Question

Recursive competitive Equilbrium. Dynamic Programming with "Guess and Check"

Consider the following infinitely lived representative agent economy. The agents have preferences over consumption sequences given by. ∑β^tU(c_t) ( the sum is from t=0 to t= ∞) , 0 < β <1,
where U is strictly increasing and strictly concave. The agents supply one unit of labor inelastically. The aggregate production function is F(K,N) = K^α N^1−α , 0 <α <1
Capital stock fully depreciates in each period, K_t+1 = I_t , where I_t is investment. There is no uncertainty.

The agent’s problem in the recursive competitive equilibrium is to solve V (k,K) = max _c,y [lnc + βV{y,H (K)}]
subject to c + y ≤ R(K)k +W (K),
where R(K) = αK^α−1(the rental rate on capital for the period) and W (K) = (1−α )K^α (the wage rate for the worker. Note that N=1 defined above, so the wage is the deveritive of the production function wrt labor (n), but then N=1 is subbed back in) are the pricing functions, and H (K) is the perceived law of motion for the aggregate capital stock.

Guess that the value function and the perceived law of motion for aggregate capital take the form
V (k,K) = a0 + a1 lnK + a2 ln( a3 + k/K) , H (K) = hK^α .
Verify that the value function satisfies the guess.
Solve for the coefficients a0 ,a1,a2 and a3 in terms of the constant h and the underlying parameters.

Verify that the aggregate law of motion for capital satisfies the guess. Find the constant h such that the expectations about the law of motion for the aggregate capital are rational.

Obtain the coefficients a0 ,a1,a2 and a3 in terms of the underlying parameters only (i.e. by substituting out the constant h ).

Verify that the recursive competitive equilibrium is efficient, i.e. that the value function satisfies V(K,K) = (1/1− β)[ ln(1−αβ ) + (αβ/1−αβ)*lnαβ] + (α/1−αβ)lnK, which is the value function in the planning problem.