Please answer all the questions.
Q1. Suppose a firm has the Cobb-Douglas production function Q = F(X, Y) = XαY1-α, where 0 < α < 1, K is capital, and L is labor. Please use this production function to answer the following questions.
(1) Does this production function exhibit increasing, constant, or decreasing returns to scale? Please use your words to explain why you think it belongs to one of which.
(2) Please draw the set of three isoquants that support your answer in part (1). For example, if you think it is a constant returns to scale, then draw three isoquants that represent constant returns to scale, and so on.
(3) What are the characteristics of these isoquants?
(4) Please find MPL and MPK.
(5) Please show diminishing marginal productivities for both K and L.
(6) Given the total cost outlay such as C = wL + rK , where w is the wage of labor and r is
the rental price of capital. Please use the same Cobb-Douglas function given by the whole problem set to determine the amount of labor (L*) and capital (K*) that the firm should use in order to maximize output. Please also find the maximum output.
Q2. A widget manufacturer has a production function as below:
Q = 5K + 3L
(1) Based on this form of the production function, are the capital and labor perfect substitutes or perfect complements?
(2) Please graph the isoquant maps for Q=15, Q=45, and Q=60. What is the RTS along these isoquants?
(3) If the wage rate (w) is $1 and the rental rate on capital (r) is $1, what cost-minimizing combination of K and L will the manufacturer employ for the three different production levels in part (2)? What is the manufacturer's expansion path? What are the total costs for the three different production levels in part (2)?
(4) How would your answer to part (3) change if r rose to $2 with w remaining at $1? What is the manufacturer's expansion path? What are the total costs for the three different production levels in part (2)?
Hint: Please refer to the lecture notes on 8/1. Your total cost should be numbers.
Q3. Suppose that the Acme Gumball Company has a fixed proportions production function that requires it to use four gumball presses and two workers to produce a unit of gumball per unit of time.
(1) Please write down the production function in the short-run using K represents gumball presses and L represents labor. According to this production function, are the capital and labor perfect complements or perfect substitutes? Please also draw the set of isoquants that represent this form of the production function in part (1)?
(2) Assume Acme can produce any number of gumballs (Q) they want using this technology. Please find the total cost function in this case.
(3) Please find the average and marginal cost of gumball production.
(4) Please graph the average and marginal cost curves for gumballs assuming r=2, w=3.
(5) Now graph these curves for r = 1, w = 3. Have these curves in part (5) shifted? Have these curves shift up or shift down?
Q4. The short-run total cost function for a firm producing skateboards is
TC = 20??2 - 10?? + 80
where q is the number of skateboards per week.
(1) What is the fixed cost?
(2) What is the variable cost?
(3) Please calculate the average cost function for skateboards.
(4) Please calculate the marginal cost function for skateboards.
(5) At what level of skateboard output does average cost reach a minimum? What is the average cost at this level of output?
Q5. A firm producing hockey sticks has a production function given by Q = A√KL where A is a parameter for technology, K is capital and L is labor.
In the short run, the firm's amount of capital equipment is fixed at K=K¯ (where K¯ represents a random constant.) The rental rate for K is r, and the wage rate for L is w.
(1) Please use your words to explain why it is a Cobb-Douglas production function. Does it exhibit increasing, decreasing or constant returns to scale?
(2) Please calculate the firm's short-run total cost function.
(3) Please calculate the short-run average cost function.
(4) Please calculate the short-run marginal cost function.
(5) What does the SMC curve intersect the SAC curve? That is, at what level of hockey sticks does average cost reach a minimum? What is the average cost at this level of output? What is the average cost at this level of output?
#Hint: This question is similar to your assignment. The only difference is now you are not asked to work with numbers but parameters or exogenous variables.
Q6. Elizabeth M Suburbs makes $400 a week at her summer job and spends her entire weekly income on new running shoes and designer jeans, because these are the only two items that provide utility to her. Furthermore, Elizabeth insists that for every pair of jeans she buys, she must also buy a pair of shoes (without the shoes the new jeans are worthless.) Therefore, she buys the same number of pairs of shoes and jeans in any given week, and she has the utility function
U= min {S, J}.
(1) If jeans cost $20 and shoes cost $20, what is Elizabeth's budget constraint? How many shoes and jeans will she buy to achieve her maximum utility level? What is her maximum utility?
(2) Suppose that the price of jeans rises to $30 a pair. What is Elizabeth's budget constraint? How many shoes and jeans will she buy to achieve her maximum utility level? (You may need to round the numbers as pairs of jeans or pairs of shoes are indivisible.) What is her maximum utility?
#Hint: You cannot use the Lagrangian method to answer this question because utility function is discontinuous. Please refer to the lecture notes to find the optimal consumption bundle (S*, J*) to achieve the maximum utility subject to budget constraint with two consumed goods being complements.