chapter thirty

Chapter Thirty

Production

Exchange Economies (revisited)

No production, only endowments, so no description of how resources are converted to consumables.

General equilibrium: all markets clear simultaneously.

1st and 2nd Fundamental Theorems of Welfare Economics.

Now Add Production ...

Add input markets, output markets, describe firms’ technologies, the distributions of firms’ outputs and profits …

Now Add Production ...

Add input markets, output markets, describe firms’ technologies, the distributions of firms’ outputs and profits … That’s not easy!

Robinson Crusoe’s Economy

One agent, RC. Endowed with a fixed quantity of one

resource -- 24 hours. Use time for labor (production) or

leisure (consumption). Labor time = L. Leisure time = 24 - L. What will RC choose?

Robinson Crusoe’s Technology

Technology: Labor produces output (coconuts) according to a concave production function.


Production function

Labor (hours)

Coconuts

240


Labor (hours)

Coconuts

Production function

240

Feasible productionplans

Robinson Crusoe’s Preferences

RC’s preferences:

– coconut is a good

– leisure is a good


Leisure (hours)

Coconuts

More preferred

240


Leisure (hours)

Coconuts

More preferred

24 0

Robinson Crusoe’s Choice

Labor (hours)

Coconuts


Production function

240


Labor (hours)

Coconuts


Production function

240

Leisure (hours)24 0


Labor (hours)

Coconuts

Production function

240

Leisure (hours)24 0

C*

L*


Labor (hours)

Coconuts

Production function

240

Leisure (hours)24 0

C*

L*

Labor


Labor (hours)

Coconuts

Production function

240

Leisure (hours)24 0

C*

L*

Labor Leisure


Labor (hours)

Coconuts

Production function

240

Leisure (hours)24 0

C*

L*

Labor Leisure

Output


Labor (hours)

Coconuts

Production function

240

Leisure (hours)24 0

C*

L*

Labor Leisure

MRS = MPL

Output

Robinson Crusoe as a Firm

Now suppose RC is both a utility-maximizing consumer and a profit-maximizing firm.

Use coconuts as the numeraire good; i.e. price of a coconut = $1.

RC’s wage rate is w. Coconut output level is C.

Robinson Crusoe as a Firm

RC’s firm’s profit is = C - wL. = C - wL C = + wL, the

equation of an isoprofit line. Slope = + w . Intercept = .

Isoprofit Lines

Labor (hours)

Coconuts

24

C wL Higher profit; 1 2 3

Slopes = + w 3 21

0

Profit-Maximization

Labor (hours)

Coconuts


Production function

240

Profit-Maximization

Labor (hours)

Coconuts

Production function

240

Profit-Maximization

Labor (hours)

Coconuts

Production function

24

C*

L*0

Profit-Maximization

Labor (hours)

Coconuts

Production function

24

C*

L*

Isoprofit slope = production function slope

0

Profit-Maximization

Labor (hours)

Coconuts

Production function

24

C*

L*

Isoprofit slope = production function slope i.e. w = MPL

0

Profit-Maximization

Labor (hours)

Coconuts

Production function

24

C*

L*

Isoprofit slope = production function slope i.e. w = MPL = 1 MPL = MRPL.

0

Profit-Maximization

Labor (hours)

Coconuts

Production function

24

C*

L*


*

* * * C wLRC gets

0

Profit-Maximization

Labor (hours)

Coconuts

Production function

24

C*

L*


* * * C wL

* Given w, RC’s firm’s quantitydemanded of labor is L*Labor

demand

RC gets

0

Profit-Maximization

Labor (hours)

Coconuts

Production function

24

C*

L*


* Given w, RC’s firm’s quantitydemanded of labor is L* andoutput quantity supplied is C*.

Labordemand

Outputsupply

* * * C wLRC gets

0

Utility-Maximization

Now consider RC as a consumer endowed with $* who can work for $w per hour.

What is RC’s most preferred consumption bundle?

Budget constraint is C wL * .


Labor (hours)

Coconuts

*

240

C wL * .Budget constraint


Labor (hours)

Coconuts

*

240

C wL * .Budget constraint; slope = w


Labor (hours)

Coconuts

More preferred

240


Labor (hours)

Coconuts

*

240

C wL * .Budget constraint; slope = w


Labor (hours)

Coconuts

*

Budget constraint; slope = w

240

C wL * .


