Problem A.
Suppose that the instantaneous per capita utility of a single representative household is given by u(c)=c1-θ-11-θ,θ≠1 & θ≥0. Population is constant. Since we are considering a representative household, household uses its financial asset to provide capital services to firms. Besides investing in physical capital, households also invest in human capital (H=h.L), ih. Thus, the asset accumulation constraint of the household is given by: a˙=r.a+w.h-c-ih. Thus, household income, both wage income and capital income, is generated from production. The discount rate is ρ>0.
If we assume that the technology displays constant returns to scale, it is Cobb-Douglas, and that households supply inelastically human capital input to production, then Y(t)=K(t)α*[H]1-α, α<1. Here, K is physical capital stock and H is human capital or efficiency units of labor, which accumulate in the same way as physical capital. c(t) is the per capita consumption. It is assumed that and K(0) and H(0) are given and human capital is non-negative. The capital depreciates at the rate δk and human capital depreciates at δh. Firms are perfectly competit
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