The economy is populated by two firms. One firm produces cars (c), the other produces beans (b). The production function of each firm i = {c,b} is Y; = KL Cars are capital intensive, hence ce > os. The price of cars is pe and the price of beans is p. Firms take all prices as given. The unit cost of labor is w and the unit cost of capital is r. The total amount of labor and capital in the economy are, respectively, L and K. 1. Write down the optimality condition for capital and labor of each firm 2. Derive the capital-to-labor ratio of each firm

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Chapter10: Cost Functions
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The economy is populated by two firms. One firm produces cars (c), the other
produces beans (b). The production function of each firm i = {c, b} is Y;
K L-4. Cars are capital intensive, hence a > an. The price of cars is pe
and the price of beans is po. Firms take all prices as given. The unit cost of
labor is w and the unit cost of capital is r. The total amount of labor and
capital in the economy are, respectively, L and K.
1. Write down the optimality condition for capital and labor of each firm
2. Derive the capital-to-labor ratio of each firm
3. Define an equilibrium for this economy
4. The quantities used by the firm producing cars as a fraction of the total
quantities in the economy are K. = nKK and Le = n1,L, for some values
0 < nK <1 and 0 < nL < 1. Use the labor and capital market clearing to
find analogous expressions for K, and Lp. How many equilibrium condi-
tions are you left with? What are equilibrium values you are left to solve
for?
Transcribed Image Text:The economy is populated by two firms. One firm produces cars (c), the other produces beans (b). The production function of each firm i = {c, b} is Y; K L-4. Cars are capital intensive, hence a > an. The price of cars is pe and the price of beans is po. Firms take all prices as given. The unit cost of labor is w and the unit cost of capital is r. The total amount of labor and capital in the economy are, respectively, L and K. 1. Write down the optimality condition for capital and labor of each firm 2. Derive the capital-to-labor ratio of each firm 3. Define an equilibrium for this economy 4. The quantities used by the firm producing cars as a fraction of the total quantities in the economy are K. = nKK and Le = n1,L, for some values 0 < nK <1 and 0 < nL < 1. Use the labor and capital market clearing to find analogous expressions for K, and Lp. How many equilibrium condi- tions are you left with? What are equilibrium values you are left to solve for?
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