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

***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?…

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.