of consumption. Then equation (12) is E t [p t+1 ] = 1 p t E t uc Ct+1 di;t+1 (30) where 1 = 1 + > 1 where is the subjective rate of time preference. The solution (17) to this equation is referred to as the fundamental solution, which we denote as f t : 2 3 1X p t = f t E t 4 j u C Ct+j di;t+j 5 (31) j=1 The sum in (31) converges if the marginal utility-weighted dividends are expected to grow more slowly than the time preference discount factor. Multiperiod Market Equilibrium 20/ 27

Rational Asset Price Bubbles cont d There are other solutions to (30) of the form p t = f t + b t where the bubble component b t is any process that satis es E t [b t+1 ] = 1 b t = (1 + ) b t (32) This is easily veri ed by substitution into (30): E t [f t+1 + b t+1 ] = 1 (f t + b t ) E t uc Ct+1 di;t+1 E t [f t+1 ] + E t [b t+1 ] = 1 f t + 1 b t E t uc Ct+1 di;t+1 E t [b t+1 ] = 1 b t = (1 + ) b t (33) where in the last line of (33) uses the fact that f t satis es the di erence equation. Since 1 > 1, b t explodes in expected value: Multiperiod Market Equilibrium 21/ 27

Bubble Examples lim E t [b t+i ] = lim (1 + ) i +1 if bt > 0 b t = i!1 i!1 1 if b t < 0 (34) The exploding nature of b t provides a rationale for interpreting the general solution p t = f t + b t, b t 6= 0, as a bubble solution. Suppose that b t follows a deterministic time trend: Then the solution b t = b 0 (1 + ) t (35) p t = f t + b 0 (1 + ) t (36) implies that the marginal utility-weighted asset price grows exponentially forever. Multiperiod Market Equilibrium 22/ 27

Bubble Examples cont d Next, consider a possibly more realistic modeling of a "bursting" bubble proposed by Blanchard (1979): b t+1 = 1+ q b t + e t+1 with probability q z t+1 with probability 1 q with E t [e t+1 ] = E t [z t+1 ] = 0. The bubble continues with probability q each period but bursts with probability 1 q. (37) This process satis es the condition in (32), so that p t = f t + b t is again a valid bubble solution, and the expected return conditional on no crash is higher than in the in nite bubble. Multiperiod Market Equilibrium 23/ 27

Likelihood of Rational Bubbles Additional economic considerations may rule out many rational bubbles: consider negative bubbles where b t < 0. From (34) individuals must expect that at some future date > t that p = f + b can be negative. Since marginal utility is always positive, P it = p t =u C (C t ) must be expected to become negative, which is inconsistent with limited-liability securities. Similarly, bubbles that burst and start again can be ruled out. Note that a general process for a bubble can be written as b t = (1 + ) t b 0 + tx (1 + ) t s " s (38) s=1 where " s, s = 1; :::; t are mean-zero innovations. Multiperiod Market Equilibrium 24/ 27

Likelihood of Rational Bubbles cont d To avoid negative values of b t (and negative expected future prices), realizations of " t must satisfy This is due to " t (1 + ) b t 1, 8t 0 (39) " t = b t (1 + ) b t 1 b t = " t + (1 + ) b t 1 > 0 " t (1 + ) b t 1 For example, if b t = 0 so that a bubble does not exist at date t, then from (39) and the requirement that " t+1 have mean zero, it must be the case that " t+1 = 0 with probability 1. Hence, if a bubble currently does not exist, it cannot get started. Multiperiod Market Equilibrium 25/ 27

Likelihood of Rational Bubbles cont d Moreover, the bursting and then restarting bubble in (37) could only avoid a negative value of b t+1 if z t+1 = 0 with probability 1 and e t+1 = 0 whenever b t = 0. Hence, this type of bubble would need to be positive on the rst trading day, and once it bursts it could never restart. Tirole (1982) considers an economy model with a nite number of agents and shows that rational individuals will not trade assets at prices above their fundamental values. Santos and Woodford (1997) consider rational bubbles in a wide variety of economies and nd only a few examples of the overlapping generations type where they can exist. If conditions for rational bubbles are limited yet bubbles seem to occur, some irrationality may be required. Multiperiod Market Equilibrium 26/ 27

Summary If an asset s dividends are modeled explicitly, the asset s price satis es a discounted dividend formula. The Lucas endowment economy takes this a step further by equating aggregate dividends to consumption, simplifying valuation of claims on aggregate dividends. In an in nite horizon model, rational asset price bubbles are possible but additional aspects of the economic environment can often rule them out. Multiperiod Market Equilibrium 27/ 27