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Econ 101A – Problem Set 4
Due on Gradescope by 11:59pm, March 17th
The problem set must be uploaded and submitted on Gradescope by 11:59pm, March 17th. Show your work, write down (legibly) the steps that you use to get a solution. If you cannot solve a problem fully, write down a partial solution. You are welcome to brainstorm with other students, but you must write a solution in your own words and hand in your own PS answers.
Problem 1. CES Production Functions. In this exercise, we consider a firm that produces a widget in quantity Q, with the Constant Elasticity of Substitution production function:
Q = [Kα + L
α
]
β
where K is capital input, L is labor input and α and β are fixed positive parameters.
1. Show that, whenever α × β > 1, this production function exhibits increasing returns to scale.
2. What is the marginal rate of technical substitution? Explain what this rate means intuitively.
3. The firm can hire capital and labor at given prices r and w respectively and minimizes cost of producing each quantity Q. Express the ratio of its cost-minimizing input choices, L/K, as a function of the input price ratio r/w.
4. Find the firm’s cost function C(r, w, Q) using cost minimization. (Hint: The correct answer takes the form: C(r, w, Q) = Qx
[r
z + w
z
]
1/z where x = αβ
1
and z are numbers for you to find. Use this as a guide for collecting and simplifying expressions in the steps of your algebra.)
5. Plot (approximately) the total cost curve, marginal cost curve, and average cost curve of this firm when α = 1/2 and β = 2.
6. Assume that the firm produces in a perfectly competitive market where the price of widgets is p. Derive the first order condition for the second step of the firm’s problem, finding the optimal scale of production. The derivation should be straightforward if you use the notation: C(r, w, Q) = QxB, where we have used B to replace the long expression B := [r
z + w
z
]
1/z (x and z are as in question (4)). Is the second order condition satisfied?
7. What is the firm’s supply function? Can you explain why, intuitively, linking back to question (1) and to your graphs in question (5) ?
Problem 2. Perfect Competition. Consider a figs farm in the Philippines. The farm has a production function
y = θLαKβF
η
,
where
α + β + η = 1, α, β, η > 0,
and L is labor, K is capital, and F is farm land. The price of labor is w, the price of capital is r, the price of land is s, with all of the prices positive. In the short-run, however, the quantity of land farmed is fixed to F , so there effectively are only two factors of production with respect to which the firm maximizes.
In this exercise, we characterize the short-run solution to the Filipino farmers problem for the case of perfect competition.
1. Write down the cost minimization problem with respect to L and K and the first order conditions with respect to L and K
2. Solve for L
∗