We solve the 2 × 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 maximization:
max h
(yn1)
σ−1
σ + (yn2)
σ−1
σ
i σ
σ−1
s.t. pn1yn1 + pn2yn2 = Xn,
where n = N, S indexes the country, and Xn 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 =
(pn1)
1−σ + (pn2)
1−σ
1
1−σ
(c) Under the assumption of free trade, law of one price holds so we have pN1 = pS1 = p1,
and pS1 = 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:
aN1
aN2
<
aS1
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.
1
