Econ 712 Problem Set #4: Asset Prices and Martingales in Economics, Assignments of Introduction to Macroeconomics

Download Econ 712 Problem Set #4: Asset Prices and Martingales in Economics and more Assignments Introduction to Macroeconomics in PDF only on Docsity! Econ 712 Problem Set #4 TA: Kyoung Jin Choi October 9, 2007 1 Asset Prices and Martingales 1.1 Lucas Economy The price is given by pst = ∞∑ j=1 Et [ βj u′(ct+j) u′(ct) dt+j ] , where ct is the equilibrium consumption. By the law of iterated expectation, we have p̄st = Et[p s t+1] = ∞∑ j=1 Et [ βj u′(ct+j+1) u′(ct+1) dt+j+1 ] , It is easily seen that the assertion is not true. This can be seen for even the simple case such as a linear utility function u(c) = u1c + u0 with constant u0 and u1. In general, pst = ∞∑ j=1 βjEt [dt+j ] 6= ∞∑ j=1 βjEt [dt+j+1] = Et[pst+1]. unless we impose a certain conditions to the dividend stream. 1.2 Arbitrary {dt}? Rewrite the price as pst = ∞∑ j=0 βjEt [ u′(ct+j) u′(ct) ] Et[dt+j ] + βjcovt ( u′(ct+j) u′(ct) , dt+j ) . 1 Assume that for each integer i ≥ 0, j ≥ 1, Et [ u′(ct+i+j) u′(ct+i) ] = k (constant) (1.1) Et[dt+j ] = d (constant) (1.2) dt+j is independent of ct+j (1.3) Under assumptions (1.1), (1.2) and (1.3), we can say that the assertion holds since Et[p̄st+1] = ∞∑ j=0 βjEt [ u′(ct+j+1) u′(ct+1) ] Et[dt+j+1] + βjcovt ( u′(ct+j+1) u′(ct+1) , dt+j+1 ) = ∞∑ j=1 βjkEt[dt+j+1] = dkβ 1− β = p s t If any of the assumption is viloated, then the assertion is not satisfied at all. But, if the utility function is linear, (1.2) is automatically satisfied. Moreveore, we can drop (1.3), i.e., covt ( u′(ct+j+1) u′(ct+1) , dt+j+1 ) = covt (1, dt+j+1) = 0. In conclusion, the assertion never holds in general. But, even if the dividend process has an arbitrary correlation with the equilibrium consumption, the forcastibility is satisfied for the risk-neutral representative agent world with constant mean dividend stream. Note (1.1) is the sufficient condition in order that the price of bond satisfies the assertion. 1.3 Martingale {Xt} is a martingale is equivalent to that Xt = Et[Xt+1] for all t. The proof is done by induction and the law of iterated expectation. (Why?) Thus, it is sufficient to show Et[Yt+1] = Yt. 2 Put λ and µ be the Lagrange multiplier for problem of agent A and agent B, respectively. Then, FOCs are u′(cA0 ) = λq0, u ′(cB0 ) = µq0 1 2 βtu′(cAt (X0)) = λqt(X0), 1 2 βtu′(cBt (X0)) = µqt(X0) 1 2 βtu′(cAt (X1)) = λqt(X1), 1 2 βtu′(cBt (X1)) = µqt(X1) By dividing the left ones by the right ones respectively, we have u′(cA0 ) u′(cB0 ) = λ µ u′(cAt (X0)) u′(cBt (X0)) = λ µ u′(cAt (X1)) u′(cBt (X1)) = λ µ Note that the right hand sides are all constants. By the market clearing condition, we have1 cA0 = c A t (X0) = c A t (X1) := c A (constant), ∀t ≥ 1 and cB0 = c B t (X0) = c B t (X1) := c B (constant), ∀t ≥ 1. Now, the prices of Arrow-Debreu securities are q0 = u′(cA) λ = u′(cB) µ qt(X0) = u′(cA) 2λ βt = u′(cB) 2µ βt qt(X1) = u′(cA) 2λ βt = u′(cB) 2µ βt. 1The result comes from the fact that each equation has the same form: u′(x) = λ µ u′(2− x) 5 Putting these into the budget constraints (3.5) and (3.6), we can obtain cA0 = c A t (X0) = c A t (X1) = 1 and c B 0 = c B t (X0) = c B t (X1) = 1, which means that each agent has the same amount of consumption every period. Then the prices are given by invoking usual Euler equations R−1t+1 = E0 [ βu′(cAt+1) u′(cAt ) ] = E0 [ βu′(cBt+1) u′(cBt ) ] = β, ∀t (3.7) sA−0 = s B −0 = 1 + ∞∑ j=1 βjE0 [ u′(cAj ) u′(cA0 ) dAj (X) ] = 1 + β 1− β = 1 1− β (3.8) sA0 = s B 0 = ∞∑ j=1 βjE0 [ u′(cAj ) u′(cA0 ) dAj (X) ] = β 1− β . (3.9) Note that R−1t+1 is the price of 1-period bond and s A −0 and s B −0 are the time-0 prices of stock A and B before the time-0 dividend is given and sA0 and s B 0 are the prices after the time-0 dividends are given. Once X is realized, the price of the tree with no dividend is 0 and the price pot of the tree with dividend of 2 is pot = ∞∑ j=1 βjEt [ u′(cAt+j) u′(cAt ) 2 ] = 2β 1− β , t ≥ 1, (3.10) which is constant. 3.2 Incomplete Markets After the realization of X, the price of the dead tree will be 0 afterward. The owner of dead tree consumed 1 at period 0 and never be able to consume after that since he doen’t afford to buy and sell the goods. After the realization there is only one agent who has constant endowment 2 at each period, which means the marginal rate of substitution of the owner of the existing tree keeps constant. Thus, the price of the existing tree is pot = 2β 1−β same as (3.10). The price of the bond is β just as in (3.7). 6 The prices at time 0 are given by R−11 = E0 [ β u′(c1(X) u′(c0) ] = β 2 ( u′(0) u′(1) × 0 + u ′(2) u′(1) × 2 ) = βu′(2) u′(1) . (3.11) and pA0 = p B 0 = E0 [ β u′(c1(X)) u′(c0) (d1(X) + po1) ] = β 2 u′(2) u′(1) ( 2 + 2β 1− β ) = ( β 1− β ) u′(2) u′(1) (3.12) Note that the incomplete market stock price (3.12) is lower than the complete market stock price (3.9). This is because the risk averse agent prefers 1 unit of consumption with probability 1 to consumption with mean 1. The incomplete market interest rate R1 of (3.11) is higher than the complete market interest rate R1 of (3.7). This is because the bond is more valuable under this environment. 3.3 Inequality and Complete Markets This model deals with the effect of future income shock. Notice that there is no aggregate uncertainty in this model. The aggregate endowment is always 2 (constant). If the market is complete, then agents can keep their consumption constant no matter how the uncertainty realized by buying and selling their future endowment (or Arrow-Debreu securities) since there is no market risk. However, it does not work in the incomplete market. Even if there is no aggregate risk in the market, a certain inequality derived by a shock can last forever without proper financial instruments. 7