= Optimal inputs: A firm has a production function, y Py = 20, pe = 4, Pk much profit do they earn? In(l, k) in a market with prices 5. Find the firm's profit maximizing choice for y, l, and k. How
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- 2 The prices of inputs x1; x2; x3; x4 are p1 = 2; p2 = 2; p3 = 5; and p4 = 2. If the production function is given by f(x1, x2, x3, x4) = min{x1, x2} + min{x3, x4}, what is the minimum cost of producing one unit of output?.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?1. Consider a production function of corn in sales with two inputs defined by f(x₁, x₂) = 2/3 1/3 X1 X₂ and the price of the input x₁ is $100 and the price of input x2 is also $100. Suppose that a farmer has a budget of 400,000 for two inputs, find the maximum production.
- Input either "increase" or "decrease" where relevant: A decrease in the price of a complementary good will cause its complement’s equilibrium price to ...... and the equilibrium quantity to .....\beta < 1. 16. A firm uses two inputs, X1 and X2, to produce a good that is described by the production function: Q = f (X1, X2) = X1/2X1/4 12 The firm sells its output at $80 per unit: P = $ 80. The cost of input 1, X1, is $4. The cost of input 2. X2, is $2. (a) Solve for the profit - maximizing input mix, output, and profit. (b) Check the second-order conditions to verify that the profit is at a maximum1. A firm’s production function is , where Ldenotes the size of the workforce. Find the value of MPLin the case when: (a) L=1, (b) L=10, (c) L=100, (d) L=1000 Does the law of diminishing marginal productivity apply to this particular function? 2. Show that the price elasticity of demand is constant for demand functions of the form where A and n are positive constants. 3. The demand and total cost functions of a good arerespectively and a) Find expressions for TR, (profit) , MR, and MC in terms of Q. b) Solve the equation and hence determine the value of Q which maximizes profit. c) Verify that, at the point of maximum profit, MR=MC. 4. The cost of building an office complex, x floors high, in a prime location in Accra is made up of three components: (a) GH¢10 million for the land (b) GH¢1/4 million per floor (c) Specialized costs of GH¢10000x per floor. How many floors should the office complex contain if the average cost per floor is to be minimized? 5. The supply…
- Problem 5 Suppose that a firm has a production function f(K, L) = 12L¹/3 K¹/3 and that w and r are input prices and p is output price. The firm does not change those prices. a) Show that the firm's profit π is a concave function of (K, L). b) Find (K*, L*) that maximizes the profit, as a function of (w,r, p).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 < 1.6 A production function is given by f(x1,x2)=(max{x1,x2})^0.5. Prices of inputs are p1=10 and p2=20. What is the lowest cost of producing 10 units of output?
- 3) Suppose a production function is y = x,"² x,", and w, is the price of x, and w, is the price of X2, and P is the price of output. a) Find the conditional factor demands. b) Find the cost function. c) Show that dx,/dw, = dx,/dw,.Production. Assume a production with two variable inputs (L, K). Show graphically the following and explain: (a) fixed-proportions production function (b) inputs are perfect substitutesGiven a firm which uses 2 inputs, X1 and X2; to produce a good that is described by the production function: Q = f (X1 X2) = X11/2 X21/4 The firm sells its ouput at N$80 per units. Given cost of input 1, X1; is N$4; and the cost of input 2, X2; is N$2 Solve for the profit maximizing input mix, output, and profit.