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Exam 12: Production With Multiple Inputs

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Question 1

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Cobb-Douglas production function have decreasing returns to scale.

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Question 2

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If producer choice sets are convex and a production plan satisfies the condition that the (marginal) technical rate of substitution is equal (in absolute value) to the ratio of input prices, then the production plan is profit maximizing.

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Suppose there are different ways of producing computer chips.If you hire one worker (for the day) for each machine that you rent (for the day), you can produce 10 chips per day with each worker/machine pair for the first 60 machine/worker pairs.For the next 60 worker/machine pairs (assuming still that you hire them as pairs for the day), you are able to produce 20 chips per day with each of the additional pairs.Once you have 120 worker/machine pairs, you can only get 5 additional chips per day for each additional pair.But hiring 1 worker for each machine is not the only way to produce computer chips.Suppose you are starting from a production plan where you are using exactly as many workers as machines to produce a given level of chips.The technology is such that, starting at the production plan where you are using the same number of workers as machines, you can replace 1 or more workers with two machines (for each worker) and get just as many chips produced.Alternatively (and again starting at the production plan where you use exactly as many workers as machines), you can replace 1 or more machines with 2 workers (for each machine) and get just as many chips produced. a.On the template below, illustrate all the different ways that 600 chips can be produced per day.(Hint: The isoquant you should draw is composed of two line segments.) b.On the same graph, illustrate the different ways of producing 400 chips and the different ways of producing 1,800 chips.(Label each isoquant with the relevant output quantity). c.Is this production technology homothetic? d.If machines cost $100 per day, for what range of daily wages will you decide to use exactly as many workers as machines? e.Suppose both machines and workers cost $100 per day.Illustrate the long run cost curve for this firm.f.Illustrate the long-run marginal and average cost curves.

Just as indifference maps represent consumer tastes, so isoquant maps represent a producer tastes.

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

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

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

Technologically efficient production plans are also economically efficient.

In considering "returns to scale" we assume that production processes are homothetic.

Quasiconcave production functions give rise to convex producer choice sets.

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.

Which of the following is possible in a 2-input production technology.

In one-input models, all technologically efficient production plans are economically efficient and vice versa.

Decreasing returns to scale production functions must be concave.

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

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

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

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?

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

Assuming an interior solution, a production plan is profit maximizing if and only if all marginal revenue products are equal to input prices.