1

Firm Theory: production functions, cost curves and

profit maximization

2

Remarks

Switching gears: “Theory of the Firm” Embarking on an analysis of the firm. Note:

– There are lots of different types of firms.– There are lots of ways to organize

entrepreneurial activity.– There are lots of firm objectives.

3

What We Assume

For our analysis we assume that:– we have an owner manager,– who has a business,– with the primary and only objective to

maximize economic profit.

4

What’s a Business? A business is an organization producing goods or

services, also called a firm. A business, or firm, is assumed to maximize its profits. Examples of businesses: Microsoft, Kinko’s, the Campus

Store (a business within Cornell University). Examples of organizations that are not businesses in this

sense: Cornell University as a whole, US Department of Defense.

Organizations that produce goods and services but are legally prohibited from using the profit motive are excluded from our formal model of a firm.

5

What’s a Market?

A collection of buyers and sellers organized for the purpose of exchanging goods and services for money.

Markets can be global, national, regional, or local depending upon the item being bought and sold.

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Profit Maximization

profit = total revenue - total cost total revenue:

– determined by the level and nature of competition in your market

total cost:– costs are determined by factor market

prices and the firm’s technology or production function

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Economic Profits Economic profits are the difference between

total revenue and total costs. Economic total costs include the opportunity

costs of all inputs to the production process–in particular, the opportunity costs of the owner’s time and physical capital (equipment and space).

Whenever we talk about profit we mean economic profit.

8

Accounting Profits

Accounting profits are defined as total sales revenue (the same as total revenue in the economic profits definition) minus operating costs (costs of goods sold + administrative and sales costs for those who know some accounting).

Accounting Profits = Sales Revenue - Accounting Costs

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Production & Cost Structures

There are lots of ways to describe production and costs.

You need to understand them all. For example:

– total, fixed and variable concepts– average and marginal concepts– long run and short run concepts– all related to each other

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Jonathan’s New York State Apple Farm

The farm is a business organized to grow and sell apples.

The owner/proprietor, Jonathan, tries to maximize his profits from the business.

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The U.S. Apple Market Americans consume 19 lbs. of apples per person annually, for a

total consumption of 5 billion lbs. More than 6 billion lbs. of apples are grown in the US each year

(mostly in Washington, New York and Michigan). Only about 234 million lbs. are imported, while more than 1.3

billion lbs. are exported. There are about 1,700 apple farms in the State of New York, with

an average of 96 acres of orchards per farm and about $92,000/year in revenue from apple sales.

80% of the New York apple farms are individual or family owned. Markets like the US apple market are classic examples of

competitive product markets.

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Production Functions The production function shows the input requirements

for each level of production. For some businesses the production function is

relatively simple--a few processes with little substitution.

For some businesses the production function involves thousands of different processes and millions of substitution possibilities.

The production function is the economists summary of the input requirements for each level of production.

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Jonathan’s Apple Farm Production Function

The table describes Jonathan’s inputs for the annual production of apples shown in the first column.

Jonathan's Apple Farm Production Function

Apples (tons/year)

Land (acres)

Labor (hired)

Proprietor's time (hours)

0 100 0 1,10050 100 2,500 1,100

100 100 3,700 1,100150 100 5,000 1,100200 100 6,800 1,100250 100 10,000 1,100300 100 15,000 1,100350 100 27,000 1,100

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Fixed Factors

A fixed factor is one that does not vary as the quantity produced increases or decreases.

Some factors are fixed in the short run (managerial time).

Some factors are fixed in the medium run (cultivated acreage).

No factors are fixed in the long run.

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Jonathan’s Fixed Factors

Jonathan has two fixed factors

– His cultivated acreage (100 acres)

– His own managerial time (1,100 hours)

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Variable Factors

A variable factor is one that must be increased in order to increase output.

The classic variable factor is labor. Variable factors usually exhibit

diminishing marginal productivity--the amount of extra product generated by each additional unit of the input, holding other inputs constant, declines.

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Jonathan’s Variable Factors

Jonathan must vary his labor input to increase his production of apples.

At first this variation is modest going from 50 tons/year to 100 tons/year requires an additional 1,200 hours

Going from 200 to 250 tons/year requires an additional 3,200 hours.

Jonathan cannot increase the size of his farm, his acreage is fixed.

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Jonathan’s Marginal Product of Labor

The graph shows how the marginal product of labor rises, then falls for Jonathan’s apple farm. Marginal product is in red.

