Economics 45 International Theory and Policy Practice Problem Set 3 Fall 007 Suggested Answers. (a) Both country s in this question have the same preferences and the same technologies. The basis for trade in this case is due to the difference in endowments. To find the equilibrium free trade price we will need to follow the same procedure as in problem set, but for the entire world rather than just one country. i. Find the world demand functions: Since both country and country have the same preferences, we know by symmetry with question 3.b) that we can write country s demand for each good as: D (p, p, I ) = D (p, p, I ) = p 0.5 I + p p 0.5 p 0.5 I + p p 0.5 where we now let the superscript refer to the country and the subscript refer to the good in question. If we let I w, D w, Dw denote world income, world demand for good and world demand for good, using our solution from question 3.b) we can write world demand as: D w (p, p, I w ) = D w (p, p, I w ) = ii. Finding world supply functions: p 0.5 I w + p p 0.5 p 0.5 I w + p p 0.5 Following the same strategy as previously we know that in country SK = K + K = 6 and S L = L + L = 4. Using the conditional factor demands we can solve for y and y as before: y = 0.8 y = 0.6. We can then define total world production of each good, y w and yw as: y w = y + y = 0.7 + 0.8 =.5 y w = y + y = 0.4 + 0.6 =
iii. Finding equilibrium prices Setting total world supply equal to total world demand as in question 3.b) and letting p = we can solve for p =.383. (b) Following the exact same procedure as in question 3.c) we can find that r = 0.067 and w = 0.35. Since the production functions are identical in each country, the factor prices will be the same in each country. (c) To calculate the necessary utility levels we need to find the autarky prices in country, the income levels in both countries and the indirect utility function for each country. i. Calculate autarkic equilibrium prices in country Following the same procedure as in question 3.b) we can find that the autarkic equilibrium prices for country are p =.59, r 0.074 and w 0.78. ii. Calculate equilibrium incomes Country Autarkic Income, IA = r A S K + w A S L = 0.0435(5) + 0.369(3) =.345 Free Trade Income, IF = r F SK + w F SL = 0.067(5) + 0.35(3) =.535 Country Autarkic Income, IA = r A S K + w A S L = 0.074(6) + 0.78(4) =.5566 Free Trade Income, IF = r F SK + w F SL = 0.067(6) + 0.35(4) =.630 iii. Calculate the utility levels Country U (c, c ) = (c. + c. ) (. ) = ( ( I p 0.5 ). ( + p p 0.5 + Inserting, the equilibrium prices and income we find that: U A(c, c ) = 7.05 Under free trade prices and income we calculate that: U F (c, c ) = 7. I p 0.5 + p p 0.5 ). ).
Country Repeating the process for country we find: U A(c, c ) = 0.03 U F (c, c ) =.5 Clearly, U A < U F for each country. That is, both countries gain from trade.. Recall the following autarkic values in Country from Problem Set : p A =.565 w A = 0.3694 r A = 0.0435 The income of Consumer in Country in autarky is I A = 5r A = (5)(0.0435) = 0.75. So his/her autarkic consumptions are: d A = 0.75 = 0.48 da + (.565).5 = and his/her autarkic utility is: U A = (0.48. + 0.0656. ) /. =.7993. 0.75 (.565.5 )( + (.565).5 ) = 0.0656 The income of Consumer in Country in autarky is I A = 3w A =.08. So his/her autarkic consumptions are: and his/her autarkic utility is: d A = 0.585 d A = 0.3343 U A = 4.67. Recall the following free trade values from Assignment : p F =.383 w F = 0.349 r F = 0.067 The income of Consumer in Country in trade is I F = 5r F = (5)(0.067) = 0.3085. So his/her free trade consumptions are: d F = 0.3085 = 0.605 df + (.383).5 = and his/her trade utility is: U F = (0.605. + 0.070. ) /. = 4.. 0.3085 (.383.5 )( + (.383).5 ) = 0.070 The income of Consumer in Country in trade is I F = 3w F = 0.9447. So his/her free trade consumptions are: d F = 0.495 d F = 0.705 3
