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Unformatted text preview: ECON 121 SQ2: STANDARD TRADE MODEL: TWO GOODS (X&Y), TWO INPUTS (L & K) AND TWO COUNTRIES (A & B) 1.The production functions for goods X and Y are given by X = Lx1/5Kx 4/5 and Y = Ly4/5 Ky1/5 . a. What does it mean to call a good “capital intensive”? Is it possible for a particular good to be both capital intensive and labor intensive? b. Which good is capital intensive?....labor intensive? Justify your answer finding the optimal combinations of inputs for each good assuming TC = $50, w =$2 per L and r= $2 per K. Find the capital‐ to‐labor ratio at each of these optimal combinations. c. How would your answer to part (b) change if TC = $100? d. Show the above results on a graph, showing each good’s expansion path. e. Show that any given w/r, good X is the capital intensive good. f. Suppose we assume that country A’s endowment is L=1 and K =32 and that country B’s endowment is L = 32 and K = 1. Which country is considered labor abundant? (OK, easy question). …capital abundant? What does this imply about the general shape of their respective PPFs? (You can use the intercept “trick”). g. Consistent with the intercepts, suppose we assume the following equations for the PPFs. For A: 256 = X2 + 64Y2 . For B: 256 = 64X2 + Y2. Find the MRTs for each PPF. What does the MRT tells about the shape of the PPF? h. Assume the utility function is U=X0.5 Y0.5 in both countries. Find the pretrade production and consumption values of X and Y for each country. Find the utility at this bundle and the implicit relative price for good X in each country. In which country would expect the higher price of X? Show results on graph. i. Suppose that free trade now becomes possible. Which direction will trade flow and how will the price of X change in each country? Show results on PPF in junction with a supply and demand graph. Indicate all key variables on your graph. 2. Show that in the no‐trade scenario, equilibrium requires that the slope of the IC (MRS ) equals the slope of the PPF (MRT) which equals the slope isorevenue (or CPF or budget line) Px/Py. 3. Suppose when Russia opens to trade, it imports automobiles, a capital‐intensive good. According to our Standard Trade Model, is Russia capital abundant or labor abundant? 4. Suppose there are two inputs—labor and land. Country Z imports manufactured goods and exports agricultural goods. According to our model, we expect Z to labor intensive or land intensive? 5. Explain why the “trade triangles” formed by the heavy shaded and dashed lines must be the same for each country in equilibrium. Y Country A yc yp xc xp X Y Country B Yp Yc Xp Xc X Answers a. We say that X is “capital intensive” if (Kx/Lx ) > (Ky/Ly ) at any given set of input prices. We say that X is “labor intensive” if (Lx/Kx ) > (Ly/Ky ) at any given set of input prices. A particular good cannot be both capital intensive and labor intensive since labor intensity is found by taking the reciprocal of (K/L). It follows that if (Kx/Lx ) > (Ky/Ly ) it true then it follows that (Lx/Kx ) < (Ly/Ky ) must be true. b. To find the optimal input combination for good X: i. Find MRTS for X. MRTSx = slope of isoquant= MPLx/MPKx = (∂X/∂L)/(∂X/∂K) = (a/b)(K/L) = (1/5)/(4/5) KX/LX = ¼ KX/LX ii. Set MRTS = w/r and solve for K. (i.e., set slope of isoquant = slope of isocost) ¼ KX/LX = w/r =2/2 = 1 or KX = 4LX iii. plug this result into the isocost equation and solve for Lx: 50 = 2LX + 2KX = 2LX + 2(4LX) = 10LX LX = 5 iv. Find KX. KX = 4LX = 4(5) = 20. KX/LX = 20/5 = 4. To find the optimal input combination for good Y: i. Find MRTS for Y. MRTSy = slope of isoquant= MPLy/MPKy = (∂Y/∂L)/(∂Y/∂K) = (a/b)(K/L) = (4/5)/(1/5) Ky/Ly = 4 Ky/Ly ii. Set MRTS = w/r and solve for K. (i.e., set slope of isoquant = slope of isocost) 4Ky/Ly = w/r =2/2 = 1 or Ky = 1/4Ly iii. plug this result into the isocost equation and solve for Ly: 50 = 2Ly + 2Ky = 2Ly + 2(1/4Ly) = 2.5Ly Ly = 20 iv. Find Ky. Ky= 1/4Ly = 1/4(20) = 5. Ky/Ly = 5/20 = 1/4. c. Repeat the above steps with TC = 100. You should get for X: LX =10 and KX = 40 and KX/LX = 40/10 = 4. For Y: Ly = 40, Ky= 10. Ky/Ly = 10/40 = 1/4. d. Capital (K) Expansion Path for X (slope = Kx/Lx = 4) 50 40 25 X* Expansion Path for Y (slope = Ky/Ly = 1/4 20 X=15.15 10 5 Y* Y 5 10 20 25 40 50 Labor (L) e. Producer optimization requires that ¼ KX/LX = w/r and 4 Ky/Ly = w/r. Therefore, ¼ KX/LX = 4 Ky/Ly so at any given w/r, KX/LX = 16 Ky/Ly . Thus, KX/LX always exceeds Ky/Ly at any given w/r. f. (K/L)A = 32/1 > (K/L)B = 1/32 so A is capital abundant and B is labor abundant. In country A, L = 1 and K = 32 and in country B, L = 32, K = 1. We can use this information to find the respective intercepts of the PPF for each country. For example, suppose country A only produces good X (so Y = 0). Then, X = Lx1/5 Kx 4/5 = (1)1/3 (32) 4/5 = 16. Now suppose A produces only good Y (so X=0). Then, Y = Ly4./5 Ky1/5 = (1)4/5 (32)1/5 = 2. Using this information I can draw in the following PPF. Notice that the curve is “stretched out” along the horizontal axis. This is because A is capital abundant and X is the capital intensive good. Y Country A 2 16 X Suppose country B only produces good X (so Y = 0). Then, X = Lx1/5 Kx 4/5 = (32)1/3 (1) 4/5 = 2 Now suppose B produces only good Y (so X=0). Then, Y = Ly4./5 Ky1/5 = (32)4/5 (1)1/5 = 16. B’s PPF curve is “stretched out” along the vertical axis. This is because B is capital abundant and X is the capital intensive good. Y 16 Country B 2 X g. For A: 256 = X2 + 64Y2 MRT = fx /fy = 2X/128Y = X/64Y. This tell us the opportunity of cost of X in terms of Y at any given value for X and Y. Notice that as we slide down the curve, X rises and Y falls. Since X is in the numerator and Y is in the denominator that means MRT rises. In other words, the marginal cost of producing X in terms of Y rises as we produce more of good X. Equivalently, this means the PPF is concave. For B: 256 = 64X2 + Y2 MRT = fx /fy = 128X/2Y = 64X/Y. Once again, not as we slide down the PPF, the MRT rises. h. For A: First, we need to find MRS and MRT, set them equal to each other (since we want to be where the IC is tangent to the PPF), and solve for Y. MRS =Y/X and MRT = X/64Y. Y/X = X/64Y 64Y2 = X2 or 8Y = X, Y = 1/8X. Second, plug this result into the PPF and solve for X. 256 = X2 + 64Y2 by substitution 256 = X2 + X2 X = √128 = 8√2 (or 11.31 approx.). Third, find Y. Y = 1/8X = √2 = 1.41. Next, U = (8√2)0.5 (√2)0.5 = 4. The equilibrium price (Px/Py) = MRT = MRS = Y/X = (√2)/(8√2) = 1/8 Y per X. For B: First, MRS =Y/X and MRT = 64X/Y. Y/X = 64X/Y Y2 = 64X2 or Y = 8X. Second, plug this result into the PPF and solve for X. 256 = 64X2 + Y2 by substitution 256 =64X2 + 64X2 X = √2 = 1.41.. Third, find Y. Y = 8X = 8√2 = 11.31. Next, U = (√2)0.5 (8√2)0.5 = 4. The equilibrium price (Px/Py) = MRT = MRS = Y/X = (8√2)/(√2) = 8Y per X. We would expect the higher price of X in country B. Country B is labor abundant and therefore “capital poor”. The relative scarcity of capital in B means that goods that are capital intensive, good X in this case, will be relatively expensive. Y Pre‐trade situation in country A ICU=4 slope = MRT = MRS = (PX/PY) = (1/8)Y per X 2 √2 8/√2 16 X Y slope = MRT = MRS = (PX/PY) = 8Y per X 16 8/√2 ICU=4 √2 2 X i. The price of good X is relatively high in country B compared to country A 8Y/X > (1/8)Y/X. Country B would therefore