The Firm and Profit Maximization

Overheads

Neoclassical firm -

A neoclassical firm is an organization that controls the

transformation of inputs (resources it owns or purchases)

into outputs (valued products that it sells),

and earns the difference between what it receives

in revenue, and what it spends on inputs

We define the production function as

f represents the relationship between outputs and inputs

xj is the quantity used of the jth input

(x1, x2, x3, . . . xn) is the input bundle

n is the number of inputs used by the firm

y represents output

f(x) maxy

[y: (x, y) is an element of the production set]

f(x1 , x2 , x3 , ) maxy ε P(x)

[y]

Returns (Profit)

The profit from a production plan is the revenue obtained

from the plan minus the cost of inputs used to implement it

π p y w1x1 w2x2

π p y Σn

i 1wixi

π Revenue Costs

Objectives of the firm

We typically assume that a firm existsin order to make money

A firm that wants to make money is called a for-profit firm, or a profit maximizing firm

The firm maximizes the returns from thetechnologies it controls taking into account:

Given the profit max assumption

The demand for final consumption goods

Opportunities for buying and selling factors / products

The actions of other firms in the market

The profit max problem

π(p , w1, w2 , ) maxx, y

p y Σn

i 1wixi such that y ε P (x)

π(p , w1 , w2 , ) maxy

py C(y , w1 , w2 , )

π(p, w1, w2 , ) maxx

pf (x1 , x2 , , xn) Σn

i 1wixi

What is profit?

Profit is revenue minus costs or

π Revenue Costs

R C

Explicit and implicit costs

Explicit costs

1. purchase of expendable inputs including labor time

2. purchase of capital services (usually rent or lease)

Implicit costs

1. value of produced expendables(feed for a cattle producer)

2. value of services provided by owned capitalincluding financial capital and charges such as implicitrent, depreciation, compensation for operator labor, etc.

Accounting profit

Accounting cost

Accounting profit

+ depreciation on plant & equipmentexplicit costs

total revenue - accounting cost

Economic profit

Economic cost

Economic profit

+ all implicit costsexplicit costs

total revenue - economic cost

Why do we use economic profit?

To reflect total costs and revenues for a decision

To account for all resources used in production

Why do profits exist?

All factors of production receive a payment

Expendables receive their market price

Labor receives wages

Land receives rent

Capital receives interest

Firm owners receive profits

What are profits?

Profits are the returns to

Innovation

Risk

production

delivery

new products

The Profit Maximizing Output Level

Profit = Revenue - Cost = R - C

π Revenue Cost

pf (x1 , x2 , , xn) Σn

i 1wixi

py C(y , w1 , w2 , ,wn)

Demand

The individual demand curve facing a firm tells us, for different prices, the quantity of output that customers will choose to purchase from the firm

The demand curve facing the firm show us themaximum price the firm can charge to sellany given amount of output

Example Inverse Demands

p = 320 - 20y

p = $1.85

Total Revenue

Revenue is the total income that comes from thesale of the output (goods and services) of a givenfirm or production process

Revenue R(p , y) py pf (x1 , x2 , , xn )

Y Price FC VC C TR Profit0.00 320 120 0.00 120.00 0 -120.00

1.00 300 120 64.00 184.00 300 116.00 2.00 280 120 130.00 250.00 560 310.00 3.00 260 120 204.00 324.00 780 456.00

4.00 240 120 292.00 412.00 960 548.00

5.00 220 120 400.00 520.00 1100 580.00

Y Price FC VC C TR Profit0.00 320 120 0.00 120.00 0 -120.00 1.00 300 120 64.00 184.00 300 116.00 2.00 280 120 130.00 250.00 560 310.00 3.00 260 120 204.00 324.00 780 456.00 4.00 240 120 292.00 412.00 960 548.00 5.00 220 120 400.00 520.00 1100 580.00 6.00 200 120 534.00 654.00 1200 546.00 7.00 180 120 700.00 820.00 1260 440.00 8.00 160 120 904.00 1024.00 1280 256.00 9.00 140 120 1152.00 1272.00 1260 -12.00 10.00 120 120 1450.00 1570.00 1200 -370.00 11.00 100 120 1804.00 1924.00 1100 -824.00 12.00 80 120 2220.00 2340.00 960 -1380.00 14.00 40 120 3262.00 3382.00 560 -2822.00 16.00 0 120 4624.00 4744.00 0 -4744.00

Total cost

Cost TC C(x1 , x2 , , w1 , w2 , )

Σn

i 1wixi

C(y , w1 , w2 , )

Maximizing profit

Choose the level of output wherethe difference between TR and TCis the greatest

Y Price FC VC C TR Profit0.00 320 120 0.00 120.00 0 -120.00

1.00 300 120 64.00 184.00 300 116.00 2.00 280 120 130.00 250.00 560 310.00 3.00 260 120 204.00 324.00 780 456.00

