Cost and Production

8/8/2019 Cost and Production

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COST AND PRODUCTION

Instructional Goals: You will understand:

The difference between opportunity costs and accounting costs. The importance of sensitivity analysis. How to derive long run cost curves from production functions by

minimizing long run costs using both marginal and incrementalanalysis.

How to derive short run cost curves from short run productionfunctions.

How to perform a shut down analysis. How to use Break-even analysis as a rule of thumb.

And the relevance of these concepts to operational decision making.These concepts will also have relevance to more complex questions ofmarketing policy and strategy.

Opportunity costs vs. accounting costs

Costs are bad things endured or good things lost. Cost always meanscost to do something. You cannot have a cost without a cost objective.Most of the confusion about costs reflects a failure to be clear aboutcost objectives. Nevertheless, where economists and accountants areconcerned, there is a second and equally critical source of confusion

about costs: economists and accountants use the term "cost" to meandifferent although related things.

Economists define cost in terms of opportunities that are sacrificedwhen a choice is made. Hence, economic costs are simply benefits lost(and, in some cases, benefits are merely costs avoided). Economiccosts are subjective -- seen from the perspective of a decision makernot a detached observer -- and prospective. Moreover, economic costis a stock concept -- economic costs are incurred when decisions aremade. Economic cost estimates are used for making decisions aboutpricing, output levels, buying or making, alternative marketing

tactics/strategies, product introductions and withdrawals, etc.

Accountants define cost in terms of resources consumed. Hence, froman accountants standpoint, costs are objective -- seen from theperspective of a detached observer -- and retrospective. Accountantsusually define costs as flows. Accounting costs reflect changes instocks (reductions in good things, increases in bad things) over a fixedperiod of time. Accounting cost measures are used in the evaluation of


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managerial performance (usually together with information on income)and as a basis for economic cost estimation.

There are two kinds of mistakes you can make when you useaccounting costs to estimate economic costs: you can include cost

measures that should be ignored; or, you can ignore costs that shouldbe included. You should ignore costs that will not vary as a result ofyour decision; you should include all costs that will vary as a result ofyour decision.

Example of including costs that should be ignored: sunk costs.

The DOE developed a cyclotron to enrich uranium. It spent billions onresearch and development, and almost had a fully operational machine(a rare "success") but they never brought it on-line because Congressrequired them to charge a high enough price for enriched uranium to

recover the cost of capital. If they brought it on-line, they would have"priced themselves out of the market."

Examples of ignoring costs that should be considered: the"hidden cost" problem.

opportunity costs of capital (like all costs, opportunity costdepend on the question being asked)opportunity cost of office space.

EXERCISES:----------------------------------------------------

Dan Connor, Artie Zimmer, and Bob Bruss each earn $25 per hour(including vacations, social security, and other benefits) as automobilemechanics at a car dealership. They are considering opening a shopthat specializes in fast-service oil changes. The projected annual costof the building and equipment is $60,000. On average, an oil changerequires $6 of materials (oil and oil filter). The average price of an oilchange is $20 in the shops that currently provide this service. Dan,Artie, and Bob think that they could each perform as many as six oilchanges per hour and could charge at least $25 per oil change if theycould guarantee to return the customer's car in fifteen minutes or less.

What are the accounting and economic (or opportunity) costfunctions for this oil-changing business?Opportunity costs are the costs of the foregone or next bestalternative.One should consider opportunity costs, not accounting costswhen making decisions.


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You are trying to determine whether it makes sense to buy a house orto continue renting for $1100. You start looking at houses but decidethat you will not purchase a house unless it is more profitable than

renting. What price house is equivalent to $1100 in rent? Make severalassumptions:

1. Assume that any transactions costs incurred in buying and sellingthe house will exactly be offset by the capital gains on the house

2. Assume that you borrow the purchase price at 7.25%. Yourinvestments also earn 7.25%.

3. Assume that you do not pay any principle on the mortgage, onlyinterest.

HINT: the annual rent equivalent for a $350,000 house is $25375.

Now you think that you will make enough money to put yourself in a33% tax bracket. Redo the calculation above, assuming a 33% taxbracket, and explain in words how this affects your willingness to payfor a house.

Suppose that you think there is a 75% chance that you will makeenough to put yourself in a 33% tax bracket, and a 25% chance thatyou will stay in the 0% tax bracket. Compute the expected value of

monthly rent equivalent payment on a $250,000 house.

Note this is a version of what is known as sensitivity analysis. Whenyou are uncertain about the future, it is important to redo yourforecasts, computations, or spreadsheets, using a variety of scenarios:a best case, middling, and worse case scenario.

Long run production functions and cost functions

Q = f(K, L). Quantity is a function of the inputs used to produce it: inthis example capital and labor. Quantity is measured as a rate of

production (flow) as are capital and labor, e.g. the amount of carswashed per day is a function of the amount of labor and capital usedeach day.

The production function specifies a technically efficient use oflabor and capital necessary to produce output, i.e. no resourcesare "wasted."


