The quantity, Q, of a certain product manufactured depends on the quantity of labor, L, and of capital, K, used according to thefunction0=900LK¹³.Labor costs $100 per unit and capital costs $50 per unit.What combination of labor and capital should be used to produce 36,000 units of the goods at minimum cost? What is that minimumcost?Round your answers to two decimal places.

Step 1: Ask:We must find the optimal combination of labour (L) and capital (K) to produce a minimum…

Answered: The quantity, Q, of a certain product…

ENGR.ECONOMIC ANALYSIS

14th Edition

ISBN:9780190931919

Author:NEWNAN

Publisher:NEWNAN

Chapter1: Making Economics Decisions

Section: Chapter Questions

Problem 1QTC

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Suppose the production function for widgets is given by: Q = f (K, L) = 2 ∗ KL − K2/2 − L2/2 (a) Suppose L=5 (is fixed), derive an expression for and graph the total product of capital curve (the production function for a fixed level of labor) and the average productivity of capital curve. (b) At what level of capital input does the average productivity reach a maximum? How many widgets are produced at this point? (c) Again, assuming L=5, derive an expression for and graph the MPK curve. At what level of capital input does MPK =0? (d) Does this production function exhibit constant, increasing or decreasing returns to scale?
Suppose that Flamerock Tires must decide where to produce one million tires: the US, where wages are 30 and the cost of capital is 5; or China, where wages are 5 and the cost of capital is 25. Production in each location follows the same technology (production function) given by: Q = f(L, K) = L^0.25 K^0.75 Computationally solve the cost - minimizing input levels in each location to produce the goal of 1 million tires
Imagine that the production function for tuna cans is given by q = 6K + 4L where q = Output of tuna cans per hour K = Capital input per hour L = Labor input per hour a. Assuming capital is fixed at K = 6, how much L is required to produce 60 tuna cans per hour? To produce 100 per hour? b. Now assume that capital input is fixed at K = 8; what L is required to produce 60 tuna cans per hour? To produce 100 per hour? c. Graph the q = 60 and q = 100 isoquants. Indicate the points found in part a and part b. What is the RTS along the isoquants?
Given the total cost equation: level of Q yields the minimum cost of production? TC = 32 + 2Q², what level of total average
In the short-run, we assume that capital is a fixed input and labor is a variable input, so the firm can increase output only by increasing the amount of labor it uses. In the short-run, the firm's production function is q = f(L, K), where is output, L is workers, and K is the fixed number of units of capital. Given a specific equation for production: or, when K = 29, Calculate the q = 8LK+ 5L² q = (8L × 29) + 5L² X - Average Product = 3 (13) ₁³. 102 and enter the values in the table to the right. (Round your answers to the nearest integer.) E Labor 624 0 6 8 10 12 Output or Product 0 481 987 1,500 2,005 2487 2,928 Production Marginal Product -- 241 253 257 253 241 221 Average Product -- 241 247 251 249
Assume instead that pharmacists and robots dispense prescriptions according to the following production function: Y = 10*KO.8L0.2 where Y is the number of prescriptions dispensed; L is the number pharmacist hours, and K is the number of robot hours. In addition, $10 worth of materials is used for each prescription. a. What is this type of production function called, and what are we assuming about the relationship between robots and pharmacists by using this production function? b. Derive the cost - minimizing demands for K and L as a function of output, the wage rate and the rental rate of capital. c. Use these results to derive the total cost function: costs as a function of y, r, w, and the $10 materials cost. d. Pharmacists earn $32 per hour. The rental rate for robots is $64 per hour. What are total costs as a function of Y? e. Does this technology exhibit decreasing, constant, or increasing returns to scale?f. The pharmacy plans to produce 40,000 prescriptions per week. At the…
Consider the production function q=√L+8K^(3). Starting from the input combination (3,6), does the production function exhibit increasing, constant or decreasing returns to scale if inputs double?
A firm has two opportunities for a new plant location, one is in China and the other is inMexico. The firm's production function is given by q = L 0.5 K 0.5 , In China, the cost of laboris w=$15 and the cost of capital is r=$5. In Mexico, w=$10 and r=$10. The firm wants toproduce 100 units of output. Which location should the firm choose for their new plant?Explain why.Note: Please round the optimal amounts of capital and labor at each location to the nearest whole number when making your calculations.Hint: cost-minimization rule.
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The per-unit cost of an item is its average total cost (= total cost/quantity). Suppose that a new cell phone application costs $250,000 to develop and only $0.60 per unit to deliver to each cell phone customer. Instructions: Round your answers to two decimal places. What will be the per-unit cost of the application if it sells 100 units? Per-unit cost = $1 What will be the per-unit cost of the application if it sells 1000 units? Per-unit cost $[ What will be the per-unit cost of the application if it sells 1 million units? Per-unit cost $
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The Cobb-Douglas production function is a classic model from economics used to model output as a function of capital and labor. It has the form f(L, C) = c₂LC1C²2 C2 are constants. The variable L represents the units of input of labor and the variable C represents the units of input of capital. (a) In this example, assume co 5, C₁ = 0.25, and c2₂ 0.75. Assume each unit of labor costs $25 and each unit of capital costs $75. With $90,000 available in the budget, develop an optimization model for determining how the budgeted amount should be allocated between capital and labor in order to maximize output. = = where co, C₁, and Max s.t. L, C ≥ 0 A ≤ 90,000 (b) Find the optimal solution to the model you formulated in part (a). What is the optimal solution value (in dollars)? Hint: Put bound constraints on the variables based on the budget constraint. Use L ≤ 3,000 and C ≤ 1,000 and use the Multistart option as described in Appendix 8.1. (Round your answers to the nearest integer when…

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