Suppose that the XYZ Corp. (which is a profit-maximizing enterprise) produces “gadgets” according to the following production function: Q = 300K + 100KL + 2000L - L^2 , where Q is the number of gadgets per year, K is the amount of capital (machines) that are used, and L is the number of workers employed per year. Gadgets sell for $100 each. a) If XYZ has 10 machines, and workers cost $100,000/year (including benefits and direct ancillary expenses), how many workers should XYZ hire? b) If the workers cost $120,000, how many workers should XYZ hire? c) If workers cost $100,000 but gadgets sell for $80 each, how many workers should XYZ hire? d) If XYZ instead has 20 machines, how many workers should it hire when workers cost $100,000 and gadgets sell for $100?

Suppose that the XYZ Corp. (which is a profit-maximizing enterprise) produces “gadgets” according to the following production function: Q = 300K + 100KL + 2000L - L^2 , where Q is the number of gadgets per year, K is the amount of capital (machines) that are used, and L is the number of workers employed per year. Gadgets sell for $100 each. a) If XYZ has 10 machines, and workers cost $100,000/year (including benefits and direct ancillary expenses), how many workers should XYZ hire? b) If the workers cost $120,000, how many workers should XYZ hire? c) If workers cost $100,000 but gadgets sell for $80 each, how many workers should XYZ hire? d) If XYZ instead has 20 machines, how many workers should it hire when workers cost $100,000 and gadgets sell for $100?