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The cost function specifies an economically efficient use ofresources, i.e. the firm chooses the least cost combination ofinputs, to produce a given output.

This yields the long run cost function: total costs (C) = g(Q),which depends on the prices of inputs. This function can be a

pretty good proxy for the opportunity cost of delivering Q, atleast where we measure costs in units of present value: i.e., thechange in present value or owners equity caused by somespecified action (and where for purposes of measurement theattendant increase in wealth is excluded from the computation ofequity).

The following long-run functional relationships traditionally obtain in the

single product case: Cost varies as a function of total production volume (V),

the rate of output (x), the date of first delivery (T), and the date of completion

of the full production run (m), where x(t) denotes the rate of output at moment

t. Moreover, as the total quantity of units produced increases, the cost of futureoutput tends to decline because production-knowledge increases as a result of

production experience (this proposition is known as the learning or progress

curve"). That is: dC / dT | x = x0 , V = V0 , < 0. This relationship probably

holds for most products produced in large batches using traditional mass-

production methods. Moreover it is usually assumed that:

1. dC / dx(t) | T = T0, V = V0> 0

2. d2C / dx(t)2 | T = T0, V = V0 > 0

3. dC / dV | x = x0, T = T0 > 0

4. d2C / dV2| x = x0, T = T0< 0

Many of these functional relationships have been attenuated by the rise of

computer assisted design and manufacturing technology and modern

information.

Long run cost minimization: marginal analysis

Here we will assume that the firm can choose any level of capital andlabor to produce output, Q. More capital leads to more output; less

capital to less output. More labor leads to more output; less labor toless output. This is known as a variable proportions productiontechnology because labor can substitute for capital, and vice-versa, inproduction.

The marginal product of labor is the additional output from oneextra unit of labor.


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The marginal product of capital is the additional output from onemore unit of capital.

The cost minimization rule for producing a given quantity: choose laborand 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 theinput tells you how much output you can produce for a dollar. If it costsmore to produce output using labor than it does using capital, then selllabor and buy capital. This allows you to produce the same amount atlower cost. Only when the costs of production using each input are thesame 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)/(priceof 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 inputmix.

Example: 1985: John Deere acquisition of Versatile. John Deere had anold fashioned production line for making farm tractors. Very high fixed

costs, but low marginal costs. Versatile had a "garage" style productionfacility 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


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larger than or equal to 7 units, John Deere has lower costs ofproduction.

The long run decision between these two production processeswould depend on how many units you thought you would sell. Ifyou anticipated selling 7 or more units, use the John Deere

production process, if you thought you would sell 6 or fewerunits, use the Versatile production process.

Sensitivity analysis

Suppose you were uncertain about how many units you anticipatedselling. Sensitivity analysis allows you to build in uncertainty to youranalysis by determining the costs of various scenarios.

Suppose you thought that your uncertainty was best described by atrinomial random variable:

1. with p1 = .4, Q = 72. with p2 = .3, Q = 103. 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), theaverage cost of the expected output is not the same as the expectedaverage cost, because Versatile has a large advantage at smalloutputs, while Deere has a small advantage at high outputs. The tablebelow shows how to compute expected average costs, which is theusually right criterion to use for deciding which technology to adopt.

Short run production functions

A long run production function relates the output produced to theinputs used, e.g. Q = f(capital, labor). In the short run, some inputscannot be varied, so the firm does not have as much flexibility as inthe long run. In this case, the short run production function is afunction of only the inputs that can be varied. Suppose that capital isfixed in the short run. Then Y = g(labor)


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The "usual" shape of the short run production function:

1. In the short run, output at first increases at an increasing rate withincreases 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 longrun, all inputs can be varied, and this greater flexibility allows you toachieve lower costs, i.e. h(Q) is greater than or equal to g(Q).


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A Numerical example

Assume, the following short run production function: Fixed costs =$20/hour, Labor costs $5/hour * The marginal productivity of laborincreases, then is constant, and then decreases.

Suppose that the output sells for $1 per unit and that the firm can sellall it wants at a price of $1 (infinite elasticity, or a perfectly competitivefirm). How much labor should the firm hire?

ANSWER: Keep hiring as long as the benefit of hiring another worker isgreater than the cost of another worker. The benefit equals themarginal 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.


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Can we derive the optimal production decision (how much laborto produce) from the cost curves instead of the productioncurves?

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 somewherebetween 10 and 11 laborers (between 75 and 84 units). To determinewhether 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 variablecost. In the short run, you don't have to cover your fixed costs, but youmust make enough to cover your variable costs. If not, then you canshut down. You will still have to pay your fixed costs, but at least youcan avoid paying your variable costs (which are greater than revenue).

Break-even analysis

Assumptions: constant price, constant average variable cost. (P-AVC) issometimes called "contribution margin" because it represents profitper unit sold (ignoring fixed costs).

Set profits equal to zero to solve for how much output would berequired to "break even." Another way of asking the same question isto ask how much quantity would be required to produce enough profitto 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)


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Cost and Production