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Exam 12: Production With Multiple Inputs

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Question 1

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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).

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Question 2

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Suppose that,at a given production plan,the marginal product of labor is 6 and the marginal product of capital is 3.In a graph with labor on the horizontal and capital on the vertical axis,this implies that the technical rate of substitution at that production plan is a. -1/2 b. -2 c. -18 d. None of the above

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Question 3

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Profit functions are homogeneous of degree zero.

Technologically efficient production plans are also economically efficient.

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.

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

Conditional input demands are homogeneous of degree zero in input prices.

Cost functions must be homogeneous of degree 1 in (input and output)prices.

Suppose that you are given a cost function c(w,r,x)=2w^1/2r^1/2x^3/2 where w is the wage rate for labor,r is the rental rate of capital and x is the output level. a.Does the production process that gives rise to this cost function have increasing,decreasing or constant returns to scale? b.Derive the marginal cost function. c.Calculate the supply function for the firm - i.e.the function that tells us for every combination of input and output prices,how much the firm will optimally produce.How does output by the firm change as input and output prices change? d.If the cost function had been c(w,r,x)=2w^1/2r^1/2x^1/2 instead,how would your answer to (c)change? How can that make any sense?

A price taking firm employs each of its inputs into production until its marginal product is equal to 1.

If a production technology has diminishing marginal product of all inputs throughout,then the producer choice set is convex.

An increasing returns to scale production function could be quasiconcave.

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.

Decreasing returns to scale production functions must be concave.

Quasiconcave production functions give rise to convex producer choice sets.

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.

Assuming convex producer choice sets,the (marginal)technical rate of substitution is equal (in absolute value)to the ratio of input prices at any profit maximizing production plan.

If a production technology has increasing returns to scale throughout,then the marginal cost curve lies below the average cost curve throughout.

Profit is constant along an isoquant.

In 2-input production models,constant returns to scale implies horizontal marginal cost curves.