The Cobb-Douglas function for a new product is given by: N (L.K) = 1.19L0.7²K0.18 Where C* =Rs. 750, w =Rs. 30, r=Rs. 40 Determine the amount of labor and capital that the firm should use in order to maximize output. What is this level of output? Also interpret the value of lagrange multiplier, 2.
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- A company has the Cobb-Douglas production function z = 400x0.6 0.4 where x is the number of units of labor, y is the number of units of capital, and z is the units of production. Suppose labor costs Php 2000 per unit, capital costs Php 4000 per unit, and the total cost of labor and capital is limited to Php 1,000,000. Use the method of Lagrange Multipliers to solve the following problems. 1) Find the number of units of labor and the number of units of capital that maximize production. 2) How much should be allocated to labor and capital to maximize production? 3) What is the maximum production?Q2. A firm's production function is Q = 9LZ+1)/10KY+1)/10 %3D • Find out MPL and MPK. • Find out APL when K=16 L= 64.The AVC and the AC curve can intersects each other at any point True/False
- Let S represent the amount of steel produced (in tons). Steel production is related to the amount of labor used (L) and the amount of capital used (C) by the following function. 2040.30 0.70 In this formula L represents the units of labor input and C the units of capital input. Each unit of labor costs $50, and each unit of capital costs $100. (a) Formulate an optimization problem that will determine how much labor and capital are needed in order to produce 55,000 tons of steel at minimum cost. min s.t. S = $ L, C≥ 0 = 55,000 (b) Solve the optimization problem you formulated in part (a). What is the optimal solution value (in dollars)? Hint: Use the Multistart option as described in Appendix 8.1. Add lower and upper bound constraints of 0 and 5,000 for both L and C before solving. (Round your answers to three decimal places.) at (L, C) =Constrained Optimization using Lagrange Multiplier: Cost MinimizationA clothing company produces two goods namely pants (P) and sweaters (S) whose combined cost function is C = 6P2 + 10S2 − PS + 30.In a day, the clothing company is limited by a production quota of 34 units of output only. Inequation form: P + S = 341. State the cost minimization problem of the clothing company.You are the manager of a firm and you are required to optimize the Cobb- Douglas function given the following parameters. The maximum amount of money available to spend is $1600 where the price of K = 12 and the price of L = 6. That is P = 12 and P = 6. The function is given as q = K04L0.6 What is the Lagrangian? of 2 estion O a K04L0.0 + X(1600 + 12K + 6L) O b. K0AL00 - A(1600 – 12K – L) Oc K04L0.0X(1600 + 12K + 6L) O d. K04L0.0 + A(1600 – 12K – 6L)
- Q5The rate of change of the function f(x) =x + 2 /1 − 8xwith respect to x when x = 1. (i) The number of units Q of a particular commodity that will be produced with K thousand dollars of capital expenditure is modeled by Q(K) = 500 K^2/3. Suppose that capital expenditure varies with time in such a way that t months from nowthere will be K(t) thousand dollars of capital expenditure,whereK(t) =2t4 + 3t + 149 /t + 2 Required:(a) What will be the capital expenditure 3 months from now? How many units will be producedat this time? (b) At what rate will production be changing with respect to time 5 months from now?Will production be increasing or decreasing at this time?Let S represent the amount of steel produced (in tons). Steel production is related to the amount of labor used () and the amount of capital used () by the following function: S- 20 L0 C0.7 In this formula L represents the units of labor input and Cthe units of capital input. Each unit of labor costs $50, and each unit of capital costs $100. (a) Formulate an optimization problem that will determine how much labor and capital are needed in order to produce 50,000 tons of steel at minimum cost. If your answer is zere, enter "p" Min 40 120 s.t. 20 S0000 (b) Solve the optimization problem you formulated in part a. Hint: When using Excel Solver, start with an initial >oand C0 Do not round intermediate calculations. Round your answers to the nearest whole number 2991 2318 Cost S 397448 ide FeedbackLet S represent the amount of steel produced (in tons). Steel production is related to the amount of labor used (L) and the amount of capital used (C) by the following function. S = 20L 0.30c0.70 In this formula L represents the units of labor input and C the units of capital input. Each unit of labor costs $50, and each unit of capital costs $100. (a) Formulate an optimization problem that will determine how much labor and capital are needed in order to produce 55,000 tons of steel at minimum cost. Min 50L + 100C s.t. ,0.300.70 20L = 55,000 L, C 2 0 (b) Solve the optimization problem you formulated in part (a). (Hint: When using Excel Solver, start with an initial L > 0 and C > 0. Round your answers to the nearest integer.) at (L, C) = ( 2467, 2880 2$
