## Dynamic Heckscher-Ohlin Model

This problem is from Tim Kehoe. It showed up on the trade prelims in 2013, 2015 (fall), 2016 (fall) and 2017(spring).

### Setup

Consider a two-sector growth model in which the representative consumer has the utility function $$\sum_{t=0}^\infty \beta^t \log (c_{1t}^{a_1}, c_{2t}^{a_2})$$ The investment good is produced according to $$k_{t+1} = d x_{1t}^{a_1} x_{2t}^{a_2}$$ Feasible consumption/investment plans satisfy the feasibility constraints $$c_{1t}+x_{1t}=\phi_{1}(k_{1t},l_{1t})=k_{1t}$$ $$c_{2t}+x_{2t}=\phi_{2}(k_{2t},l_{2t})=l_{2t}$$ where $$k_{1t}+k_{2t}=k_{t}$$ $$l_{1t}+l_{2t}=1$$ The initial value of $k_t$ is $\bar{k_0}$. All of the variables specified above are in per capita terms. There is a measure $L$ of consumer/workers.

### Problem (a): Setup Social Planner's Problem

Write out a social planner’s problem for this economy. Explain how the solution to this social planner’s problem is related to that of the one-sector social planner’s $$\sum_{t=0}^\infty \beta^t \log c_t$$ $$\text{s.t.: } c_t + k_{t+1} = dk_t^{a_1}$$ $$c_t, k_t \geq 0$$ $$k_0 = \bar{k_0}$$ [You can write done a proposition or propositions without providing a proof or proofs, but be sure to carefully relate the variables in the two-sector model to the variables in the one-sector model.]

##### Solution:
The social planner's problem for the two sector economy is to choose $C \equiv\{c_{1t}, c_{2t}, x_{1t}, x_{2t}, k_{1t}, k_{2t} \}_t$ to solve
$$\max_{C}\sum_{t=0}^{\infty}\beta^{t}\log\left(d c_{1t}^{a_{1}}c_{2t}^{a_{2}}\right)$$ subject to the constraints: \begin{gather} c_{1t}, c_{2t}, x_{1t}, x_{2t} \geq 0, \quad \forall t \tag{NonNeg}\\ c_{1t} + x_{1t} = k_{1t} \quad \forall t \tag{Resource 1} \\ c_{2t} + x_{2t} = 1 \quad \forall t \tag{Resource 2} \\ k_{1,t+1} + k_{2,t+1} = d x_{1t}^{a_1} x_{2t}^{a_2} \tag{Capital} \\ k_{10} \quad given \tag{initial k} \\ \end{gather}

Assume that $a_1 + a_2 = 1$

Proposition 1: If $C \equiv\{c_{1t}, c_{2t}, x_{1t}, x_{2t}, k_{1t}, k_{2t} \}_t$ is a solution to the two-sector social planner's problem, then $\{c_t, k_t\}_t$ is the solution to the one-sector social planner's problem if for each time period $t$: $$c_t = d c_{1t}^{a_1} c_{2t}^{a_2}$$ $$k_{t+1} = k_{1,t+1} = d x_{1t}^{a_1} x_{2t}^{a_2}$$