and his/her trade utility is: U F = 9.558. Since U A < U F, consumer prefers trade and since U A > U F, consumer prefers autarky. This occurs because the real income of consumer is higher in trade than in autarky while the reverse is true for consumer. Consumer s real income is higher in trade because when the countries open for trade, the relative price of good to good rises in Country (because that country has a comparative advantage in production of good ). This causes an increase in demand for factor which is used intensively in production of good which puts upward pressure on the price of factor which is the source of consumer s income. The reverse happens for good and the price of factor. 3. (a) After solving for the conditional factor demands we can write the full employment conditions in each country as follows: 7X H S + Y H S = 0 3X H S + 8Y H S = 30 7X F S + X F S = 0 3X F S + 8Y F S = 0 We can solve the full employment conditions to find the supply of each good in each country. Now since aggregate income equals national product, aggregate demand for good X in Country J = {H, F } is given by: (. XD J = p X ) I J = (. p X ) (px XS J + p Y YS J ). The excess demand function for good X in Country J is given by ( ) EX J py =. YS J.8XS J. () p X In autarky, E J X = 0 which implies p JA X p JA Y = ( ) ( Y J ) S 4 XS J. Substituting in for the supplies gives the following autarkic equilibrium prices p HA X p HA Y = 9 4 p F A X p F A Y = 6 Since the relative price of good X to good Y is higher in Country H than in Country F in autarky, Country H must have a comparative advantage in the production of good Y. Check which industry is more capital intensive and see if this matches up with the capital-labour ratios in each country. 4
(b) Using the indirect utility function from Question 4), substituting in the incomes for the consumers in Country H (I H = (wh )(0) and I F = (rh )(30), and solving for equilibrium price of labour (w H ) and capital (r H ) using the unit cost function, we can solve for the following indirect utility functions: V HA =.55 V HA = 0.78 V F A =.80 (c) Using the excess demand function given by (9) and the balanced trade condition (EX H + EF X = 0) implies that p X p Y ( Y H = S + YS F ) XS H + = 9 XF S 4. (d) After resolving for the real factor prices given the equilibrium free trade price ratio we can sub them into the indirect utility functions and find: V H = 0.78 V H =.6 Since there is a single consumer in Country F we know that her income will equal the value of national production: I F = p X X F S + p Y Y F S. V F =.85 (e) If you sum up the two utilities in Country H in autarky and free trade you should find that aggregate utility is higher in trade. Does Country F also gain from trade? Does the Gains from Trade Theorem hold here? (f) Consumer in Country H loses while Consumer in Country H gains in moving from autarky to trade. This is a good check of your understanding of the Hecksher-Ohlin model. How is the change in relative prices related to the change in returns to factors (see the Stolper-Samuelson Theorem assuming we have time to cover this before the midterm)? What role does the ownership of factors play here? 4. There are two ways to solve this question: (a) Hard way: Solve for price ratio in each country and determine which country has a comparative advantage in each good. (b) Easy way: Apply the Hecksher-Ohlin Theorem. Since the model s environment matches that of the Hecksher-Ohlin theorem we know that the country that has a relatively high capital-labour ratio should export the good that uses capital relatively intensively. 5
We can calculate the capital-labour ratios in each country as: K H L H = K F L F = 3. Country F clearly has a relatively high capital-labour ratio compared to country H. Examining the production functions we would expect that industry Y is relatively more capital-intensive than industry X since both are constant returns to scale Cobb-Douglas production functions, but capital is raised to the power of 0.6 in industry Y but only to 0.4 in industry X. We can also see this further by considerly the marginal products of the two production functions: r = X ( ).6 LX = 0.4 K X w = X = 0.6 r = Y = 0.6 w = Y = 0.4 K X ( ).4 KX ( ).4 LY ( ).6 KY If we examine the ratio of w to r in each industry we see: w r = 3 w r = 3 Given the assumptions of the model we know that the w r ratio will equalize across industries in the same country. Solving for the capital-labour ratio in each country we find: which implies = 9K X 4 K X K X = w 3r = 3w r or > K X The Hecksher-Ohlin theorem then implies that country H has a comparative advantage in good X. Similarly, it implies that country F has a comparative advantage in good Y. 6