---
tags: micro, lecture_note, book
---
$$
% My definitions
\def\ve{{\varepsilon}}
\def\dd{{\text{ d}}}
\newcommand{\dif}[2]{\frac{d #1}{d #2}} % for derivatives
\newcommand{\pd}[2]{\frac{\partial #1}{\partial #2}} % for partial derivatives
\def\R{\text{R}}
$$

# Trade

This note is based on MWG 15.D
## Setting
We use the $2 \times 2$ production model, which means two inputs and two outputs, in the whole section. In this economy, there are $\bar{z}_1, \bar{z}_2$ endowment of two inputs/factors.

### Technology
There are two firms that use two inputs/factors, $z= (z_1, z_2)$, to produce two different outputs. Both firms have a constant return to scale production function
$$f_j(z)=f_j(z_1, z_2), \quad j=1, 2$$

The prices of the final good are exogenous and fixed at $p=(p_1, p_2)$. The prices of input is $w=(w_1, w_2)$, which is endogenous in the model.

### Profit-maximization
We assume all firms are (output and input) price-taker. Firm $j$'s problem is

$$
\max_{z_j \ge 0} \quad p_j f_j(z_j) - w \cdot z, \quad j=1, 2,
$$

where $z_j=(z_{1j}, z_{2j})$ is the input demand of firm $j$.

The cost functions of firm $j$ with factor price $w=(w_1, w_2)$ and output leve $q_j$ is

$$
c(w_1, w_2, q_j), \quad j=1,2
$$

By the CRTS assumption, we can use the unit cost function,

$$
c(w_1, w_2)=c(w_1, w_2, q_j=1), \quad j=1,2
$$

## Equilibrium
Given the fixed output prices $p$, an equilibrium for the factor markets consists of input price vector $w^*=(w_1^*, w_2^*)\gg 0$ and a feasible factor allocation
$$
(z_1^*, z_2^*) = ((z_{11}^*, z_{21}^*), (z_{12}^*, z_{22}^*) ),
$$

such that each firm maximizes profit under $(p, w^*)$ and all the factor markets clear, that is,

$$
z_{l1}^* + z_{l2}^* = \bar{z}_l,\quad l=1, 2
$$

### Solution

Assuming interior solution, by the concavity of the firm's production functions, first-order conditions are both necessary and sufficient for the optimal factor demands. Hence, the factor allocation $(z_1^*, z_2^*)$ and factor price $(w_1^*, w_2^*)$ constitute an equilibrium if and only if:

\begin{align}
p_j \pd{f_j(z_j^*)}{z_{lj}}&= w_l^*, l=1, 2, j=1, 2\\
z_{l1}^* + z_{l2}^* &= \bar{z}_l,\quad l=1, 2
\end{align}

### Equilibrium with cost functions
We can also use cost function $c_j(w, q_j), j=1,2$ to define the equilibrium. The factor price $w^*=(w_1^*, w_2^*)\gg 0$ and output levels $(q_1^*, q_2^*)\gg 0$ constitute an equilibrium if and only if

\begin{align}
&p_j =\pd{c_j(w^*,q_j^*)}{q_{j}}, j=1, 2\\
&\sum_{j} \pd{c_j(w^*,q_j^*)}{w_{l}}= \sum_{j} z_{lj} = \bar{z}_l,\quad l=1, 2,
\end{align}

where the second condition uses Shephard's lemma.

## Factor Intensity Assumption

Fixed the output price $p$, let $a_{11}(w)$ be the demand of factor $1$ for firm/good $1$ at factor price $w$; let $a_{21}(w)$ be the demand of factor $2$ for firm/good $1$ at factor price $w$.

The production of good $1$ is **relative more intensive** in factor $1$ than is the production of good $2$ if

$$
\frac{a_{11}(w)}{a_{21}(w)}> \frac{a_{12}(w)}{a_{22}(w)}
$$

at all factor prices $w=(w_1, w_2)$.

In other words, we evaluate intensity by the input ratio between factor $1$ and $2$.

### Equlilbrium with CRTS

Assume interior equilibrium and the economy produces two goods (So it is not **specialized**), the equilibrium must satisfy,

$$
c_1(w_1, w_2)=p_1, \quad c_2(w_1, w_2)=p_2,
$$

where we use $c_1(w_1, w_2)=c_1(w_1, w_2, q_1=1)$. The equations mean that the input unit cost equals to output unit price.

