Calculate the Lagrange multiplier of the cost minimization problem with a CobbDouglas production function, and show that it is equal to the derivative of the cost function with respect to output (the marginal cost)
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Calculate the Lagrange multiplier of the cost minimization problem with a CobbDouglas production function, and show that it is equal to the derivative of the cost function with respect to output (the marginal cost).
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- A firm has production function F(K, L) = 1/4 (K1/2 + L1/2) . The wage rate is w = 1 and the rental rate of capital is r = 3. (a) How much capital and labor should the firm employ to produce y units of output? (b) Hence find the cost of producing y units of output (the firm’s cost function). (c) Differentiate the cost function to find the marginal cost, and verify that it is equal to the value of the Lagrange multiplierConsider the production functions of three different Firms utilizing inputs labor (L) and capital (K) in producing goods X, Y, and Z given below. The three firms face the same fixed price for labor and capital at 5 per unit and 10 per unit, respectively. X = KL2 – L3; Y = 10K1.5L0.5; Z = K0.5L0.5 C = wL + rK Using Lagrange Multiplier Method, calculate the cost-minimizing values of L, K, and C of Firm Y if it decides to produce 1,000 units of good Y. Graphically illustrate the expansion path of Firm Y if it decides to double its target output Derive the short-run cost function of Firm Z if 25 units of capital are employed. Suppose that good Z is sold at a perfectly competitive price of 10 per unit, calculate Firm Z’s profit and discuss if the Firm Z should continue to operate Derive the long-run cost function of Firm YConsider a firm that maximizes profit function x^2y subject to the input constraint 2x^2+y^2=3 where x, y are inputs in the production. Find the extra profits garnered by the firm if the input constraint is relaxed by changing 3 to 3.3. Consider only positive values for optimal inputs and the multiplier.
- Asap In a perfectly competitive market, the market price of a toy is $100, with labour costs of $5 and capital costs of $8. The Cobb-Douglas function is below, Q = K0.2L0.5 A ) Find the profit-maximizing quantity Q and its associated profit using the Lagrangian multiplier approach. B) To ensure that the firm maximises profit, how would you develop the objective function, constraint, and Lagrangian?Q.No.3. (a) What is the input use level for total value product maximization for the following function? y = x1 + 0.1x12 - 0.05x13 + x2 + 0.1x22 - 0.05x23 Q.No.3. (b) Briefly make a comparison between output maximization criteria and profit maximization criteria with respect to necessary and sufficient conditions?1. Inputs K, L, R and M cost £10, £6, £15 and £3 respectively per unit. What is the cheapest way of producing an output of 900 units if a firm operates with the production function Q = 20K0.4L0.3R0.2M0.25? 2. Make up your own constrained optimization problem for an objective function with three variables and solve it. 3. A firm faces the production function Q = 50K0.5L0.2R0.25 and is required to produce an output level of 1,913 units. What is the cheapest way of doing this if the per-unit costs of inputs K, L and R are £80, £24 and £45 respectively?
- If the production function of a firm is given by Q=K^2L^3 and the input prices are r= 8 per unit and w= 2 per unit find the levels of labor and capital that maximize the level of output for a total outlay of 240 birr?If the production function of a firm is given by Q=K2L3 and the input prices are r=Birr 8 per unit and w= Birr 2 per unit, find the levels of labor and capital that maximize the level of output bfor a total outlay of birr 240.2.4 Water is produced and sold by the government. Demand for water is represented by the linear function Q=50-2P. The total cost function for water production is also a linear function: TC(Q)= 100+ 100. You will also need to work out both the average cost of production, denoted by AC(Q), equal to the total cost of producing a quantity of output divided by that quantity of output, TC(Q)/Q, and the marginal cost of production, denoted by MC(Q), which is the additional cost incurred to produce one more unit. a. What fee should the government charge per unit of water in order to reach the efficient allocation? b. How much should it charge if it wishes to maximize profit from the sale of water? C. What is the value of the efficiency loss that results from charging the price in part b rather than the price determined in part a?
- Economics Suppose that production for good X is characterized by the following production function, Q = 4K0.5L0.5, where K is the fixed input in the short run. If the per-unit rental rate of capital, r, is $12 and the per-unit wage, w, is $20, then the average total cost of using 25 units of capital and 49 units of labor is Multiple Choice $6.25. $9.14.Correct $10.07. incalculable since there is insufficient information to determine the average total costs.Suppose a tax is set at t’ in the figure. How many units of emissions are ABATED and is abatement achieved cost effectively? Suppose a regulation has just been implemented requiring that coal-fired power plants pay a tax of $80 on each unit of emissions (1 unit is 1 metric ton). Before the tax was imposed, a plant called Old River Power had an emissions level of 100 units. Given a marginal abatement cost function of 200 – 2E, under the tax policy what will Old River Power pay the regulator in emissions taxes?Student tuition at ABC University is $500 per semester credit-hour. The state supplements school revenue by matching student tuition dollar for dollar. In other words, for each dollar a student pays in tuition, the state pays a dollar that is also considered revenue. Average class size for a typical three credit-hour course is 25 students. Labor costs are $3,000 per class, materials costs the school $25 per student per class and overhead costs are $15,000 per class. For this problem, ABC University measures output as the total amount of revenue generated. a) What is the multifactor productivity performance for a course at ABC University? b) If instructors work an average of 10 hours per week for 16 weeks for each three-credit hour class of 25 students, what is the labor-hours productivity ratio? c) Suppose next year, the tuition increases to $550 per semester credit hour but the state only pays $100 per semester hour instead of the matching it had previously paid. The instructors…