# Social Planner problem: two period

The production function is $$F(K_t,N_t)=AK_t^\alpha N_t^{1-\alpha}$$ and depreciation $$(\delta)$$ is equal to 1. The given preferences are as follows: $$U(c_1,l_1,c_2,l_2)=\gamma log(c_1)+(1-\gamma)log(l_1)+\beta[\gamma log(c_2)+(1-\gamma)log(l_2)]$$

From which $$\gamma\in(0,1)$$,$$\beta\in(0,1)$$. We assume $$\forall t:l_t + n_t ≤ 1$$ and all non-negativities on all variables. How would we solve the Social Planner?

Since $$\delta=1$$ we would have the constraint for period 1: $$c_1+k_2=AK_1^\alpha N_1^{1-\alpha}$$, from which $$c_1=AK_1^\alpha N_1^{1-\alpha} -k_{2}$$. For period 2, $$c_2=AK_2^\alpha N_2^{1-\alpha}$$ since we don't have anymore capital for next period. Now we need to find the allocation $$\{c_1,c_2,l_1,l_2\}$$ by doing the Lagrangian.

$$U(c_1,l_1,c_2,l_2)=\gamma log(c_1)+(1-\gamma)log(l_1)+\beta[\gamma log(c_2)+(1-\gamma)log(l_2)]-\lambda_1[c_1-AK_1^\alpha N_1^{1-\alpha} -k_{2}]-\lambda_2[c_2-AK_2^\alpha N_2^{1-\alpha}]$$

Finding F.O.Cs wrt $$c_1,c_2,k_2:$$

{$$c_1$$}: $$\frac{\gamma}{c_1}=\lambda_1$$

{$$c_2$$}: $$\frac{\beta\gamma}{c_2}=\lambda_2$$

{$$k_2$$}: $$\lambda_2A\alpha k_2^{\alpha-1}N_2^{1-\alpha}=\lambda_1$$

Generating $$\frac{\lambda_1}{\lambda_2}$$ we would get $$c_2=A\alpha k_2^{\alpha-1}N_2^{1-\alpha}c_1\beta$$

That we substitute in the original constraint to get $$k_2=c_1\alpha\beta$$ to which we substitute in $$k_2+c_1=Ak_1(1-l_1)^{1-\alpha}$$ Some algebra and we get $$c_1=\frac{Ak_1^\alpha(1-l_1)^{1-\alpha}}{\alpha\beta+1}$$ $$c_2=k_2^{\alpha}N_2^{1-\alpha}$$

For $$l_t$$ we substitute $$c_1,c_2$$ in the FOCs that equal each $$\lambda$$ respectively along with the FOCs of $$l_1,l_2$$ and get the following, again after the algebra: $$l_1=\frac{1-\gamma}{\gamma(\alpha\beta+1)(1-\alpha)+1-\gamma}$$ $$l_2=\frac{1-\gamma}{\gamma(1-\alpha)(1-\alpha)+1-\gamma}$$

Phew! This was hella lot of algebra and I don't know if I made a mistake or not. I hope this is correct but feel free to point my errors.

• Define the Arrow-Debreu equilibrium for this economy. Find the equilibrium prices and quantities.

Besides the market clearing and prices and allocations conditions how do we continue from here :)

• Is there a relationship between $n_t$ and $N_t$? The planner's problem usually involves maximizing utility subject to resource constraint(s). So you should determine what those are. Feb 24 at 14:29
• So I guess $n_t=1-l_t$ to which we substitute in $c_t=Ak_t^\alpha (1-l_t)_t^{1-\alpha} -k_{t+1}$ and from this we substitute $c_1$ and $c_2$ in the utility? I'm a bit confused with all of this :\ Feb 24 at 16:21
• The problem you provided seems under-specified. For example, where does $K_1$ come from? What's the aggregation rule that takes lower case variables ($c_t,n_t,k_t$) to the upper case variables ($C_t,N_t,K_t$)? Feb 26 at 2:46
• For Arrow-Debreu equilibrium, you need to incorporate market-clearing conditions in each period: wages should clear the labor market, rents should clear the capital market, and prices should clear the goods market. Feb 26 at 2:55
• For AD would we have the following: max $u(c_1,l_1)+\beta u(c_2,l_2)$ s.t $p_1(c_1+k_2)\leq p_1(r_1k_1+w_1n_1)$ and $p_2(c_2)\leq p_2(r_2k_2+w_2n_2)$? And then again with the Lagrangian by finding focs w.r.t $c_t,l_t,k_2$? Feb 26 at 11:14