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# 经济代写|国际经济学代写International Economics代考|Derivation of the Edgeworth Box Diagram and Production Frontiers

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## 经济代写|国际经济学代写International Economics代考|Derivation of the Edgeworth Box Diagram and Production Frontiers

We will now use the knowledge gained from Figure 3.8 to derive the Edgeworth box diagram and, from it, the production frontier of each nation. This is illustrated in Figure 3.9 for Nation 1 and in Figure 3.10 for Nation 2.

Our discussion will first concentrate on the top panel of Figure 3.9. The dimensions of the box in the top panel reflect the total amount of $L$ (measured by the length of the box) and $K$ (the height of the box) available in Nation 1 at a given time.

The lower left-hand corner of the box $\left(O_X\right)$ represents the zero origin for commodity $\mathrm{X}$, and X-isoquants farther from $O_X$ refer to greater outputs of $\mathrm{X}$. On the other hand, the top right-hand corner $\left(O_Y\right)$ represents the zero origin for commodity $\mathrm{Y}$, and $\mathrm{Y}$-isoquants farther from $O_Y$ refer to greater outputs of Y.

Any point within the box indicates how much of the total amount of labor available $(L)$ and how much of the total amount of capital available $(K)$ are used in the production of $\mathrm{X}$ and Y. For example, at point $A, L_A$ and $K_A$ are used to produce $50 \mathrm{X}$, and the remaining quantities, or $L-L_A$ and $K-K_A$, are used in the production of $60 \mathrm{Y}$ (see Figure 3.9).

By joining all points in the box where an $\mathrm{X}$-isoquant is tangent to a $\mathrm{Y}$-isoquant, we get the nation’s production contract curve. Thus, the contract curve of Nation 1 is given by the line joining $O_X$ to $O_Y$ through points $A, F$, and $B$. At any point not on the contract curve, production is not efficient because the nation could increase its output of commodity without reducing its output of the other.

For example, from point $Z$ in the figure, Nation 1 could move to point $F$ and produce more of $X$ (i.e., $95 \mathrm{X}$ instead of $50 \mathrm{X}$ ) and the same amount of $Y$ (both $Z$ and $F$ are on the isoquant for $45 \mathrm{Y}$ ). Or Nation 1 could move from point $Z$ to point $A$ and produce more of $Y$ (i.e., $60 \mathrm{Y}$ instead of $45 \mathrm{Y}$ ) and the same amount of $X$ (both $Z$ and $A$ are on the isoquant for $50 \mathrm{X})$. Or Nation 1 could produce a little more of both $\mathrm{X}$ and $\mathrm{Y}$ and end up on the contract curve somewhere between $A$ and $F$. (The isoquants for this are not shown in the figure.) Once on its contract curve, Nation 1 could only expand the output of one commodity by reducing the output of the other. The fact that the contract curve bulges toward the lower right-hand corner indicates that commodity $\mathrm{X}$ is the $L$-intensive commodity in Nation 1 .

## 经济代写|国际经济学代写International Economics代考|Some Important Conclusions

The movement from point $A$ to point $B$ on Nation 1’s contract curve (see Figure 3.9) refers to an increase in the production of $\mathrm{X}$ (the commodity of its comparative advantage) and results in a rise in the $K / L$ ratio. This rise in the $K / L$ ratio is measured by the increase in the slope of a straight line (not drawn) from origin $O_X$ to point $B$ as opposed to point $A$. The same movement from point $A$ to point $B$ also raises the $K / L$ ratio in the production of $Y$. This is measured by the increase in the slope of a line from origin $O_Y$ to point $B$ as opposed to point $A$.

The rise in the $K / L$ ratio in the production of both commodities in Nation 1 can be explained as follows. Since $\mathrm{Y}$ is $K$ intensive, as Nation 1 reduces its output of $\mathrm{Y}$, capital and labor are released in a ratio that exceeds the $K / L$ ratio used in expanding the production of $X$. There would then be a tendency for some of the nation’s capital to be unemployed, causing the relative price of $K$ to fall (i.e., $P_L / P_K$ to rise).

As a result, Nation 1 will substitute $K$ for $L$ in the production of both commodities until all available $K$ is once again fully utilized. Thus, the $K / L$ ratio in Nation 1 rises in the production of both commodities. This also explains why the production contract curve is not a straight line but becomes steeper as Nation 1 produces more $\mathrm{X}$ (i.e., it moves farther from origin $O_X$ ). The contract curve would be a straight line only if relative factor prices remained unchanged, and here, factor prices change. The rise in $P_L / P_K$ in Nation 1 can be visualized in the top panel of Figure 3.9 by the greater slope of the common tangent to the isoquants at point $B$ as opposed to point $A$ (to keep the figure simple, such tangents are not actually drawn). We will review and expand these results in the appendix to Chapter 5 , where we prove the factor-price equalization theorem of the Heckscher-Ohlin trade model.
Problem Explain why as Nation 2 moves from point $A^{\prime}$ to point $B^{\prime}$ on its contract curve (i.e., specializes in the production of $Y$, the commodity of its comparative advantage), its $K / L$ ratio falls in the production of both $\mathrm{X}$ and Y. (If you cannot, reread Section A3.4.)

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