Explain the concept of a maximum - value function. Explain the EnvelopeTheorem for unconstrained and constrained optimization. Give aneconomic example of these concepts.

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ENGR.ECONOMIC ANALYSIS

14th Edition

ISBN:9780190931919

Author:NEWNAN

Publisher:NEWNAN

Chapter1: Making Economics Decisions

Section: Chapter Questions

Problem 1QTC

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Given TC = 100 + 60Q – 12Q2 + Q3 . Find a. The | equations of the TVC, AVC, and MC functions. b. The level of output at which AVC and MC are minimum, and prove that the AVC and MC curves are U-shaped. c. Find the AVC and MC for the level of output at which the AVC curve is minimum.
suppose noW that the firm must pay a fixed cost of ē > 0 in order to produce (e.g., this could be the cost of buying a manufacturing plant). If the firm chooses not to produce at all, it does not incur the fixed cost. If it chooses to produce any strictly positive quantity, it must incur the full fixed cost of č. Suppose p = w = r = 4. When will the firm choose to produce a strictly positive amount of the good Y? Write down the firm's supply and profit functions.
In optimization we convert a problem to which calculus problem? Group of answer choices a. Find an optimal domain of a function and then find limits at the endpoints of the domain. b. Find optimal equations for a given situation. c. Find extremum values of a function defined on a domain.
Given the following data on input and output levels. Suppose the output price is $5 and input price is $10. Find the values of AVP and MVP when X = 4: X 0 2 4 6 8 10 12 Y 0 100 250 450 600 700 750 $312.50 and $375 $20 and $375 $265.50 and $475 $20 and $750
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***PLEASE NOTE - An answer is NOT needed for parts A, B and C; these are included to assist with answering part D. Only an answer for part D is required, but it is derived from the previous answers*** Given: A farmer raises peaches using land (K) and labor (L), and has an output of ?(?,?)= ?0.5?0.5 bushels of apples. a. Find several input combinations that give the farmer 6 bushels of apples. Sketch the associated isoquant on a graph, with L on the x-axis and K on the y-axis. b. In the short run, the farmer only has 4 units of land. What is his short-run production function? Graph it for values of L from 0 to 16, with L on the x-axis and output on the y-axis. What is the name of the slope of this curve? c. Assuming the farmer still only has 4 units of land, how much extra output does he get from adding 1 extra unit of labor if he is already using only 1 unit of labor? How much extra output does he get from adding 1 extra unit of labor if he is already using 4 units of labor?…
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?
Constrained Optimization: Cobb-Douglas Production Function:1. Based from the factor shares of the two inputs, what will happen to the number of output if it the firm decides to triple both the amount of labor and capital?2. State the optimization problem of the firm.3. Solve for the formulas of the Marginal Product of Labor (MPL), and Marginal product ofCapital (MPK)4. Using your knowledge of the tangency condition in Producer’s theory, find the combinationof K and L that the firm should use to produce the maximum possible output. Do not solvethe problem using the Lagrangian method.Note: The tangency conditions just states that the slope of the production function must beequal to the slope of the isocost function.5. What is the maximum possible output that the firm could earn given the constraint it faces?
Increasing returns and imperfect competition: (a) The production function for the word processor is Y = X – 100 million if X is larger than 100 million, and zero otherwise. By spending $100 million, you create the first copy, and then $1 must be spent distributing it (say for the DVD it comes on). For each dollar spent over this amount, you can create another copy of the software. (b) The production function is plotted in Figure 6.5. Output is zero whenever X is less than 100 million. Does this production function exhibit increasing returns? Yes. We spend $100 million (plus $1) to get the first copy, but doubling our spending will lead to 100 million copies (plus 2). So there is a huge degree of increasing returns here. Graphically, this can be seen by noting that the production function “curves up” starting from an input of zero, a common characteristic of production functions exhibiting increasing returns. (Constant returns would be a straight line starting…
Let y = f(x1, x2)=x11/2 + x1x2 be a firm’s production function, where x1≥0, x2≥0. Write down the firm’s production possibility set, and its input requirement set. Is this production function concave, quasi-concave? Is this production function homogenous? Find its returns to scale when x1=1, and x2=1.
3. Suppose that the feasible region of a cost minimization linear programming problem has three corners points of (5,8), (10,5), and (4,10). If the objective function is given as: Minimize Z = 2X + Y Which of the following represents an iso-cost line? Select one: a. X + 2Y = 10 b. X – Y = 10 c. 2X – Y = 10 d. 2X + Y = 10 e. none of the other options.

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Explain the concept of a maximum - value function. Explain the Envelope Theorem for unconstrained and constrained optimization. Give an economic example of these concepts.