1) The Labor-Leisure Trade-off
Sheila must decide how to allocate her 24-hour day between non-wage activities, leisure, and wage activities, labor. For every hour of labor that she supplies she is paid by her employer $w. Sheila can purchase bottles of beer at a price of $1 per bottle (and beer is the only consumption good). She has no non-wage income. Finally assume that Sheila's preferences over leisure and beer can be represented by the following utility function:
u(l, b) = ln b + 2 ln l.
a. Find Sheila's demand functions for leisure and beer. What is Sheila's supply curve for labor? Is her supply of labor upward sloping or downward sloping?
b. Suppose that Sheila's wage is $12. Given the demand functions that you found in part (a), what are Sheila's demands for leisure and beer when her wage is $12? What is her supply of labor? What is the elasticity of Sheila's supply of labor at a wage of $12?
c. Suppose that Sheila's wage rises to $15. Given the demand functions that you found in part (a), what are Sheila's demands for leisure and beer when her wage is $15?
d. Putting hours of leisure on the x-axis and bottles of beer on the y-axis, illustrate in an indifference curve diagram Sheila's optimal choice of leisure and beer when her wage is $12. Illustrate in your diagram the effect of the increase in wages on her optimal choice of leisure and beer. Make sure to clearly indicate the income and substitution effects in your diagram.
Which effect (the income or substitution) is larger on her demand for leisure?
2) Foreign or Domestic Production
If a firm manufactures in its home country, it faces input prices for labor and capital of w and r and produces q units of output using L units of labor and K unites of capital. Abroad, the wage and cost of capital are half as much as at home. If the firm manufactures abroad, will it change the amount of labor and capital it uses to produce q? What happens to its cost of producing q?
3) Cost Functions
For each of the following cost functions, find the average fixed cost, marginal cost, average variable cost, and average cost.
a. C (q) = 10 + 10q
b. C (q) = 10 + q2
c. C (q) = 10 + 10q ? 4q2 + q3
4) Costs and Production
Assume a firm is using labor and capital as inputs. The production function is given by:
f (K, L) = 5K 3 L3
. The rental cost for capital is r = $5, and the wage is w = $25.
a. Does the production function exhibit decreasing, increasing or constant returns to scale?
b. What are the marginal products of labor MPL and of capital MPK ?
c. In the short-run, capital is fixed at K = 125. Draw the expansion path.
Express L as a function of q, the quantity produced.
d. What is the total cost function T C (q)? What is the fixed cost? What is the variable cost? Find the average cost function, the average fixed cost function, the average variable cost function and the marginal cost function. Draw them in two separate graphs, for total costs and average costs. They do not have to be to scale but have to represent the relationships between the cost curves.
e. Now consider the long run. Find the expansion path. That is, for every q, find the optimal inputs (L, K ). You can express the optimal L and K as functions of q.
f. You want to produce q = 100. Do you expect the cost to be higher in the short run or the long run?