Consider a neoclassical economy with a representative household with preferences at time t = 0 given by

There is no population growth and labor is supplied inelastically. Assume that the aggregate production function is given by Y(t) = F(A(t)K(t), L(t)), where F satisfies the standard assumptions (constant returns to scale, differentiability, and the Inada conditions).

(a) Define a competitive equilibrium for this economy.

(b) Suppose that A(t) = A(0) for all t, and characterize the steady-state equilibrium. Explain why the steady-state capital-labor ratio is independent of θ.

(c) Now assume that A(t) = exp(gt)A(0), and show that a BGP (with constant capital share in national income, and constant and equal rates of growth of output, capital, and consumption) exists only if F takes the Cobb-Douglas form, Y(t) = (A(t)K(t))αL(t)^1-α.

(d) Characterize the BGP in the Cobb-Douglas case. Derive the common growth rate of output, capital, and consumption.