Economics 245 — Fall 2011International TradeProblem Set 1September 29, 2011Due: Thu, October 13, 2011Instructor: Marc-Andreas MuendlerE-mail: [email protected] Dornbusch-Fischer-Samuelson’s Ricardian Model with Un-balanced TradeConsider a version of the Dornbusch-Fischer-Samuelson model of Ricardian with transport costsand a non-zero trade balance. There is a continuum of goods indexed with z ∈ [0, 1]. There aresymmetric iceberg transportation cost so κ melts away and 1/(1 − κ) units of a product need to bemade for one unit to arrive abroad.Consumers have homothetic preferences with consumption basket Cd≡ exp{R10ln cd(z)dz}.Using a result from question 3, demand for a product z is c(z) = P C/p(z) with the ideal priceindex P = exp{R10ln p(z)dz}.Labor is only factor of production and makes a product z under unit labor requirements a(z).Define the Home country’s comparative advantage in industry z with A(z) ≡ a∗(z)/a(z). As-sume without loss of generality that z strictly indexes the industries with the Home’s strongestcomparative advantage so that A0(z) < 0.1. Using the condition w a(z) ≤ w∗aast(z)/(1 − κ) for home production, determine the cut-off industry zHup to which the home country produces. Similarly, using the conditionw∗a∗(z) ≤ w a(z)/(1 − κ) for foreign production, determine the cutoff industry zFup towhich the foreign country produces. Show that zH> zFfor κ > 0 and A0(z) < 0.2. To simplify exposition, consider the functional form A(z) = exp{1 − 2z}. Show that thesize of the nontraded sector zH− zFcan then be expressed as zH− zF= − log(1 − κ) > 0.3. In equilibrium, global consumption expenditure must equal global income so that PC +P∗C∗= wL + w∗L∗(“market clearing”). Home income equals global expenditure on homeproduced goods so wL = zHP C + zFP∗C∗, and a similar expression applies to the foreigncountry. Define the home trade balance as T B = wL − P C = −T B∗6= 0, that is the1excess output over absorption. Make good 1 the numeraire, a foreign produced good, sothat w∗= p∗(1)/a∗(1) = 1/a∗(1). Show that the global “market clearing” condition andT B = wL − P C 6= 0 implyww∗=log(1 − κ)T BL∗/a∗(1)+ zFL∗/L1 + log(1 − κ) − zF≡ B(zF).4. Using the cutoff for foreign production w∗a∗(zF) = w a(zF)/(1 − κ), show that this rela-tionship and B(zF) above result in a unique equilibrium. (Hint: Establish monotonicity andtheir limits.)5. How does the equilibrium with a non-zero trade balance differ from that derived under azero trade balance? How does an increase in the home trade balance T B affect the locationof industries? How does the increase affect the size of the nontraded sector under A(z) =exp{1 − 2z} and in general?2 Heckscher-Ohlin Model with Two Countries, Two Industriesand Two FactorsThere are two industries 1 and 2 and two factors of production K and L. Capital earns a rentalrate r and labor a wage w. Each industry i’s production function Qi= AFi(Ki, Li) is homoge-neous of degree one. The foreign country’s production function is identical up to a Hicks-neutralproductivity parameter: Qi= A∗Fi(Ki, Li).1. A function f(x, y) is homogeneous of degree α if f (λx, λy) = λαf(x, y) for any λ > 0.Differentiate the production function Qi= AFi(Ki, Li) with respect to Li. Is the marginalproduct A ∂Fi(Ki, Li)/∂Lihomogeneous, if so of what degree? Use your result to showthat A ∂Fi(Ki/Li, 1)/∂Li= A ∂Fi(Ki, Li)/∂Li.From now on, define fi(ki) ≡ Fi(Ki, Li)/Li= Fi(Ki/Li, 1), where ki≡ Ki/Liis industryi’s capital-labor ratio.2. State the input rules, by which each factor’s income r and w equals the marginal revenueproduct in each local industry. Derive the wage-rental ratio ω ≡ w/r as a function of kiandshow that ω is not a function of A (or A∗). Derive the total differential dω/dkias a functionof kifor each industry and show that it is not a function of A (or A∗).23. Use the input rule for labor in each industry to show the relationship between the productprice ratio p ≡ P1/P2and the two industries’ marginal products in terms of labor. How doesthe slope of the production possibility frontier relate to the two industries’ marginal productsin terms of labor? How does the slope of the production possibility frontier relate to thetwo industries’ marginal products in terms of capital? Show that the slope of the productionpossibility frontier is not a function of A (or A∗).4. Show thatdpdω=ω (k2− k1)(ω + k1)(ω + k2)pω.Does this relationship allow you to state the Stolper-Samuelson theorem?5. Suppose international product markets are fully integrated but that capital and labor arecompletely immobile across borders. So both countries face the same product price ratiop ≡ P1/P2. Using the above findings, how does an industry i’s capital-labor ratio in thehome country kidiffer from the same industry’s capital-labor ratio in the foreign country k∗i?Using the input rules, show that the relative wages w/w∗and relative rental rates r/r∗areequal in free-trade equilibrium. What does the result imply about factor price equalization(FPE)?From now on, assume Cobb-Douglas production functions qi= Afi(ki) = (ki)αiand q∗i=A∗fi(k∗i) = (k∗i)αi, and assume that industry 2 is more capital intensive with α2> α1.6. Rederive the wage-rental ratio ω ≡ w/r as a function of kiand show that ki= [αi/(1−αi)]ω.Can there be a factor-intensity reversal? For a Cobb-Douglas production function, does itmatter whether A is Hicks-neutral or capital augmenting (such as in qi= (Aki)αi)?7. Show thatp =(α2)α2(1−α2)1−α2(α1)α1(1−α1)1−α1ω(α2−α1).Based on this result, state the Stolper-Samuelson theorem in its weak form for the Cobb-Douglas production function.8. Use the factor market clearing conditions L1+ L2=¯L and K1+ K2=¯K together withproduction to derive as an intermediate step the relative supply relationship as a function ofrelative price p:Q1Q2=κ11 − κ1α2α11p,3where κ1≡ K1/¯K = 1 − κ2. Finally, use the above p-ω-relationship and the equilibriumlevel of κ1(as a function of p) to establishQ1Q2=1 −α1α21−α21−α1α1α2−α11−α21−α11p−1α2−α1¯L¯Kα1α21−α21−α1α2α2−α11−α21−α11p−1α2−α1¯L¯K− 1·1−α21−α11p.Based on this result, and under the assumption of homothetic preferences, derive the Heck-scher-Ohlin theorem for the