2. Suppose the production function for widgets is given by:f (K, L) 2* KL - K2/2- L2/2Q1(a) Suppose L-5 (is fixed), derive an expression for and graph the total product ofcapital curve (the production function for a fixed level of labor) and the averageproductivity 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. Atwhat level of capital input does MPK =0?(d) Does this production function exhibit constant, increasing or decreasing returnsto scale?

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Answered: 2. Suppose the production function for…

ENGR.ECONOMIC ANALYSIS

14th Edition

ISBN:9780190931919

Author:NEWNAN

Publisher:NEWNAN

Chapter1: Making Economics Decisions

Section: Chapter Questions

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13. Suppose you have a production technology given by f(x1, x2) = min{2x₁, x2} and you are producing at the point where x₁ = 10 and x₂ = 20. (a) Explain in words what we mean (generally) by the ‘marginal product' of an input in production. (b) For the production technology in this question and the initial point x₁ = 10 and x2 = 20, what is the marginal product of a small increase in input 1? (c) Suppose input 2 increases and you are now at the initial point x₁ = 10 and x2 = 30. Relative to your answer in part (b), does the marginal product of input 1 decrease, increase, or stay constant? Explain briefly.
Refer to the graph below: y2 The graph above illustrates a production set of a firm. (a) Which assumption(s) do(es) the production set violates? Explain. (b) Suppose the firm decided to scale all inputs up or down by some amount t > 0. What will happen to the level of output?
13. Suppose you have a production technology given by f(x1,x2) = min{2æ1, x2} and you are producing at the point where x1 = 10 and x2 = 20.
Suppose the production function is given by Q = 5K+2L. What is the marginal product of capital when 5 units of capital and 10 units of labor are employed?
Let S represent the amount of steel produced (in tons). Steel production is related to the amount of labor used (4) and the amount of capital used (C) by the following function. S- 20100.70 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. -55,000 L.C20 (b) Solve the optimization problem you formulated in part (a). (Hnt: When using Excel Solver, start with an initial L>O and Co. Round your answers to the nearest integer.) at (L. C) =
Suppose the production function is given by G = 10K - 8L. What is the average product of capital when 2 units of capital and 10 units of labor are employed?
6
Suppose the representative firm of the economy has a production function of the form F(K, N) = AKO.5N 0.5 The marginal product of labor is then given by MP N = 0.5AK0.5N –0.5 . The current capital 3. stock is K = 40. (a) Holding fixed capital at 40, draw a graph of output as a function of labor. What are some important features of this graph? (b) If A = 4 what is the Labor Demand Curve, ND(w), as a function of the real wage w? (c) Suppose labor is supplied inelastically with NS(w)= 10. What is the equilibrium wage w, employment level N, and full employment output Y ? (d) Suppose that productivity unexpectedly increases to A = 6. What is the new equilibrium wage w, employment level N, and full employment output Y ? (e) Depict this change graphically. Denote the original labor supply and demand curves by NS and ND respectively. Denote the new labor demand curve after the productivity shock by ND' . (Does not need to be to scale).
Suppose that production function of certain devices is determined by: q = k * L - 0.8 k2 - 0.2 L2 Where q represents annual quantity of devices produced, k represents annual quantity of capital factor, and L represents annual quantity of labor factor. (a) Suppose k = 10; draw a graph with average and total labor productivity. What is level of labor (L) factor at which average productivity reaches a maximum? How many devices are produced at that point? (b) Assuming again that k = 10, plot Marginal Product (MP) curve. At what level do we have that labor (L) factor is MP = 0? (c) Suppose capital (k) factor were to increase to k = 20. How would your previous answers change?
2. Consider a Cobb-Douglas production function with three inputs. K is capital (the number of machines), L is labor (the number of workers), and H is human capital (the number of college degrees among the workers). The production function Y = K2/6 L3/6 H1/6 a) Derive an expression for the marginal product of labor. How does an increase in the amount of human capital affect the marginal product of labor? (Hint: The marginal product of labor MPL is found by differentiating the production function (Y) with respect to labor (L)) b) Derive an expression for the marginal product of capital. How does an increase in the amount of human capital affect the marginal product of capital? (Hint: The marginal product of capital MPK is found by differentiating the production function (Y) with respect to capital (K)).
1. Suppose the production function is Q = 8L + 15K where Q is the quantity of output, L is the quantity of labor used in production, and K is the quantity of capital used in production. What can be said about this production function? It has Decreasing Returns to Scale It has Constant Returns to Scale It has Increasing Returns to Scale There isn’t enough information to determine the Returns to Scale for this production function 2. You’re dreaming of what to do during a nice summer day. You could mow the lawn which you would pay someone $15 to do for you. You could go for a walk which you value at $11. You could also take a nap and ignore everyone and everything else which you would pay $29 to do. If you to take a nap, what is your opportunity cost? Group of answer choices $29 $26 $15 $11
3. Frisbees are produced according to the production function q = 2K+L where q = output of frisbees per hour, K =capital input per hour, L =labor input per hour. a) If K = 10, how much L is needed to produce 100 frisbees per hour? b) If K = 25, how much L is needed to produce 100 frisbees per hour? c) Graph the q = 100 isoquant. Indicate the points on that isoquant defined in part a and part b. What is the RTS along this isoquant? Explain why the RTS is the same at every point on the isoquant. d) Graph the q = 50 and q = 200 isoquants for this production function also. Describe the shape of the entire isoquant map. e) Suppose technical progress resulted in the production function for frisbees becoming q = 3K+ 1.5L. Answer part a through part d for this new production function and discuss how it compares to the previous case.

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