Wealth in the utility function

Question

Suppose there is a representative household of unit mass who lives forever. Preferences are given as: $$\sum\beta^tu(c_t,k_{t-1})$$ Technology is given as: $$k_{t+1}=AF(K_t,L_t)+(1-\delta)K_t-c_t$$ The production function has constant returns to scale.

The household’s dynamic program is: $$V (k, z) = \max u(c, z) + \beta V (k', k) + \lambda [R k + w − c − k']$$ or alternatively $L=\beta^tU(c_t,k_{t-1})+\lambda_t[R_t k_t + w_t − c_t − k_{t+1}]$

FOC: $U_c(c,z)=\beta R'U_c(c',z')+\beta^2U_z(c'',z'')$, where $z'=k$.

My question is why is steady-state capital stock written like this? $$1=\beta[f'(k)+1-\delta]+\beta^2\frac{u_z(f(k)-\delta k,k)}{u_c(f(k)-\delta k,k)}$$

Answer 1

Two points:

1. As Richard Hardy points out in the comment, there is not really any depiction of wealth in the question.

2. I am very curious which model this is. I can not think of any models that include lagged capital in the utility function. The form of the Euler is however very similar to a Cash-in-Advance model (which is interesting).

Following the way you have written the Bellman, but getting rid of the multiplier, since according to the problem, the constraint binds.

$V[k_t, k_{t-1}] = U[AF[k_t] +(1-\delta)k_t, k_{t-1}]+\beta V[k_{t+1}, k_t]$

FOC(with respect to $(k_{t+1})$):

$U_1[c_t, k_{t-1}]=\beta V_1[k_{t+1}, k_t]$

Envelope Condition 1 (with respect to $k_t$):

$V_1[k_t, k_{t-1}] = U_1[c_t, k_{t-1}](AF_1[k_t] + 1-\delta) + \beta V_2[k_{t+1}, k_t]$

Envelope Condition 2 (with respect to $k_{t-1}$):

$V_2[k_t, k_{t-1}] = U_2[c_t, k_{t-1}]$

Iterate Envelope conditions forward by one period and use the FOC to get (assuming finite consumption):

$U_1[c_t, k_{t-1}]/U_1[c_{t+1}, k_{t}] = \beta (AF_1[k_{t+1}] + 1-\delta) + \beta^2 U_2[c_{t+2}, k_{t+1}]/U_1[c_{t+1}, k_{t}]$

Now impose steady state (denoting at steady state $k_{t+1}=k_t=k$) and you have:

$1 = \beta (AF_1[k] + 1-\delta) + \beta^2 \frac{U_2[AF[k]-\delta k, k]}{U_1[AF[k]-\delta k, k]} $