like to import good X from country A. Likewise, Y is relatively expensive in country A, so A would like to import good Y from country B. As the supply of good X rises in country B this will drive down its price in that country. In country A, the increased competition from B’s consumers will drive the price of X up in A. The new price will be somewhere between 8Y and 1/8Y per X. In other words, 1/8Y per X < (Px/Py)world < 8Y/X and this means that the slope of the isorevenue curve or budget line (recall that it is simply (Px/Py) ) will get flatter in country B and steeper in country A with the following outcomes: Y Country A IC* IC 2 √2 Yp Pre‐trade (slope if 1/3 Y per X) 8/√2 xp 16 X Country A will increase its production of X and decrease its production of Y. It will import good Y and export X (not shown on the above graph). Country B Y IC* yp 16 8/√2 IC after‐trade (slope is something less than 3Y per X) Pre‐trade (slope is 3Y per X) xp √2 2 X Country B will decrease its production of X and increase its production of Y. It will export good Y and import X (not shown on the above graph). Y Y COUNTRY A (capital abundant) COUNTRY B (labor abundant) consumption pt. after trade Yc production point after trade 16 IC* Yp IC consumption pt. after trade Imports of Y exports of Y 8√2 2 Yc IC* √2 IC Yp production point after trade Xc 8√2 Xp 16 X Xp √2 Xc 2 X imports of X exports of X (Px/Py) (Px/Py) (Px/Py) WORLD MARKET FOR X (MCx/MCy)B Smkt = Σ(MCx/MCy) (MCx/MCy)A 8Y/X (Px/Py)mkt (Px/Py)market (1/8)Y/X Dx Dmkt = ΣDi Dx Xc 8√2 Xp X Xp √2 Xc X Xmkt X exports imports 2. Suppose the relative price of X is such that the following situation arises: Y ICU Consumer wants to max utility and so want to be where (Px/Py) = MRS. Yd producer wants to maximize profit and so wants to where (Px/Py) = MRT = (MCx/MCy) Ys Xd XS 16 X At this (Px/Py), we can see that XS > Xd (surplus of X) and that Yd > Ys (shortage of Y). This leads to Px being bid down and Py being bid up. This, in turn, results in (Px/Py) falling. A in this relative price causes the quantity of X supplied to fall (moving up PPF) and the quantity demanded of X to rise. This process will occur until Y Pre‐trade situation in country A ICU slope = MRT = MRS = (PX/PY) Ys=Yd Xs=Xd X At this point there is an equilibrium in both markets, thus (Px/Py) has reached its equilibrium level. 3. According to our model, a country will import the good that is intensive in the input that the country is NOT abundant in. Therefore, we would conclude that Russia is labor abundant since it would be exporting labor intensive goods and importing capital intensive goods. 4. Since we would expect agricultural goods to be land intensive and manufactured goods to be labor intensive, our model would predict that Z is land abundant. 5. The base of the triangle for A represents exports of good X and the base of the triangle for B represents imports of good X. In equilibrium, exports of X by A = imports of X by B. Suppose for example that the base of each triangle was not the same. If, say, it was larger for A, then A would desire to export more of this good than B desires to import at the current price of X. There would therefore be a surplus of X in the world market which would drive (Px/Py) down. As this price falls, A would reduce its exports (base of triangle gets smaller) and B would increase its imports of the now cheaper X (base of B’s triangle increases). The same argument would be made for the height of the triangle which measures imports and exports of Y. In equilibrium, exports of Y = imports of Y. ...
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 Spring '10
 McDevitt

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