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
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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?
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- 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.what is the marginal rate of substitution when the price of coffee is $1.00 per ounce? the prof said the answer is -2.IV. Will can produce a higher grade, Gw, on an upcoming economic exam by studying. His production function depends on the number of hours he studies marginal analysis problems, A, and the number of hours he studies supply and demand problems, R. Specifically, Gw = 2.5A0.36R 0.64. His roommate David's grade production function is G) = 2.5A0.25R0.75 a. What is Will's marginal productivity from studying supply and demand problems? What is David's? b. What is Will's marginal rate of technical substitution between studying the two types of problems? What is David's?
- Problem 1 a) The local cleaner launders white clothes using the production function q = 3B + G, where B is the number of cups of a brand bleach and G is the number of cups of a generic bleach that is half as potent. a.1) In a graph, draw an isoquant. a.2) What are the marginal product of B and G? What is the marginal rate of technical substitution? TechnologyWhat 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 2Input either "increase" or "decrease" where relevant: An increase in the prices of inputs will cause the equilibrium price to ....... and the equilibrium quantity to .....
- Given the production function Y=3x+2x2-0.1x3 Compute The APP and MPPDefine marginal product and explain the law of diminishing marginal utility advocated by David Ricardo.Answer the Constrained Optimization: Cobb-Douglas Production Function:1. Based from the factor shares of the two inputs, what will happen to the number of output ifit the firm decides to triple both the amount of labor and capital?2. State the optimization problem of the firm.3. Solve for the formulas of the Marginal Product of Labor (MPL), and Marginal product ofCapital (MPK)4. Using your knowledge of the tangency condition in Producer’s theory, find the combinationof K and L that the firm should use to produce the maximum possible output. Do not solvethe problem using the Lagrangian method.Note: The tangency conditions just states that the slope of the production function must beequal to the slope of the isocost function.5. What is the maximum possible output that the firm could earn given the constraint it faces?
- 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)Consider a firm that produces output using industrial robots and skilled labor. Suppose it is the case that a reduction in the price of industrial robots causes the firm's labor demand curve for skilled workers to increase (i.e., to shift to the right). This implies that: There is diminishing marginal product of robots. O There is diminishing marginal product of skilled labor. O Robots and skilled labor are complements. O Robots and skilled labor are substitutes. The unit cost of skilled labor is below the unit cost of robots.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.