The marginal product is the extra amount of apple production (pounds/hour worked) that can be produced by an extra hour of work.

Average product (in blue) is the ratio of output to labor used. Average product of labor is usually called “labor productivity” in the business press.

Marginal Product of Labor

0.000

0.005

0.010

0.015

0.020

0.025

0.030

0.035

0.040

0.045

0 5,000 10,000 15,000

Hours of LaborT

on

s o

f a

pp

les

/ho

ur

Marginal Productof Labor (midpointformula)

Average Productof Labor

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The Average/Marginal Relation

For product curves or any other average/marginal pair of curves:– If marginal product is above average product, then

average product is rising.– If marginal product is below average product, then

average product is falling.– Therefore, marginal product equals average

product when average product is at a critical value, in this case a maximum.

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From Production Curves to Cost Curves

By combining the production function and the factor prices, we produce the business’s cost curves.

The total cost curve is the amount spent on all fixed and variable costs to produce the indicated output.

The average cost curve is the ratio of total costs to units produced.

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Marginal Cost

The most important cost concept is marginal cost.

Marginal cost measures the amount by which costs increase as output increases by one unit.

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7 Short Run Cost Curves Total values

– fc = fixed costs = PK K where K is fixed

– vc = variable costs = PLL*(x)

– srtc = short run total costs = fc+vc

Average values– afc = average fixed cost = fc/x

– avc = average variable cost = vc/x

– sratc=short run average total cost=srtc/x=afc+avc

Marginal value– srmc=short run marginal cost = srtc/x = vc/x

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Short Run Cost Curves

Short run cost curves get their shape from the marginal productivity of the variable factor (except the fixed costs, of course).

If capital is held constant (short run) then the marginal product of labor gives the short run cost curves their shape.

The levels of cost curves are determined by factor market prices along with technology.

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Jonathan’s Input Prices

Each of the entries in this table represents a price that Jonathan must pay for an input.

Notice that he “pays” for his managerial time because his next best alternative is to earn $12/hour.

He must pay rent for his land. Jonathan is a “wage taker” as well as “price-taker.”

PricesLabor's time $8.00 per hourOwner's time $12.00 per hourRent $124.00 per acre

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Jonathan’s Cost Curves

When Jonathan faces the technology and input prices shown previously, the table shows his cost structure.

Jonathan's Apple Farm Costs

Apples (tons/year) Land

Hired Labor

Proprietor's time Total Cost

Average Cost

Marginal Cost

(midpoint formula)

0 12,400 0 13,200 25,60050 12,400 20,000 13,200 45,600 912 296

100 12,400 29,600 13,200 55,200 552 200150 12,400 40,000 13,200 65,600 437 248200 12,400 54,400 13,200 80,000 400 400250 12,400 80,000 13,200 105,600 422 656300 12,400 120,000 13,200 145,600 485 1,360350 12,400 216,000 13,200 241,600 690

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Graph of Jonathan’s Cost Curves

The marginal cost of each ton of apples is shown as the red line.

The average total cost curve is shown as the blue line.

Notice that the marginal cost = average cost at minimum average cost.

Jonathan's Cost Functions

0

100

200

300

400500

600

700

800

900

1,000

0 100 200 300 400

Apples (tons/year)

$/t

on

Average Cost Marginal Cost (midpoint formula)

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Long Run Cost Curves Describing the long run cost curves requires a description of all

of the technological possibilities for operating in the industry. Now both labor and capital are considered variable. There are three long run cost curves for the firm:

– long run total cost = lrtc = PLL*(x) + PKK*(x)

– long run average total cost = lratc = lrtc/x

– long run marginal cost = lrmc = lrtc/x The most important one for us is the lratc curve. Choosing L*(x) and K*(x) will depend on the prices of capital and labor

relative to the marginal products of capital and labor and what x you expect to produce. The “bang/buck” condition would have to be met. That is, at L* and K*, the (mpL/PL )= (mpK/PK)

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Example: 3 Potential Technologies

Suppose there are three different ways for a company, System-fixer, to do business.

Firm sizes A, B and C illustrate the possibilities.

Firm A is small, using only $80 in fixed costs.

Firm B uses twice the capital.

Firm C uses three times the capital.

