Cost and Production
Copyright 1996, Cohen & Froeb
Revised on 9/5/96
Table of Contents
Opportunity costs vs. accounting costs
- 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
- 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
- opportunity costs of capital
- opportunity cost of office space.
Long run production functions and cost functions
- 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
- 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.
Long run cost minimization: marginal analysis
- 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
- 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.
Long run cost minimization: incremental analysis.
- 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.
- 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
- 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.
Short run production functions
- 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.
- The "usual" shape of the short run production function:
- In the short run, output at first increases at an increasing rate with
increases in labor (increasing returns to labor)
- Then output increases at a constant rate with increases in labor (constant
returns to labor).
- 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
- The marginal productivity of labor increases, then is constant, and
- 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.
- It is easy to see that profits are maximized with 11 laborers.
- 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.
Shut down analysis
- 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).
Break even analysis
- 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
revenue-variable costs-fixed Costs=0
P*Q-AVC*Q -fixed Costs=0