### Diversification Cone

Given output price $p$, let $\hat{w}=(\hat{w}_1, \hat{w}_2)$ be the factor prices that unit cost equals to unit price for both goods. The **diversification cone** is

$$
\{(z_1, z_2): \frac{a_{11}(\hat{w})}{a_{21}(\hat{w})} > \frac{z_1}{z_2} >\frac{a_{12}(\hat{w})}{a_{22}(\hat{w})}\}.
$$

If the endowment ratio $\bar{z_1}/\bar{z_2}$ is in the cone, then the economy will produce two goods. Otherwise, the economy will specialize in producing one good.

# Graphs

## Edgeworth box

We can represent the allocation of two factors in an Edgeworth box. By the above assumption, the factor demand market is similar to a pure exchange economy. But we need to replace indifference curves with isoquants.


Just like the Edgeworth box for a pure exchange economy, we define **Pareto set** as the intersection of all isoquants. In the above graph, the brown curve is the Pareto set, and the green curves are isoquants.

By the CRTS assumption, the Pareto set must be on one side of the diagonal, or it is the diagonal. It can be shown on the graph. As the below graph, two isoquants intersect at a point below the diagonal. By the CRTS, we can draw a ray from each origin, and the isoquants of each firm should be paralleled on the ray. (Technically, the MRS are the same for each point on the ray) Hence, having two intersections below and above the diagonal is impossible.


## Factor price

We can draw the unit cost functions of two firms on the $w_1-w_2$ plane. The equilibrium is the intersection of two unit cost functions. By the factor intensity assumption, firm $1$ is relatively more intensive on input $1$, so the curve is steeper than firm $2$.(Because a little decrease on $w_1$ is more valuable for firm $1$)

We can also draw the normal line for two curves on $(\hat{w}_1, \hat{w}_2)$, by Shephard's lemma, the slope equals $z_2/z_1$.


## Lerner Diagram/Factor demand

We can draw the isoquants of two firms on the $z_1-z_2$ plane. However, this graph can only tell us the factor price and the factor ratio of each firm, but not the real factor demand.

http://www-personal.umich.edu/~alandear/glossary/figs/Lerner/ld.html

# Theorems

## Factor price equalization theorem
Suppose two goods are tradable, and all countries have identical production technologies and price-taking behavior. Then the (untradable) factor price is equalized across nonspecialized countries.

### Explanation
If the economy is not specialized, then the factor price is only determined by output price and technology. Hence, countries with the same technology and the same output prices must have the same factor prices.

### Graph

By the above assumptions, when output price $p$ is given, the temporary factor price $\hat{w}$ is also decided, and so does the factor demand ratio for each firm. We can draw the factor demand ratio on the Edgeworth box, and the intersection is the allocation of factors. As the below graph, the economy is diversified, and the temporary factor price $\hat{w}$ is the final factor price.


However, if the factor ratio of the endowment is too skew, the two lines of the factor ratio may not intersect. In that case, the economy will specialize in one good. The temporary factor price $\hat{w}$ could not be the final factor price.

## Stolper-Samuelson Theorem
In the $2 \times 2$ production model with the factor intensity assumption, if $p_j$ increases, then the equilibrium price of the factor more intensively used in the production of good $j$ increases, while the price of the other factor decreases (assuming interior equilibria both before and after a price change).

Moreover, the factor price of intensive input is proportionally larger than the increase in the output price of the good, i.e.,

$$
\frac{d w_i}{w_i} > \frac{d p_i}{p_i}.
$$

### Proof

Since the unit input cost must equal to unit output price on equilibrium,

$$
c_1(w_1^*, w_2^*)=p_1, \quad c_2(w_1^*, w_2^*)=p_2,
$$

By the total differentiation,

\begin{align}
d p_1 = \nabla c_1(w^*) \cdot d w = a_{11}(w^*) d w_1 + a_{21}(w^*)d w_2,\\
d p_2 = \nabla c_2(w^*) \cdot d w = a_{12}(w^*) d w_1 + a_{22}(w^*)d w_2,\\
\end{align}

in matrix notation,

$$
d p =\begin{bmatrix} dp_1 \\ dp_2\end{bmatrix}= \begin{bmatrix} a_{11}(w^*) &a_{21}(w^*) \\ a_{12}(w^*) & a_{22}(w^*)\end{bmatrix} \begin{bmatrix} d w_1 \\ d w_2\end{bmatrix} = A \begin{bmatrix} d w_1 \\ d w_2\end{bmatrix}
$$