Long Run Cost Curves

Firm A (Capital investment A) Firm B (Capital Investment B) Firm C (Capital Investment C)

QuantityTotal Costs

Firm A Average

Total Cost

Marginal Cost

(midpoint formula) Quantity

Total Costs

Firm B Average

Total Cost

Marginal Cost

(midpoint formula) Quantity

Total Costs

Firm C Average

Total Cost

Marginal Cost

(midpoint formula)

0 80 0 160 0 2401 100 100.00 15.00 1 161 161.00 1.50 1 250 250.00 7.502 110 55.00 10.50 2 163 81.50 2.50 2 255 127.50 6.003 121 40.33 13.50 3 166 55.33 3.00 3 262 87.33 6.504 137 34.25 22.00 4 169 42.25 4.00 4 268 67.00 6.005 165 33.00 33.00 5 174 34.80 5.50 5 274 54.80 6.006 203 33.83 46.50 6 180 30.00 7.00 6 280 46.67 6.507 258 36.86 65.00 7 188 26.86 14.00 7 287 41.00 7.508 333 41.63 90.00 8 208 26.00 26.00 8 295 36.88 8.509 438 48.67 122.50 9 240 26.67 50.00 9 304 33.78 10.00

10 578 57.80 10 308 30.80 84.00 10 315 31.50 13.0011 11 408 37.09 125.00 11 330 30.00 30.0012 12 558 46.50 12 375 31.25 52.5013 13 13 435 33.46 80.0014 14 14 535 38.21 140.0015 15 15 715 47.67

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Question

What is the best technology for our system-fixer firm?

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Answer

It depends on how much System-fixer expects to produce and sell in the market.

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The Firm’s Long Run Average Total Cost Curve

The firm’s long run average total cost curve consists of the minimum of the three curves illustrated on the right.

System-fixer’s long run average total cost curve is size A’s (blue) until 6 units, size B’s (red) from 6 to 10 units and size C’s (brown) from 11 units onward.

The shape of the firm’s lratc curve will be determined by how the technology behaves.

The lratc is the outer envelope of the possible sratc curves.

Long Run Average Total Cost

0.00

10.00

20.00

30.00

40.00

50.00

60.00

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

Quantity (installations/week)

Co

st (

do

llar

s/in

stal

lati

on

)

Firm A Average Total CostMinimum Average Total CostFirm B Average Total CostMinimum Average Total CostFirm C Average Total CostMinimum Average Total Cost

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The Firm’s Long Run Average Total Cost Curve and MES

MES = minimum efficient scale MES = the level of output at the

minimum of the firm’s lratc curve.

When there are “economies of scale,” the lratc is declining.

When there are “diseconomies of scale,” the lratc is increasing.

At MES all the economies of scale are exhausted.

In real world empirical analysis, economists often observe a more “L-shaped” lratc curve.

Long Run Average Total Cost

0.00

10.00

20.00

30.00

40.00

50.00

60.00

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

Quantity (installations/week)

Co

st (

do

llar

s/in

stal

lati

on

)

Firm A Average Total CostMinimum Average Total CostFirm B Average Total CostMinimum Average Total CostFirm C Average Total CostMinimum Average Total Cost

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Profit Maximization

Profit () = total revenue(tr) - total cost(tc). Profit depends on the firm’s output level (x). So… (x) = tr(x) - tc(x) Define

– marginal revenue (mr) = tr/x– marginal cost (mc) = tc/x

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Profit Maximization General rules for profit maximization: If x* maximizes , then

– mr = mc at x*– x* is a profit max and not a profit min – at x* it’s worth operating

If using a short-run perspective use short run cost curves.

If using a long-run perspective uselong run cost curves.

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Running at a Profit

When the market price exceeds the firm’s average total cost at x*, the firm is running at a profit.

Great.

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Running at a Loss When the market price is less than the

firm’s average total cost at x*, the business is running at a loss.

What should the firm do in the short run?– produce x* if the firm takes in enough in

revenue to cover its variable costs.– shut down and play dead if its revenues at

x* don’t even cover variable costs.

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Running at a Loss

What should the firm do in the long run?– It should plan to go out of business if it

does not expect either the market price to rise or its costs to fall or both.

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Sunk Costs and Avoidable Fixed Costs In the real world the shut down rule is slightly different. When the firm actually operates, x>0, it makes sense to

only talk about fixed and variable costs. However, when the firm considers shutting down in the

short run (when profits are negative at the profit maximizing output level) then they have to take another look at their fixed costs.

Fixed costs=sunk costs + avoidable fixed costs Now change the shutdown rule to be:

– produce x* if the firm takes in enough in revenue to cover its variable costs + avoidable fixed costs, otherwise shut down.