Cost and Production

Copyright 1995, Froeb

Revised on 8/1/95

Table of Contents

• Opportunity costs vs. accouting costs.
• How to derive long run cost curves from production functions by minimizing long run costs using both marginal and incremental analysis.
• When making forecasts about uncertain events, always conduct a sensitivity analysis.
• How to derive short run cost curves from short run production functions.
• Numerical example of a short run production functions.
• Shut down analysis.
• Break even analysis as a rule of thumb.

• Opportunity costs are the costs of the foregone or next best alternative.
• One should consider opportunity costs, not accounting costs when making decisions. There are two mistakes you can make by using accounting costs: you can consider costs that should be ignored; or you can ignore costs that should be considered. Another way of restating this proposition is that you should consider only costs that vary with the consequences of your decision; and all costs that vary with the consequences of your decision.
• Example of considering costs that should be ignored: the sunk or fixed cost fallacy.
  • The DOE developed a cyclotron to enrich uranium. It spent billions on research and development, and almost has a fully operational machine (a rare "success") but they never brought it on-line because Congress requires them to charge a high enough price for enriched uranium to recover the cost of capital. If they brough it on-line, they would be forced to "price themselves out of the market."
• Examples of ignoring costs that should be considered: the "hidden cost" problem.
  • opportunity costs of capital
  • opportunity cost of office space.

• Q=f(K, L). Quantity is a function of the inputs used to produce it: in this example capital and labor. Quantity is measured as a rate of production (flow) as are capital and labor, e.g. the amount of cars washed per day is a function of the amount of labor and capital used each day.
• The production function specifies a technically efficient use of labor and capital necessary to produce output, i.e. no resources are "wasted."
• The cost function specifies an economically efficient use of resources, i.e. the firm chooses the least cost combination of inputs, to produce a given output.
• This yields the long run cost function: total costs=g(Q). The cost function depends on the prices of inputs.

• Here we will assume that the firm can choose any level of capital and labor to produce output, Q. More capital leads to more output; less capital to less output. More labor leads to more output; less labor to less output. This is known as a variable proportions production technology because labor can substitute for capital, and vice-versa, in production.
• The marginal product of labor is the additional output from one extra unit of labor
• The marginal product of capital is the additional output from one more unit of capital.
• The cost minimization rule for producing a given quantity: choose labor and capital such that (MP of labor)/(Price of labor)=(MP of capital)/(price of capital).
• Proof: dividing the marginal products of each input by the price of the input tells you how much output you can produce for a dollar. If it costs more to produce output using labor than it does using capital, then sell labor and buy capital. This allows you to produce the same amount at lower cost. Only when the costs of production using each input are the same are no further cost savings possible.
  • If (MP of labor)/(Price of labor) is greater than (MP of capital)/(price of capital), sell capital, buy labor.
  • If (MP of labor)/(Price of labor) is less than (MP of capital)/(price of capital), sell labor, buy capital.

• Incremental analysis considers large discrete changes in input mix, whereas marginal analysis considers small continuous changes in input mix.
• Example: 1985: John Deere acquisition of Versatile. John Deere had an old fashioned production line for making farm tractors. Very high fixed costs, but low marginal costs. Versatile had a "garage" style production facility with much lower fixed costs, but higher marginal costs.
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• It is easy to see that if production is less than or equal to 6 units, then Versatile has lower costs of production. If production is larger than or equal to 7 units, John Deere has lower costs of production.
• The long run decision between these two production processes would depend on how many units you thought you would sell. If you anticipated selling 7 or more units, use the John Deere production process, if you thought you would sell 6 or fewer units, use the Versatile production process.

• Suppose you were uncertain about how many units you anticipated selling. Sensitivity analysis allows you to build in uncertainty to your analysis by determining the costs of various scenarios.
• Suppose you thought that your uncertainty was best described by a trinomial random variable.
  • with p1=.4, Q=7
  • with p2=.3, Q=10
  • with p3=.3, Q=4; note that p3=1-p1-p2
• What's the best technology to choose? ANSWER: The expected output is 7 (.4*7+.3*4+.3*10=7), the average cost of the expected output is not the same as the expected average cost, because Versatile has a large advantage at small outputs, while Deere has a small advantage at high outputs. The table below shows how to compute expected average costs, which is the usually right criterion to use for deciding which technology to adopt.
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• A long run production function relates the output produced to the inputs used, e.g. Q=f(capital, labor). In the short run, some inputs cannot be varied, so the firm does not have as much flexibility as in the long run. In this case, the short run production function is a function of only the inputs that can be varied. Suppose that capital is fixed in the short run. Then Y=g(labor)
• The "usual" shape of the short run production function:
  1. In the short run, output at first increases at an increasing rate with increases in labor (increasing returns to labor)
  2. Then output increases at a constant rate with increases in labor (constant returns to labor).
  3. Finally, output increases at a decreasing rate with increases in labor (diminishing returns to labor).
• Short run cost functions are larger than long run cost functions because, in the short run, fixed inputs can not be varied. In the long run, all inputs can be varied, and this greater flexibility allows you to achieve lower costs, i.e. h(Q) is greater than or equal to g(Q).

Short run production function

• Fixed costs=$20/hour, Labor costs $5/hour
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• The marginal productivity of labor increases, then is constant, and then decreases.
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• Suppose that the output sells for $1 per unit and that the firm can sell all it wants at a price of $1 (infinite elasticity, or a perfectly competitive firm). How much labor should the firm hire?
• ANSWER: Keep hiring as long as the benefit of hiring another worker is greater than the cost of another worker. The benefit equals the marginal revenue of the worker (price times the marginal production), Benefit=$1*(marginal production); The cost is the wage. Cost=$5.
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• It is easy to see that profits are maximized with 11 laborers.

Cost curves

• Can we derive the optimal production decision (how much labor to produce) from the cost curves instead of the production curves? Yes: keep producing as long as the benefit of producing another unit ($1) exceeds the cost of producing another unit. Looking at the graph, you can see that $1 intersects the marginal cost curve somewhere between 10 and 11 laborers (between 75 and 84 units). To determine whether to produce at 75 or 84, you must look at the spread sheet.

• In the long run, stop producing if economic profits are negative.
• In the short run, stop producing if revenue is less than total variable cost. In the short run, you don't have to cover your fixed costs, but you must make enough to cover your variable costs. If not, then you can shut down. You will still have to pay your fixed costs, but at least you can avoid paying your variable costs (which are greater than revenue).

• Assumptions: constant price, constant average variable cost. (P-AVC) is sometimes called "contribution margin" because it represents profit per unit sold (ignoring fixed costs).
• Set profits equal to zero to solve for how much output would be required to "break even." Another way of asking the same question is to ask how much quantity would be required to produce enough profit to cover fixed costs.
  revenue-variable costs-fixed Costs=0
  P*Q-AVC*Q -fixed Costs=0
  (P-AVC)*Q-fixed Costs=0
  Q=fixed Costs/(P-AVC)