We solve the 2 x 2 Ricardian model with a constant elasticity of substitution (CES) demand. Assume that the demand in country n of the goods i, yni, is the result of the utility maximiza- tion: max x [(Uml) + (Un2)²=11 s.t. Pnlnl + Pn2Yn2= Xn, where n = N, S indexes the country, and X, is the total expenditure of the country. σ > 1 denotes the elasticity of substitution across goods. (a) Write down the Lagrangian function and solve for the optimal consumption bundle for country n. (b) The ideal price index is defined as the cost of one unit of utility. Show that with CES utility function, the price index takes the form: Pn = [(Pm1) + (P2)¹- (c) Under the assumption of free trade, law of one price holds so we have PN₁ = Ps₁ = P1; and psi = Ps2 = p2. Solve for the world relative demand of the two goods. (d) Same as in the lecture, we assume the North has comparative advantage in goods 1, so: aN2 asi as2 Derive the world relative supply of the two goods. (e) Derive the autarky real wage in the North. (f) Derive the real wage in the North under trade, and compute its gains from trade.
We solve the 2 x 2 Ricardian model with a constant elasticity of substitution (CES) demand. Assume that the demand in country n of the goods i, yni, is the result of the utility maximiza- tion: max x [(Uml) + (Un2)²=11 s.t. Pnlnl + Pn2Yn2= Xn, where n = N, S indexes the country, and X, is the total expenditure of the country. σ > 1 denotes the elasticity of substitution across goods. (a) Write down the Lagrangian function and solve for the optimal consumption bundle for country n. (b) The ideal price index is defined as the cost of one unit of utility. Show that with CES utility function, the price index takes the form: Pn = [(Pm1) + (P2)¹- (c) Under the assumption of free trade, law of one price holds so we have PN₁ = Ps₁ = P1; and psi = Ps2 = p2. Solve for the world relative demand of the two goods. (d) Same as in the lecture, we assume the North has comparative advantage in goods 1, so: aN2 asi as2 Derive the world relative supply of the two goods. (e) Derive the autarky real wage in the North. (f) Derive the real wage in the North under trade, and compute its gains from trade.
Chapter1: Making Economics Decisions
Section: Chapter Questions
Problem 1QTC
Related questions
Question
Transcribed Image Text:We solve the 2 x 2 Ricardian model with a constant elasticity of substitution (CES) demand.
Assume that the demand in country n of the goods i, yni, is the result of the utility maximiza-
tion:
max
x [(Uml) + (Un2)²=11
s.t. Pnlnl + Pn2Yn2= Xn,
where n = N, S indexes the country, and X, is the total expenditure of the country. σ > 1
denotes the elasticity of substitution across goods.
(a) Write down the Lagrangian function and solve for the optimal consumption bundle for
country n.
(b) The ideal price index is defined as the cost of one unit of utility. Show that with CES
utility function, the price index takes the form:
Pn = [(Pm1) + (P2)¹-
(c) Under the assumption of free trade, law of one price holds so we have PN₁ = Ps₁ = P1;
and psi = Ps2 = p2. Solve for the world relative demand of the two goods.
(d) Same as in the lecture, we assume the North has comparative advantage in goods 1, so:
aN2
asi
as2
Derive the world relative supply of the two goods.
(e) Derive the autarky real wage in the North.
(f) Derive the real wage in the North under trade, and compute its gains from trade.
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