Question

Consider a specific version of the static representative agent economy. In particular, we will dispense with capital and let the production function satisfy y=zn. The household's utility function is given by u=2θc^1/2+2l^1/2 where θ > 0 and the budget constraint is now just c=w(1-l). Verify that this utility function satisfies u1, u2>0 and u11 ,u22<0

Answer 1

The utility function is given by, U = 2*θ*C^1/2 + 2L^1/2. The 1^st order differentiations are given below.

=> U_C = δU/δC = 2*θ*(1/2)*C^1/2-1 = θ*C^(-1/2), => U_C = θ*C^(-1/2) > 0.

=> U_L = δU/δL = 2*(1/2)*L^1/2-1 = L^(-1/2), => U_L = L^(-1/2) > 0.

So, here the 1^st order differentiations are positive. Now, the 2^nd order differentiations are given below.

=> U_C = θ*C^(-1/2), => U_CC = θ*(-1/2)*C^(-1/2-1) = (-θ/2)*C^(-3/2).

=> U_CC = (-θ/2)*C^(-3/2) < 0.

=> U_L = L^(-1/2), => U_LL = (-1/2)*L^(-1/2-1) = (-1/2)*L^(-3/2).

=> U_LL = (-1/2)*L^(-3/2) < 0.

So, here the 2^nd order differentiations are negative.