4.00 240 120 292.00 412.00 960 548.00

5.00 220 120 400.00 520.00 1100 580.00 6.00 200 120 534.00 654.00 1200 546.00 7.00 180 120 700.00 820.00 1260 440.00 8.00 160 120 904.00 1024.00 1280 256.00 9.00 140 120 1152.00 1272.00 1260 -12.00 10.00 120 120 1450.00 1570.00 1200 -370.00 11.00 100 120 1804.00 1924.00 1100 -824.00 12.00 80 120 2220.00 2340.00 960 -1380.00

Y Price FC VC C TR Profit0.00 320 120 0.00 120.00 0 -120.00 1.00 300 120 64.00 184.00 300 116.00 2.00 280 120 130.00 250.00 560 310.00 3.00 260 120 204.00 324.00 780 456.00 4.00 240 120 292.00 412.00 960 548.00

5.00 220 120 400.00 520.00 1100 580.00

6.00 200 120 534.00 654.00 1200 546.00 7.00 180 120 700.00 820.00 1260 440.00 8.00 160 120 904.00 1024.00 1280 256.00 9.00 140 120 1152.00 1272.00 1260 -12.00 10.00 120 120 1450.00 1570.00 1200 -370.00 11.00 100 120 1804.00 1924.00 1100 -824.00 12.00 80 120 2220.00 2340.00 960 -1380.00 14.00 40 120 3262.00 3382.00 560 -2822.00 16.00 0 120 4624.00 4744.00 0 -4744.00

Marginal cost (MC)

Marginal cost is the increment, or addition,to cost that results from producingone more unit of output

MC ΔC(y,w)Δy

ΔTC(y,w)Δy

y Price FC VC C AFC AVC ATC MC TR MR Profit0.00 320 120 0.00 120.00 0 -120.00

64.00 300.00 1.00 300 120 64.00 184.00 120.00 64.00 184.00 300 116.00

66.00 260.00 2.00 280 120 130.00 250.00 60.00 65.00 125.00 560 310.00

74.00 220.00

3.00 260 120 204.00 324.00 40.00 68.00 108.00 780 456.00

88.00 180.00

4.00 240 120 292.00 412.00 30.00 73.00 103.00 960 548.00108.00 140.00

5.00 220 120 400.00 520.00 24.00 80.00 104.00 1100 580.00134.00 100.00

6.00 200 120 534.00 654.00 20.00 89.00 109.00 1200 546.00166.00 60.00

7.00 180 120 700.00 820.00 17.14 100.00 117.14 1260 440.00204.00 20.00

8.00 160 120 904.00 1024.00 15.00 113.00 128.00 1280 256.00248.00 -20.00

9.00 140 120 1152.00 1272.00 13.33 128.00 141.33 1260 -12.00298.00 -60.00

10.00 120 120 1450.00 1570.00 12.00 145.00 157.00 1200 -370.00

MC ΔC(y,w)Δy

(412 324)(4 3)

881

88

y Price FC VC C AFC AVC ATC MC3 260 120 204.00 324.0 40.00 68.00 108.00

88.004 240 120 292.00 412.0 30.00 73.00 103.00

108.005 220 120 400.00 520.0 24.00 80.00 104.00

Marginal Revenue (MR)

MR ΔR(y,p)Δy

ΔTR(y,p)Δy

Marginal revenue is the increment, or addition,to revenue that results from producingone more unit of output

Marginal revenue is the change in total revenuefrom producing one more unit of output

y Price FC VC C AFC AVC ATC MC TR MR Profit

0 320 120 0.00 120.0 0 -120.064.00 300.0

1 300 120 64.00 184.0 120.00 64.00 184.00 300 116.066.00 260.0

2 280 120 130.00 250.0 60.00 65.00 125.00 560 310.074.00 220.0

3 260 120 204.0 324.0 40.0 68.0 108.0 780 456.0

88.00 180.04 240 120 292.0 412.0 30.0 73.0 103.0 960 548.0

108.00 140.05 220 120 400.00 520.0 24.00 80.00 104.00 1100 580.0

134.00 100.06 200 120 534.00 654.0 20.00 89.00 109.00 1200 546.0

166.00 60.07 180 120 700.00 820.0 17.14 100.00 117.14 1260 440.0

204.00 20.08 160 120 904.00 1024.0 15.00 113.00 128.00 1280 256.0

y Price TR MR Profit3.00 260 780 456.0

180.04.00 240 960 548.0

140.05.00 220 1100 580.0

100.06.00 200 1200 546.0

MR ΔTR(y,p)Δy

(960 780)(4 3)

