Exam 12: Production With Multiple Inputs

Production technologies A and B can have the same-shaped isoquant map, with technology A having decreasing returns to scale and technology B having increasing returns to scale.

Increasing returns to scale production technologies cannot give rise to convex producer choice sets.

An increasing returns to scale production function could be quasiconcave.

Profit functions are homogeneous of degree zero.

Profit is constant along an isoquant.

There are two economically meaningful ways of slicing two-input production frontiers.

Consider a firm that uses labor and capital to produce output x using a homothetic production technology that has increasing returns to scale when output lies between 0 and x_A, constant returns to scale when output lies between x_A, and x_B, and decreasing returns to scale when output exceeds x_B (where 0<x_A<x_B).Although the different parts of the question repeatedly refer to the isoquant graph you first draw in (a), you should probably re-draw the graph several times - each time only with the portions you need for the question -- to indicate the different items that are asked for in the remaining parts of the question (rather than indicating all your answers on literally the same graph). a.On a graph with labor on the horizontal and capital on the vertical axis, draw isoquants for x_A and x_B.For a given set of input prices w and r, indicate the least cost input bundle A=(l^A, k^A) for producing x_A using an isocost line.Label the slope of the isocost line and then label the slope of the isoquant in terms of the marginal product of labor and capital. b.Indicate where the least cost input bundle B for producing x_B must lie (in light of the homotheticity property of the production technology.) What does the vertical slice along which all cost-minimizing input bundles lie look like (on a graph with "inputs" on the horizontal and x on the vertical)? c.Indicate all input bundles in your isoquant graph that could be part of a profit maximizing production plan for some output price p>0. d.Suppose the actual profit maximizing production plan is (l*,k*,x*).What two conditions involving the marginal products of the inputs hold at this - and only this - production plan? e.Now suppose that a change in tax policy results in an increase of the rental price of capital r.Indicate all possible input bundles in an isoquant graph that might be long-run profit maximizing assuming no change in p or w.(Include the isoquant corresponding the initial profit maximizing output level x* as well as the isoquant that contains B (from (b)) in your graph.) Explain your reasoning.

If production technologies are homothetic, all cost-minimizing production plans lie on the same ray from the origin for a given set of input prices.

Suppose capital and labor are perfect complements in production.For output levels between 0 and 100, 2 units of labor together with 1 unit of capital produce 1 unit of output; for output levels between 100 and 200, 1 unit of labor together with 1 unit of capital produces 1 unit of output; and for output levels above 200, 1 unit of labor together with two units of capital produces one additional output.In each graph below, carefully label as much of each graph as you can. a.On a graph with labor on the horizontal axis and capital on the vertical, illustrate isoquants for 100, 200 and 300 units of output. b.Is this production technology homothetic? c.Suppose the wage and rental rates are 10.On a graph with output on the horizontal axis and dollars on the vertical, plot the total (long run) cost of producing 100, 200 and 300 units of output and illustrate the total cost curve. d.On a separate graph with output on the horizontal and dollars on the vertical axis, illustrate the (long run) marginal cost curve and the approximate shape of the long run average cost curve.

Output prices are irrelevant for a firm as it is calculating its cost curves.

We have worked a lot with homothetic production technologies.Suppose instead that a production process that uses capital and labor is quasilinear in capital and that capital is fixed in the short run.Then, assuming the firm currently profit maximizes at a given wage and rental rate, the short and long run slices of the production frontier are identical.

All economically efficient production plans are technologically efficient.

It is not sufficient for profit maximization that a production plan has all marginal revenue products equal to input prices -- because it must also be the case that the (marginal) technical rate of substitution is equal to the ratio of input prices (in absolute value).

Changing the labels on isoquants without changing the shapes of the isoquants implies no change in the underlying technology so long as the ordering of isoquants is preserved.