Question

1. Kiara’s razor-edge focus in class is the product of a careful combination of tea and coffee. Her hours of concentration per day, Q, are produced according to Q = C + 1 2 T, where C and T represent the ounces of coffee and tea she drinks in a day, respectively. Draw three isoquant curves that are consistent with 3, 6, and 9 hours of concentration, and do so by putting C on the x-axis, and T on the y-axis. Also, what is the marginal product of tea? And what is the marginal rate of technical substitution (of tea for coffee) at each point on the isoquant?

2. For each production function listed below, determine MPL , MPK , and MRTSK,L.

(a) Q = L+7K

(b) Q = 2L1/4K 1/2

(c) Q = 4L2+5K3

3. Suppose that Ben & Jerry’s company has the following production function for ice cream: Q = 5K2+30L1/3 . What is the marginal product of capital and marginal product of labor given this production function? What is the marginal rate of technical substitution of capital for labor?

4. IKEA produces their classic MALM bed frame using capital and labor under the following function: Q = min{K, 3L}.

(a) What type of production function is this?

(b) Suppose the firm is currently choosing K = 1 and L = 2 for their production. What are the M PK and M PL at that point?

(c) When prices are pK = 2 and pL = 1, is the choice in (a) the optimal allocation? Why?

(d) What can we say about the MRTS at the optimal allocation?

PART II: OPTIMAL INPUTS

1. A cost-minimizing firm’s production function is Q = LK2 . The price of labor services is w, and the price of capital services is r. Suppose that w = $15, r = $5, and the firm’s total cost is $180.

(a) What are the firm’s MPL , MPK?

(b) What is the firm’s optimal input combination of L and K? What is Q?

(c) Suppose a new management takes over the company; and they intend to significantly cut down costs by reducing production. The CEO announces that they will only produce Q = 288 units now, with their production technology unchanged. Find the new optimal choices of K and L given what the company intends to produce, and the new level of their production costs.

2. State whether the following claim is true or false, and briefly explain your answer. The following production function exhibits constant returns to scale: Q = 5L 0.45K 0.33

Answer 1

Assuming the isoquant equation is , not what is mentioned, the three isoquants will be linear in nature, as both C & T have power equal 1. For any value of Q, the curves will be as , taking T in y-axis. Also, as can be seen, all the slopes are equal to +2. The graph are as below.

The marginal product of Tea can be found as , ie or , is the marginal product of Tea. The marginal rate of technical substitution (MRTS) can be found as for , we have or (taking Q constant) or . Hence, for 1 unit of Coffee, 2 units of Tea have to be sacrificed.

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2. For any given isoquant as , the marginal product of labor is , the marginal product of capital is , and marginal rate of technical substitution of K for L is for a constant Q.

(a) Given .

or .

or .

For , we have or or .

(b) Given .

or or .

or or .

For , we have or or or or or or .

(c) Given .

or .

or .

For , we have or or or or .