Labor (hours)

Coconuts

*

240

C wL * .C*

L*

Budget constraint; slope = w


Labor (hours)

Coconuts

*

240

C wL * .C*

L*

MRS = wBudget constraint; slope = w


Labor (hours)

Coconuts

*

240

C wL * .C*

L*

Laborsupply

Budget constraint; slope = wMRS = w

Given w, RC’s quantitysupplied of labor is L*


Labor (hours)

Coconuts

*

240

C wL * .C*

L*

Given w, RC’s quantitysupplied of labor is L* andoutput quantity demanded is C*.

Laborsupply

Outputdemand

Budget constraint; slope = wMRS = w

Utility-Maximization & Profit-Maximization

Profit-maximization:

– w = MPL

– quantity of output supplied = C*

– quantity of labor demanded = L*



– w = MPL

– quantity of output supplied = C*– quantity of labor demanded = L*

Utility-maximization: – w = MRS– quantity of output demanded = C*– quantity of labor supplied = L*



– w = MPL

– quantity of output supplied = C*– quantity of labor demanded = L*

Utility-maximization: – w = MRS– quantity of output demanded = C*– quantity of labor supplied = L*

Coconut and labormarkets both clear.


Labor (hours)

Coconuts

24

C*

L*

*

0

MRS = w = MPL

Given w, RC’s quantitysupplied of labor = quantitydemanded of labor = L* andoutput quantity demanded =output quantity supplied = C*.

Pareto Efficiency

Must have MRS = MPL.

Pareto Efficiency

Labor (hours)

Coconuts

240

MRS MPL

Pareto Efficiency

Labor (hours)

Coconuts

240

MRS MPL

Preferred consumptionbundles.

Pareto Efficiency

Labor (hours)

Coconuts

240

MRS = MPL

Pareto Efficiency

Labor (hours)

Coconuts

240

MRS = MPL. The common slope relative wage rate w that implements the Pareto efficient plan by decentralized pricing.

First Fundamental Theorem of Welfare Economics

A competitive market equilibrium is Pareto efficient if

– consumers’ preferences are convex

– there are no externalities in consumption or production.

Second Fundamental Theorem of Welfare Economics

Any Pareto efficient economic state can be achieved as a competitive market equilibrium if

– consumers’ preferences are convex

– firms’ technologies are convex

– there are no externalities in consumption or production.

Non-Convex Technologies

Do the Welfare Theorems hold if firms have non-convex technologies?



The 1st Theorem does not rely upon firms’ technologies being convex.


Labor (hours)

Coconuts

240

MRS = MPL The common slope relative wage rate w that implements the Pareto efficient plan by decentralized pricing.



The 2nd Theorem does require that firms’ technologies be convex.


Labor (hours)

Coconuts

240

MRS = MPL. The Pareto optimal allocation cannot be implemented by a competitive equilibrium.

Production Possibilities

Resource and technological limitations restrict what an economy can produce.

The set of all feasible output bundles is the economy’s production possibility set.

The set’s outer boundary is the production possibility frontier.


Fish

Coconuts

Production possibility frontier (ppf)


Fish

Coconuts

Production possibility frontier (ppf)

Production possibility set


Fish

Coconuts

Feasible butinefficient


Fish

Coconuts


Feasible and efficient


Fish

Coconuts


Feasible and efficient

Infeasible


Fish

Coconuts

Ppf’s slope is the marginal rateof product transformation.


Fish

Coconuts

Ppf’s slope is the marginal rateof product transformation.

Increasingly negative MRPT increasing opportunitycost to specialization.


If there are no production externalities then a ppf will be concave w.r.t. the origin.

Why?


If there are no production externalities then a ppf will be concave w.r.t. the origin.

Why? Because efficient production

requires exploitation of comparative advantages.

Comparative Advantage

Two agents, RC and Man Friday (MF). RC can produce at most 20 coconuts

or 30 fish. MF can produce at most 50 coconuts

or 25 fish.


