ECON 30001 Assignment,
University of Melbourne,
Semester 02, 2025.
YA = LαAK
1−α
A , YM = L
β
MK
1
Consider a small open economy with two sectors, Agriculture (A) and Manufacturing (M). Produc-
tion in each sector uses capital (K) and labor (L
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) under Cobb–Douglas technologies:
−β
M ,
with α > β, so that Agriculture is relatively labor-intensive.
Firms in each sector are competitive and maximize profits:
πj = PjYj −WLj − RKj, j ∈ {A, M},
where Pj is the good’s price, W the common wage, and R the rental rate of capital. The economy has
fixed endowments of labor and capital: L¯ and K¯.
1. Part (a)
Derive the first-order conditions for labor and capital demand in each sector. Use these to obtain an
expression of relative demand for labor as a function of their relative factor prices. Plot the relation-
ship between relative demand and relative factor prices with L/K on horizontal axis and W/R in
vertical axis.
2. Part (b)
Using the factor market clearing conditions:
L¯ = LA + LM, K¯ = KA + KM
Derive an expression for the economy-wide capital-labor ratio K¯L¯ as a function of the factor price ratio
W
R and the sectoral labor allocations. Define λ =
LA
L¯ as agriculture’s share of total labor. Express your
final answer in terms of λ, α, β, and WR .
1
3. Part (c)
Your expression for λ in Part (b) shows that the labor allocation depends on both the factor en-
dowment ratio K¯L¯ and the equilibrium factor price ratio
W
R . Explain what determines
W
R in general
equilibrium.
Hint: In lectures we use θ as the share of capital in sector A.
4. Part (d)
Now instead, do a cost minimization for the producer to obtain unit costs for both sectors. Use these
conditions to obtain conditional factor demands for both sectors. Use this to obtain the relationship
between relative price of goods and relative price of factors.
)
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