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suppose an economic agent's life is divided into 2 periods, 1st youth and 2nd old age. there is a single consumption good available in both periods, utility fn is U(c1,c2) = {c1^(1-@)-1}/(1-@) + (1/1+p){c2^(1-@) -1}/(1-@) with given 0<@,1 and p>0. here 1st term is utility from consumption during youth and 2nd term represents discounted utility from consumption in old age, 1/(1+p) being the discount factor.During the period, the agent has a unit of labor which she supplies inelastically for wage rate ,w. Any savings (i.e., income minus consumption during 1st pd) earns a rate of interest r, the proceeds of which is available in old age in units of the only consumption good available. denote savings by s. the agent maximizes utility subject to her budget constraint. write down the optimization problem and find an expression for s as a function of w and r sir, i have tried to solve, pls check and correct it. setting the langrangian subject to its budget constraint. MAX U(C1,C2) -K{C1 +C2/(1+r) -W} finding 1st two partial derivatives and setting them equal to zero, we get the relation between c1 and c2 as follows C2^@/C1@ = (1+r)/(1+p) -------------equation (1), setting partial derivative with respect to K equal to zero, we get C2/(1+r) = w-C1 putting w-c1 equal to s and substituting value of c2 from equation (1) , we get s= (wz)/ 1+r+z where z stands for {(1+r)/(1+p)}^(1/@).
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