Econ 510a second half Prof Tony Smith TA Theodore Papageorgiou Fall 2004 Yale University Dept of Economics Solutions for Homework 3 Question 1 a The recursive formulation for the planning problem is v k max U c l v k c l k s t c k F k L l 1 k or v k max U F k L l 1 k k l v k l k From the way we write it we can see that the state variable is k and control variables are l k b The F O C is l U 1 c l F 2 k L l U 2 c l k U 1 c l v k From Envelope Theorem we have v k U 1 c l F 1 k L l 1 Iterate forward for one period it becomes v k U 1 c l F 1 k L l 1 Plug it into F O C we get the final optimality conditions l t U 1 c t l t F 2 k t L l t U 2 c t l t k t 1 U 1 c t l t U 1 c t 1 l t 1 F 1 k t 1 L l t 1 1 c In steady state the optimality condition becomes l k U1 c l F 1 k L l 1 1 F2 k L l U2 c l where c F k L l k We can see that k and l depends on both production technology F k t n t and utility function U c t l t In a growth model without valued leisure the steady state is determined by the equation F 1 k L 1 1 which does not depend on the utility function U c t l t Now let s compare two models First in the model with leisure choice we add an additional equation which states that the marginal rate of substitution between consumption and leisure must equal to the marginal rate of transformation Second the equation about k is the same except that the level of steady state leisure is different Third for the equation F 1 k L l 1 1 due to the difference in the steady state leisure level the steady state capital stock is also different For example if F 12 0 as in the case of Cobb Douglas production function the capital stock in the model with leisure will be lower than that without leisure choice since L l L 1 c l 1 1 the steady state conditions become d With F k n k a n 1 a and u c l 1 l k 1 k a n k a n a 1 a c 1 l 1 1 which leads to if we normalize L 1 l a 1 1 1 1 a 1 1 a 1 1 n 1 l k 1 1 a a 1 1 a 1 1 c k a n 1 a k Question 2 a A competitive equilibrium is a set of sequences c t t 0 b t t 0 q t t 0 such that 1 c t b t 1 t 0 arg max c t b t 1 t 0 s t E0 t t 0 c t c t 1 1 1 1 c t q t b t 1 b t w t c t 0 t b 0 0 t lim b t 1 t qj 0 j 0 2 b t 0 t bonds market clearing 3 c t w t t goods market clearing b To simplify notation we conjecture that in equilibrium the bond price is constant across time we will check this conjecture later Now the recursive formulation of the consumer s problem is v b t c t 1 t max c t b t 1 c t c t 1 1 1 v b t 1 c t t 1 1 s t c t qb t 1 b t t or equivalently v b t c t 1 t max b t 1 a t t qb t 1 c t 1 1 1 1 v b t 1 b t t qb t 1 t 1 Note the choice of aggregate state variable here In principle we should include the triple aggregate state A t 1 t into our state variable But here we know that A 0 since it is a representative agent economy And as long as we know about one value in the pair t 1 t we can deduce the other from the constant growth rate g Therefore we need only one aggregate endowment either t 1 or t as our aggregate state variable For example we could choose t 1 and the bond price would be q t q t 1 To save notation further we can even write q t q t 1 since this is a representative agent exchange economy t 1 t 1 and we cannot change either individual or aggregate endowment anyway Furthermore due to the special utility function here we can conjecture that the bond price is constant across time and check it later So after a long chain of reasoning we choose q t q and only include the individual triple state b t c t 1 t into our recursive formulation c F O C for this problem is b t 1 u 1 c t c t 1 v 2 t 1 q v 1 t 1 where v 1 t 1 and v 2 t 1 are partial derivatives of v b t 1 c t t 1 The envelope condition is b t v 1 t u 1 c t c t 1 v 2 t 1 c t 1 v 2 t u 2 c t c t 1 Solve for this we get v 1 t u 1 c t c t 1 u 2 c t 1 c t v 2 t u 2 c t c t 1 Iterate forward for one period and plug into F O C we get the Euler Equation u 1 c t c t 1 u 2 c t 1 c t q t 1 u 1 c t 1 c t u 2 c t 2 c t 1 u 1 c t 1 c t u 2 c t 2 c t 1 q u 1 c t c t 1 u 2 c t 1 c t c t 1 c t c t 2 c t 1 q c t c t 1 c t 1 c t Notice the similarity with normal Euler equation it is the marginal rate of substitution between consumption c t 1 and c t The difference is the involvement of two period felicity function which is due to the habit persistence d In equilibrium we must have c t t Plug into the Euler Equation we get the equilibrium bond price as u 1 t 1 t u 2 t 2 t 1 u 1 t t 1 u 2 t 1 t …