Problem 1.

Consider two countries, indexed by $j \in\{1,2\}$. The representative household in each of these two countries inelastically supplies $H_j>0$ units of labor, for $j \in\{1,2\}$. Labor can only be used domestically. There are two tradable goods, indexed by $n \in\{1,2\}$. And there is one non-tradable good in each of the two countries, indexed by $n=0$ for both countries. Everyone has the same preferences over consumption vectors $\left(c_0, c_1, c_2\right)$, determined by the utility function $\mathcal{U}\left(c_0, c_1, c_2\right)=c_0^{\beta_0} c_1^{\beta_1} c_2^{\beta_2}$. The parameters $\beta_n$ are strictly positive and $\sum_{n=0}^2 \beta_n=1$ Country $j$ has $A_j>0$ units of land. It can use $L_{0, j} \geq 0$ units of labor to produce $y_{0, j}=A_j^{1-\alpha} L_{0, j}^a$ units of its non-tradable good, where $\alpha \in(0,1)$ is a parameter. Country $j$ can also use $L_j \geq 0$ units of labor to produce $y_j=z_j L_j$ units of good $j$. Country 1 cannot produce good 2 and country 2 cannot produce good 1. Write $w_j$ for the wage and $p_{0, j}$ for the price of the non-tradable good in country $j \in$ $\{1,2\}$. The prices of the two tradable goods are $p_1$ and $p_2$, respectively. a. Give the demand curves for the consumption goods $n \in\{0,1,2\}$ in each of the two countries. You may use what you know about the preferences specified in this question, without deriving the result from scratch. b. In any equilibrium, what has to be true about the relative prices $p_1 / w_1$ and $p_2 / w_2$ ? What does profit maximization in the non-tradable goods sector tell you about the factor shares $w_j L_{0, j} /\left(p_{0, j} y_{0, j}\right)$ for $j \in\{1,2\}$ ? c. Use non-tradable goods market clearing and the definition of income as the total value of goods sold to explain why $$\frac{p_{0, j} y_{0, j}}{p_j y_j}=\frac{\beta_0}{1-\beta_0} .$$ Combine this with your answer to $\mathbf{b}$ and labor market clearing to determine $L_{0, j}$ and $L_j$. d. Use the market clearing conditions for tradables to determine $p_1 / p_2$. Determine the wage ratio $w_1 / w_2$. e. Find $p_{0,1} / p_{0,2}$, the real exchange rate, and the ratio of real per-capita income across the two countries.

Proof .

a. The demand functions for each good $n$ in each country $j$ can be derived from maximizing the utility function subject to the budget constraint:

$\max {c 0, c 1, c 2} c_0^{\beta_0} c_1^{\beta_1} c_2^{\beta_2}$ s.t. $p{0, j} c_0+p_1 c_1+p_2 c_2=w_j H_j+p_{0, j} y_{0, j}$.

The resulting demand functions are:

$c_{n, j}=\frac{\beta_n}{\sum_{m=0}^2 \beta_m}\left(w_j H_j+p_{0, j} y_{0, j}-p_n c_n\right)$, for $n \in{0,1,2}$

b. In equilibrium, the relative prices of the tradable goods must satisfy the law of one price, meaning that $p_1 / p_2 = w_2 / w_1$ because goods 1 and 2 cannot be traded across countries. Profit maximization in the non-tradable goods sector implies that the factor share of labor $L_{0,j}$ in country $j$ is given by:

$\frac{w_j L_{0, j}}{p_{0, j} y_{0, j}}=\frac{1-\alpha}{\alpha}$

c. Non-tradable goods market clearing implies that the total value of non-tradable goods produced equals the total value of non-tradable goods consumed in each country:

$p_{0, j} y_{0, j}=c_{0, j}$, for $j \in{1,2}$

Total income in country $j$ is given by $w_j H_j + p_{0,j}y_{0,j} + p_1y_{1,j} + p_2y_{2,j}$. Using the budget constraint and the demand functions, we can write total income as:

$w_j H_j+p_{0, j} y_{0, j}+p_1 y_{1, j}+p_2 y_{2, j}=\left(\sum_{n=0}^2 \frac{\beta_n}{\sum_{m=0}^2 \beta_m}\right)\left(w_j H_j+p_{0, j} y_{0, j}\right)+\frac{\beta_1}{\sum_{m=0}^2 \beta_m} p_1 y_{1, j}+\frac{\beta_2}{\sum_{m=0}^2 \beta_m} p_2 y_{2, j}$.

Simplifying and using non-tradable goods market clearing, we get:

$\frac{p_{0, j} y_{0, j}}{p_1 y_{1, j}+p_2 y_{2, j}}=\frac{\beta_0}{1-\beta_0}$.

Combining the two previous equations, we have:

$\frac{p_{0,1} A_1^{1-\alpha} L_{0,1}^a}{w_1}=\frac{p_{0,2} A_2^{1-\alpha} L_{0,2}^a}{w_2} \Rightarrow \quad \frac{p_{0,1}}{p_{0,2}}=\frac{w_1}{w_2}\left(\frac{A_2}{A_1}\right)^{1-\alpha / a}$