Economics 440 - find an expression for the wage return to

Question

Problem Set

This problem set is due on Monday, October 24th in class.

Consider the specific factors model (2 goods, 3 factors, where labor is the flexible factor) for a given small country with the following Cobb-Douglas production technologies:

Q_C = Z_CK^1/2L^1/2

Q_F = Z_FT^1/2L^1/2

where Z_C and Z_F are the productivities in both sectors and L = L_C + L_F is the aggregate endowment of labor (as usual, K and T are the aggregate endowments of capital and land, which are fixed factors while labor can move between sectors).

1. Use the two equations above to find an expression for Q_C/Q_F in terms of the parameters of the model and the equilibrium distribution of labor (L, K, T, Z_C , Z_F, and L_C and L_F).

2. Derive the MP L_C and MP L_F.

3. Find PC /PF as a function of LC and LF and the parameters of the model (L, K, T, Z_C , Z_F ).

4. Combine the answers to part (1) and part (3) to find an equation for the relative supply curve (Q_C /Q_F as a function of the relative price P_C /P_F ). What can you say about the shape of the supply curve?

5. Now assume that all consumers in the country have Leontief preferences over food and clothing - one unit of food is consumed with every unit of clothing. Find the equlibrium relative prices (PC /PF ) as a function of the parameters of the model (L, K, T, Z_C , Z_F ).
Now assume the country opens to trade at a fixed world price of pT = 1 and pT = 1 (the country is small relative to its trading partners).

6. How will the country's pattern of trade (which good will be exported or imported) depend on the parameters of the model? How does the size of the labor force L affect the pattern of trade?

7. Find an expression for the wage, return to capital, and return to land, as a function of the parameters of the model. How will an increase in the productivity of the clothing sector (an increase in ZC ) affect the welfare of all three factor owners? Contrast your response with the welfare change induced by an increase in the aggregate capital endowment K.

8. Policy makers are considering the imposition of a 10% tariff on food (this tariff would raise the price of food in the country by 10% and would not affect the price of clothing). What would be the effect of this tariff on the returns to all three factors (w, r_K , r_T )? If these returns increase, be sure to specify whether they will increase by more or less than the amount of the tariff.

9. Use your answer to the last question to predict whether each factor will have any incentives to lobby the policy makers for protection (the imposition of this tariff).

Extra Credit

The Cobb-Douglas production function has the following general form:

F (K, L) = ZK^αL^1-α

where Z > 0 is a parameter that represents overall productivity and α is any constant between 0 and 1.

Verify that the Cobb-Douglas production function above satisfies the assumption of con- stant returns to scale and diminishing returns to a single factor.