Consider a firm with the production function F(K, L) = Kº.8 Lº.2 10. (Concretely, imagine that due to technological imperfections, the first 10 units of output are unus- able, so that if the firm produces 100 units of a good, only 90 are of sufficient quality). 20 units of capital, what = If the firm is currently using L = 20 units of labor and K is the elasticity of substitution?
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- If a firm has the production function f(k, I) = (2k1/4 + 14)" what is the firm's elasticity of substitution? O a. G = O b. o = O c. o = 1 O d. a = 2 O e. o = 4 114 -/2Consider the production function Q = 2(KL)0.5 c) What is the elasticity of substitution at a point K = 1, L = 1 if we increase K by one unit?What is the elasticity of substitution for the production function f(K, L) = K¹/4 [³/4? (a) o = 1 (b) o = -1 (c) o = 1 (d) σ = 3 2
- Find the elasticity of scale and the elasticity of substitution for the CES production function (x1; x2) = (x1rho + x2tho 1/P, where 0 * p > !!!Question 2: For this problem you will analyze the elasticity of substitution and the isoquant graphs for two different production functions F₁(K, L) = 4K + 2L F₂(K, L) = K²/³ [1/3 (a) Graph the isoquant for F₁(K, L) = 4K+2L that represents an output of 8. Be sure to show your work and label the axes clearly (b) What is the marginal rate of technical substitution (MRTS) for F₁ ? (c) What is the elasticity of substitution for F₁ ? /Consider the following production function for shirts: q = v6 L3/4K/4, where L is worker-hours, and K is sewing machine-hours. a. Compute the marginal products of labor and capital, the average product of labor, and the marginal rate of technical substitution of labor for capital (i.e. how many units of capital are needed to make up for the loss of one unit of labor)? b. Are there diminishing returns to labor (that is, does the marginal product of labor decrease when labor L increases)? What about to capital? Is there diminishing marginal rate of technical substitution (MRTS)? с.
- Question 2: Elasticity of Substitution Show that the production function f (x1,x2) = a ln x, + ß In x, has elasticity of substitution equal to one. a) Find the marginal rate of technical substitution MRTS,2, express it in terms of a ratio of x2 to x1, call this r = X1 b) Take logs on both sides, c) Find the derivative of In MRTS12 with respect to In r. The inverse of this quantity is the elasticity of substitution.1. Juan Valdez owns a coffee farm in Colombia. His production function is: f (x1, x2) = (x1 – 1)0.25 x9-5. Assume the price of input 1 is r and the price of input 2 is w.. (a) Write down an expression for the technical rate of substitution. (b) Find Juan's demand for inputs conditional on the quantity y of coffee Juan wants to produce. (c) Find Juan's cost function. (d) What is the supply function of Juan's firm? 2. Show that the profit function is convex in (p, w). 3. Find the profit function for the Cobb-Douglas production function f(¤1, 12) = Ax†' x" with A > 0, a1, ¤2 > 0 and a1 + a2 0, B > 0, 0 < a < 1, and 0 + p< 1. 6. Find the profit function for the CES production function. 7. Verify Hotelling's Lemma for the CES production function with B < 1.Juan Valdez owns a coffee farm in Colombia. His production function is: f(x1,x2)=(x1−1)^0.25 x2^0.5 Assume the price of input 1 is r and the price of input 2 is w. (a) Write down an expression for the technical rate of substitution. (b) Find Juan's demand for inputs conditional on the quantity y of coffee Juan wants to produce. (c) Find Juan's cost function. (d) What is the supply function of Juan's firm?
- Suppose the production function is Cobb-Douglas and f(x1;x2)=x11/2x23/2 Write an expression for the marginal product of x1. Does marginal product of x1 increase for small increases in x1, holding x2 fixed? Explain Does an increase in the amount of x2 lead to decrease the marginal product of x1? Explain What is the the technical rate of substitution between x2 and x1? What is the type of returns to scale of this production function? (Increasing, decreasing, constant)Calculate the elasticity of substitution for the production functionDoes the value of λ change if the budget changes from $4600 to $5600?What condition must a Cobb-Douglas production function q = cKαW β satisfy toensure that the marginal increase of production is not affected by the size of thebudget?