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Economics 3563: International Trade Data Assignment 1 Chapter 6 Example 1 Initial Setup Before we can compare the real wages in autarky to the real wages in free trade, we need to find the relative quantities so that we can find relative prices. In the US the autarky amount apparel produced, Q A , is 12 million. The autarky amount of plastics produced, Q P , is 16 million. This makes the relative quantity, Q A /Q P , equal to 0.75. The relative demand (RD) curve is RD:Q A Q P = 49 −100 P A P P Rearranging and then substituting in the quantities we get P A P P = 0 .49 −(0.01) Q A Q P = 0 .49 −(0.01)(0 .75) = 0 .4825 In China the autarky amount of apparel produced is 240 million and the autarky amount of plastics produced is 20 million. The relative quantity is 12.

Plugging these into the RD curve yields P A P P = 0 .49 −(0.01)(12) = 0 .37 The world total combines output from both countries. The world total output of apparel is 240 million plus 12 million for a total of 252 million. The world total of plastics is 16 million plus 20 million for a total of 36 million. The world relative demand is 252/36 = 7. Plugging this into RD we get the world relative price P A P P = 0 .49 −(0.01)(7) = 0 .42 1 2 Setting up the Systems of Equations Our ultimate goal is to compare real wages (i.e., nominal wage divided by price) in autarky with the real wages in free trade. We will make this comparison for both factors (unskilled labor and skilled labor). We will also make this comparison for the US and then for China. Because there are two goods and two factors, there are four unknowns: w U /P A , w U /P P , w S /P A , w S /P P .

2.1 United States Since all markets are competitive, firms make zero profit in equilibrium. Hence, P A = 2 w U + 1 w S (2.1) P P = 3 w U + 3 w S (2.2) where equation 2.1 is the zero-profit condition for the apparel industry, while equation 2.2 is the zero-profit condition for the plastics industry. Divide each of equations 2.1 and 2.2 by P A and P P P A P A = 1 = 2 w U P A + 1 w S P A (2.3) P P P A = 2 .075 = 3 w U P A + 3 w S P A (2.4) P A P P = 0 .4825 = 2 w U P P + 1 w S P P (2.5) P P P P = 1 = 3 w U P P + 3 w S P P (2.6) 2.2 China The zero-profit conditions are the same as the US. The productivities are also the same. Thus the system of equations for China is identical to the system of equations for the US except that the relative autarky price is different P A P A = 1 = 2 w U P A + 1 w S P A (2.7) P P P A = 2 .7027 = 3 w U P A + 3 w S P A (2.8) P A P P = 0 .37 = 2 w U P P + 1 w S P P (2.9) P P P P = 1 = 3 w U P P + 3 w S P P (2.10) 2.3 Free Trade When we open the world to free trade, both the US and China end up with the same relative price. Thus, the free-trade system of equations as well as the 2 solutions to the system of equations is the same for both countries.

P A P A = 1 = 2 w U P A + 1 w S P A (2.11) P P P A = 2 .3810 = 3 w U P A + 3 w S P A (2.12) P A P P = 0 .42 = 2 w U P P + 1 w S P P (2.13) P P P P = 1 = 3 w U P P + 3 w S P P (2.14) 3 Solving the Systems We have three systems of equations (actually four, but, again, the free-trade system is the same for the US and China). Each of the three systems of equations is a system of four equations in four unknowns: System 1 is equations 2.3-2.6, system 2 is equations 2.7-2.10, and system 3 is equations 2.11-2.14. We could solve all these equations by hand, but doing so is cumbersome and can be easily done with a computer (see the accompanying Excel spreadsheet). To give you a sense of how the computer solves these systems, we write out each system in matrix form. Again, our ultimate goal is to solve for the real factor payments (i.e., real wages): w U /P A , w U /P P , w S /P A , w S /P P for each country in each state of the world (i.e., autarky and free trade). Writing system 1 (US-autarky, equations 2.3-2.6) in matrix form we have 2 6 6 4 1 2 .0725 0 .4825 1 3 7 7 5 4 × 1 | {z } y = 2 6 6 4 2 1 0 0 3 3 0 0 0 0 2 1 0 0 3 3 3 7 7 5 4 × 4 | {z } X 2 6 6 6 4 w U P A w S P A w U P P w S P P 3 7 7 7 5 4 × 1 | {z } a (3.1) In matrix form we have y= Xa . In order to solve for the avector (i.e., the four unknowns), we have to invert the Xmatrix, X − 1 , and premultiply both sides by this inverted matrix. Before we do that, however, note that the rows in the Xmatrix correspond to each equation, while the columns correspond to the unknowns. The numbers in the matrix come from the coefficients in the equations. But where do the zeros come from? Each of the four unknowns is actually in every equation but in some equations is multiplied by zero. For example, take equation 2.3:

1 = 2w U P A + 1 w S P A + 0 w U P P + 0 w S P P And equation 2.6:

1 = 0w U P A + 0 w S P A + 3 w U P P + 3 w S P P 3 And so on.

Inverting the matrix yields X − 1 y = a. Writing out the matrix multiplication we arrive at the following:

2 6 6 4 1 −0.33 0 0 − 1 0 .67 0 0 0 0 1 −0.33 0 0 −1 0 .67 3 7 7 5 4 × 4 | {z } X − 1 2 6 6 4 1 2 .0725 0 .4825 1 3 7 7 5 4 × 1 | {z } y = 2 6 6 6 4 w U P A w S P A w U P P w S P P 3 7 7 7 5 4 × 1 | {z } a (3.2) Excel will do the matrix inversion for you (MINVERSE function). Let’s multiply out X − 1 y , which will also be done in Excel (MMULT function). Doing the matrix multiplication we arrive at the solution to the system, i.e., we solve for the real wages in the US in autarky 2 6 6 4 0 .3092 0 .3817 0 .1492 0 .1842 3 7 7 5 = 2 6 6 6 4 w U P A w S P A w U P P w S P P 3 7 7 7 5 Doing this same process for China in autarky and then again for both coun- tries in free trade yields the following table with all of the solutions US Autarky China Autarky Free T rade w U /P A 0.3092 0.0991 0.2063 w S /P A 0.3817 0.8018 0.5873 w U /P P 0.1492 0.0367 0.0867 w S /P P 0.1842 0.2967 0.2467 Again, the accompanying Excel spreadsheet has these calculations.

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