1801

180

y Price TR MR Profit3.00 260 780 456.0

180.04.00 240 960 548.0

140.05.00 220 1100 580.0

100.06.00 200 1200 546.0

Another example Increase output from 4 to 5 units

MR ΔTR(y,p)Δy

(1100 960)(5 4)

1401

140

A note on marginal revenue and price

Marginal revenue is always less than price

The firm must lower price in order to sell more units

WHY?

p

qq0

1 2 3 4

p0

5

Revenue Marginal Revenue

Marginal Revenue = p0

The lower price applies to all units and so the revenue per unit will be less than the price

p0

q0

p1

q1

B

A

p

q

Demand

MR = A - B

Profit Max Using MR and MC

An increase in output will always increase profit

if MR > MC

An increase in output will always decrease profit

if MR < MC

The rule is then

Increase output whenever MR > MC

Decrease output if MR < MC

y Price C MC TR MR Profit3.00 260 324.00 780 456.00

88.00 180.00 4.00 240 412.00 960 548.00

108.00 140.00 5.00 220 520.00 1100 580.00

134.00 100.00 6.00 200 654.00 1200 546.00

Example

Should we increase output from 3 to 4?

Should we increase output from 4 to 5?

Should we increase output from 5 to 6?

Yes

Yes

No !

Profit Maximization Using Graphs

Profit is positive if TR > TC

0200400600800100012001400160018002000

0 2 4 6 8 10 12 14 16 18Output

$

TRC

Profit Maximization Using MR and MC

Profit on a given unit is positive if MR > MC

050100150200250300350400

0 2 4 6 8 10 12 14 16 18Output

$ MC

DemandMR

Two intersections of MC and MR

0

50

100

150

200

250

300

0 2 4 6 8 10 12 14 16 18 20Output

$

MC

DemandMR

The optimal output level occurs where MC intersects MR from below

0

50

100

150

200

250

300

0 2 4 6 8 10 12 14 16 18 20Output

$

MC

DemandMR

Why average costs are irrelevant in the short-run

The short-run decision is whether to produceone more unit or not

Only marginal cost and marginal revenueare relevant for this decision

Marginal Decision Making and Short-run Decisions

The marginal approach to profit statesthat a firm should take any action that addsmore to its revenue than to its cost

Examples where marginal decision making is relevant

advertising

cost efficiency consultant

adding a salesperson

sprucing up sales area

adding a two-year warranty to product

The shutdown ruleDo we keep producing if we are losing money?

It depends on what we mean by a loss

It depends on whetherwe are in the short-run or in the long run

It depends on which costs are fixed,which are variable, and which are sunk

Case 1 - TC > TR at all Q

TR > TVC where MR = MC

TC > TR at all Q

TR > TVC whereMR = MC

0255075100125150175200

0 1 2 3 4 5 6 7 8 9 10Output - y

Co

st

VC

CTR

Marginal Cost & Revenue Curves

0510152025303540

0 1 2 3 4 5 6 7 8 9 10Output - y

Co

st MC

MR

Total Cost & Revenue Curves

In the short-run fixed costs must be paidindependent of the level of output

At 6 units of output, total revenue more thancovers total variable costs,

leaving a residual to help cover fixed costs

So the firm should produce 6 units in the short run

The shutdown rule

In the short-run, the firm should continue to produceif total revenue exceeds total variable costs;

otherwise, it should shut down

Case 2 - TC > TR at all Q

TR < TVC where MR = MC

TR < TCat all Q

TR < TVC whereMR = MC

Total Revenue and Cost Curves

0102030405060708090

0 1 2 3 4 5 6 7Output - y

Co

st

TVC

CTR

Marginal Revenue and Cost Curves

048121620242832

0 1 2 3 4 5 6 7Output - y

Co

st MC

MR

The shutdown rule

In the short-run, the firm should continue to produceif total revenue exceeds total variable costs;

otherwise, it should shut down

Shutdown in the long-run

In the long-run the firm should exitthe industry if it has any size loss

Shutdown and fixed costs that are not sunk

In the short-run, if some of the fixed costs are not sunk,the firm may be better off to not operate when TR > TVC,if it can recover most of its sunk costs that are fixed costs,by shutting down

Suppose at 6 units of output, TFC =$50, TVC = $60 and TC = $110

But suppose that only $10 of the fixed costs are sunk,so that by shutting down the firm can recover $40.00 of fixed cost

Suppose at 6 units of output, that total revenue is equal to $95.

By operating the firm can make $35 over variable costs

The firms net loss is then just $15.00

The firm's net loss by shutting down is only $10.00

The firm is better off by shutting down

This will help cover the the fixed costs of $50

(95 - 60)

(35 - 50 or 95 - 110)

Asset Fixed Variable Total Sales Disposal Total Profit/

Cost Cost Cost Revenue Revenue Revenue LossOperate 50 60 110 95 0 95 -15Shut-down 50 0 50 0 40 40 -10

The End