By the factor intensity assumption, $|A| = a_{11}(w^*)a_{22}(w^*) - a_{12}(w^*)a_{22}(w^*)>0$, and $A$ is invertible. We have

$$
A^{-1} = \frac{1}{|A|} \begin{bmatrix} a_{22}(w^*) &-a_{21}(w^*) \\ -a_{12}(w^*) & a_{11}(w^*)\end{bmatrix} ,$$

and
$$ A^{-1} d p = dw,
$$

Assume the price of good $1$ increase, let $dp =\begin{bmatrix} 1 \\ 0\end{bmatrix}$, we have

$$
\begin{bmatrix} d w_1 \\ d w_2\end{bmatrix} = \frac{1}{|A|} \begin{bmatrix} a_{22}(w^*) \\ -a_{12}(w^*) \end{bmatrix},
$$

where $d w_1>0, d w_2 <0$.

To derive a strong version of the theorem, let

$$
\frac{d p_1}{p_1} =1 \implies dp =\begin{bmatrix} p_1 \\ 0\end{bmatrix}
$$

we have

$$
d w_1= \frac{1}{|A|} a_{22}(w^*) p_1 \implies \frac{d w_1}{w_1}= \frac{1}{|A|} \frac{1}{w_1} a_{22}(w^*) p_1
$$

By the competitive equilibrium, $p_1=c_1$. We have

\begin{align}
\frac{d w_1}{w_1} &=\frac{1}{a_{11}(w^*)a_{22}(w^*) - a_{12}(w^*)a_{22}(w^*)} \frac{1}{w_1} a_{22}(w^*) p_1\\
&> \frac{1}{a_{11}(w^*)a_{22}(w^*) } \frac{1}{w_1}a_{22}(w^*) p_1\\
&=\frac{1}{a_{11}(w^*) } \frac{1}{w_1} p_1 = \frac{c_1}{a_{11}(w^*)w_1 }
\end{align}

Assuming interior solution, we have $c_1 = a_{11}(w^*)w_1 + a_{21}(w^*)w_2$. Hence,
$$
\frac{d w_1}{w_1} >1.
$$

## Rybczynski Theorem
In the $2 \times 2$ production model with factor intensity assumption, if the endowment of a factor increases, then the production of the good that uses this factor relatively more intensively increases, and the production of the other good decreases (assuming interior equilibria both before and after the change of endowment).

Moreover, the proportional increase in the production of the good that uses the increased factor relatively more intensively is greater than the proportional increase in the endowment of the factor. Let $y$ be the output of good intensive on $z_l$, we have

$$
\frac{d y}{y} >\frac{d \bar{z}_l}{\bar{z}_l}.
$$

### Explanation

It is easily to see it on the Edgeworth box. When the one factor increases, the economy allocates more factors to the firm that is relatively more intensive on that factor. We can also observe that the factor allocate to the other firm decrease, hence the the proportional increase in the production of the good that uses the increased factor relatively more intensively is greater than the proportional increase in the endowment of the factor.


## Heckscher-Ohlin Theorem
Suppose there are two consumption goods, two factors, and two countries $A$ and $B$. Each country has technologies as in the $2 \times 2$ production model. Each country has the same technologies. The production of good $1$ is relatively more intensive in factor $1$. Country $A$ is relatively better endowed with factor $1$. Two countries have the same increasing and homogeneous utility functions.

Suppose that factors are not mobile and each country is a price taker with respect to international prices. If neither county is specialized and the international markets are clear. Country $A$ must export good $1$ and import good $2$.

### Explanation

By the $2 \times 2$ production model, country $A$ will produce relatively more good $1$. By the homogeneous utility function, the consumption ratio of good $1,2$ is the same for each country. To clear the market, country $A$ must export good $1$ and import good $2$.