1. A rm produces an output y using labor and capital, whose quantities are denot
ID: 1094704 • Letter: 1
Question
1. A rm produces an output y using labor and capital, whose quantities are
denoted respectively by xL and xK. Suppose this rm's production function
is given by f(L;K)=50?KL + K + L
Answer the following questions.
a. Derive the marginal product of labor and the marginal product of capital.
b. Is the marginal product of labor diminishing everywhere? Does it ever take negative values?
2. Let B denote the number of bicycles produced from xF units of bicycle frames and xT units of tires. Suppose that every bicycle needs exactly two tires and one frame. Draw an isoquant that represents this production process and write down a possible production function for bicycles.
3. Suppose a rm's production function is given by f(x1; x2) = 6?x1 + 8?x2. Let w1 = $1 be the price of input 1 and let w2 = $4 be the price of input2. The price of the rm's output, which the rm takes as given, is p = $8. Calculate the prot-maximizing quantity of output.
4. Suppose a rm's production function is given by f(x1; x2) = ?x1.?x2. Let w1 = $10 and w2 = $15. If the rm's goal is to maximize prots, in what proportions should it use the inputs 1 and 2?
Explanation / Answer
2=
The marginal product of capital (MPK) function is the partial derivative with respect to capital. That is, MPK=(25/3)(L/K)^(2/3). Likewise, the marginal product of labor (MPL) function is the first derivative with respect to labor. That is, MPL=(50/3)*(K/L)^(1/3).
Since the exponents on capital and labor sum to one, this production function displays constant returns to scale.
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