Microeconomic Theory

12th Edition

ISBN:9781337517942

Author:NICHOLSON

Publisher:NICHOLSON

Chapter9: Production Functions

Section: Chapter Questions

Problem 9.8P

See similar textbooks

Similar questions

Q2: Suppose that a firm's production function of output Q is a function of only two inputs, L and K and can be written Q = L+ 4K. Let w = 5 and r = 25. If the firm produces 100 units of output, how many units of capital will it use? Q3: Suppose that a firm's production function of output Q is a function of only two inputs, L and K, and can be written Q = = min ( K , 2L). Let w 5 and r = 10. If the firm produces 20 units of output, how many units of capital will it use?
Karol has a coffee shop and produces coffee drinks under the production function Q = F(K,L) = 5K + L , where q is the number of cups generated per hour, K is the number of coffee machines (capital), and L is the number of employees hired per hour (labor). Suppose Karol needs to produce 500 cups of coffee and the price of capital is 40 euro per unit and the price of labor (wage) is 20 euro per unit. If Karol minimizes costs, how much capital and how much labor should he employ?
Suppose the production function for widgets is given by q=kl -0.8k²-0.21² where q represents the annual quantity of widgets produced, k represents annual capital input, and I represents annual labor input. Suppose k = 10; at what level of labor input does this average productivity reach maximum? (please put your answer in numerical values with no comma or decimal place). How many widgets are produced at that point? (please put your answer in numerical values with no comma or decimal place). If k = 10, at what level of labor input does MPL = 0? Suppose capital inputs were increased to k = 20, at what level of labor input does this average productivity reach maximum? widgets are produced at that point? (please put your answer in numerical values with no comma or decimal place). If k = 20, at what level of labor input does MPL = 0? answer in numerical values with no comma or decimal place). Does the widget production function exhibit constant, increasing, or decreasing returns to scale?…
Question 14 Consider a newspaper with production function f(L,K)= 4min{L,K}, where L is the units of labor and K the units of capital they use. Denote by APL(L,K)=f(L,K)/L the so-called average product of labor (here f is the production function of the firm). Is the Average Product of Labor always equal to the Average Product of Capital for this firm? No. Yes.
Consider the following graph of a production function when capital is constant. (The following is a description of the figure: it shows a two-axis graph; the horizontal axis measures labor and the vertical axis measures output; for a K fixed, the graph shows that maximal production that the firm can achieve with different levels of labor; the graph starts at cero production for zero labor; then it is increasing in all of its range; three levels of labor are shown as reference; there are L1, L2, and L3; they are related as follows L1<L2<L3; the graph is convex from 0 to L1, that is, its slope is increasing; the graph is concave from L1 on, that is, its slope is decreasing; the line that is tangent to the curve at L2, passes through the origin of the graph.) Denote by APL(L,K)=f(L,K)/L the so-called average product of labor (here f is the production function of the firm). From the graph we know that for the corresponding K: APL(L1,K)=MPL(L1,K) APL(L2,K)>MPL(L2,K)…
2. Suppose the production function for widgets is given by: f (K, L) 2* KL - K2/2- L2/2 Q 1 (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 for a company manufacturing calculators, the cost, revenue, and profit equations are given by x² C = 70,000 + 30x, R=200x - A) Costs are increasing at the rate of $ (Simplify your answer.) where the production output in 1 week is x calculators. If production is increasing at a rate of 600 calculators per week when production output is 5,000 calculators. Find the rate of increase (decrease) in cost, revenue, and profit. B) Revenue is decreasing at the rate of $ (Simplify your answer.) C) Profits are (Simplify your answer.) 20 at the rate of $ P=R-C per week at this production level. per week at this production level. per week at this production level.
Prof. Smith and Prof. Jones are going to produce a new textbook. The production function for the book is: ?=?1/2?1/2?= is the number of pages in the finished book?= is the number of working hours spent by Smith?= is the number of working hours spent by JonesSmith's labor is valued at 3 TL per working hour and Jones's labor is valued at 12 TL per working hour. After having spent 900 hours preparing the first draft, Smith cannot contribute any more to the book. Jones will revise the Smith's draft to complete the book.a) How many hours will Jones have to spend to produce a finished book of 300 pages?b) What is the marginal cost of the 300th page of the finished book?
1. Suppose firms use two inputs, capital, K, and labour L to produce output. Their production functions are written as a function of K and L: F(K, L). For each of the following production functions, determine i) whether the production function exhibits increasing, decreasing, or constant returns to scale, and ii) if both inputs double, by what factor (how many times) output will increase. (a) F(K,L) = 5K² +3L² (b) F(K,L) = 8K0.4 0.3 (c) F (K, L) = K² (2K³ – L³)
Suppose that a year after opening, Kenji has developed a new system that has increased the productivity of his pizza restaurant, changing the production function to q 2 =8K 0.5 L 0.5 . Complete the fourth column of the previous table by entering the number of workers required to produce 48 pizzas per hour based on the given number of ovens and his new production function. (Note: Because it is not possible to hire fractional workers, you must round to the nearest whole worker when necessary.) Using your most recent calculations, use the orange points (square symbol) to plot the isoquant representing q 2 =48 .
2. Fresh Steps, a manufacturer of cat litter, has estimated the following production function: Q = 50 LK In this equation, Q is the output rate (quantity), L is the amount of labor Fresh Steps hires per period of time, and K is the amount of capital Fresh Steps uses per period of time. Fresh Steps pays $10 per unit of labor and $20 per unit of capital. The company's vice president for manufacturing has hired you to determine which combination of inputs the factory should use to produce 6,400 units of output per period. a. Provide a recommendation for the vice president. (Calculations required). b. Suppose that the cost of labor increases to $20 per unit. Determine the effect of this change on your answer to part (a). (Calculations required). 1 c. Determine whether the Fresh Steps factory is subject to increasing or decreasing returns to scale. (Calculations required).
22. Consider the following production function: Q = 8K³/4L¹/², with capital (K) and labour (L) as the two inputs. (a) Find the marginal products of capital and labour and comment on whether they reflect increasing, constant or decreasing marginal returns. What type of returns to scale does the production function exhibit? (b) What are the average products of capital and labour and what happens to them as inputs increase? How do the marginal and average products relate to each other? Provide an intuitive explanation as to whether this relationship is as you would expect. (c) Assuming that the prices of Labour and Capital are w and r respectively, find the optimal conditional input demands of capital and labour. (d) Suppose that the firm must keep output at the original level, Qo, but that the price of labour, w, falls. By drawing a diagram, illustrate and explain how the firm's optimal inputs would change and what this would mean for the firm's demand curve for labour.