F

C

F

C

RC

MF

20

50

30

25


F

C

F

C

RC

MF

20

50

30

25

MRPT = -2/3 coconuts/fish so opp. cost of onemore fish is 2/3 foregone coconuts.


F

C

F

C

RC

MF

20

50

30

25


MRPT = -2 coconuts/fish so opp. cost of onemore fish is 2 foregone coconuts.


F

C

F

C

RC

MF

20

50

30

25


MRPT = -2 coconuts/fish so opp. cost of onemore fish is 2 foregone coconuts.

RC has the comparativeopp. cost advantage inproducing fish.


F

C

F

C

RC

MF

20

50

30

25

MRPT = -2/3 coconuts/fish so opp. cost of onemore coconut is 3/2 foregone fish.


F

C

F

C

RC

MF

20

50

30

25


MRPT = -2 coconuts/fish so opp. cost of onemore coconut is 1/2 foregone fish.


F

C

F

C

RC

MF

20

50

30

25


MRPT = -2 coconuts/fish so opp. cost of onemore coconut is 1/2 foregone fish.

MF has the comparativeopp. cost advantage inproducing coconuts.


F

C

Economy

F

C

F

C

RC

MF

20

50

30

25

70

55

50

30

Use RC to producefish before using MF.

Use MF toproducecoconuts before using RC.


F

C

Economy

F

C

F

C

RC

MF

20

50

30

25

70

55

50

30

Using low opp. costproducers first resultsin a ppf that is concave w.r.t the origin.


F

C

Economy

More producers withdifferent opp. costs“smooth out” the ppf.

Coordinating Production & Consumption

The ppf contains many technically efficient output bundles.

Which are Pareto efficient for consumers?


Fish

Coconuts

C

F

Output bundle is ( , ) F C


Fish

Coconuts

C

F

Output bundle isand is the aggregateendowment for distribution to consumers RC and MF.

( , ) F C


Fish

Coconuts

ORC

OMFC

F

Output bundle isand is the aggregateendowment for distribution to consumers RC and MF.

( , ) F C


Fish

Coconuts

ORC

OMFC

F

Allocate efficiently;say to RC

( , ) F C

CRC

FRC

( , ) F CRC RC


Fish

Coconuts

ORC

OMFC

F

Allocate efficiently;say to RC and to MF.

( , ) F C

CRC CMF

FMF

FRC

( , ) F CRC RC

( , ) F CMF MF


Fish

Coconuts

ORC

OMFC

F

CRC CMF

FMF

FRC


Fish

Coconuts

ORC

OMFC

F

CRC CMF

FMF

FRC

MRS MRPT


Fish

Coconuts

ORC

OMFC

F

CRC CMF

FMF

FRC

O’MFC

F

( , ). F CInstead produce


Fish

Coconuts

ORC

OMFC

F

CRC CMF

FMF

FRC

C

F

( , ). F C

O’MF

CMF

Instead produceGive MF same allocation as before.FMF


Fish

Coconuts

ORC

OMFC

F

CRC CMF

FMF

FRC

C

F

( , ). F C

O’MF

CMF

Instead produceGive MF same allocation as before. MF’s utility is unchanged.

FMF


Fish

Coconuts

ORC

OMF

C

F

( , ). F C

O’MFFMF

Instead produceGive MF same allocation as before. MF’s utility is unchanged

CMF


Fish

Coconuts

ORC

OMF

FRC

C

F

( , ). F C

O’MF

CRC

FMF

CMF

Instead produceGive MF same allocation as before. MF’s utility is unchanged


Fish

Coconuts

ORC

OMF

FRC

C

F

( , ). F C

O’MF

CRC

FMF

CMF

Instead produceGive MF same allocation as before. MF’s utility is unchanged, RC’s utility is higher


Fish

Coconuts

ORC

OMF

FRC

C

F

( , ). F C

O’MF

CRC

FMF

CMF

Instead produceGive MF same allocation as before. MF’s utility is unchanged, RC’s utility is higher; Pareto improvement.


MRS MRPT inefficient coordination of production and consumption.

Hence, MRS = MRPT is necessary for a Pareto optimal economic state.


Fish

Coconuts

ORC

OMF

FRC

C

F

CRC

FMF

CMF

Decentralized Coordination of Production & Consumption

RC and MF jointly run a firm producing coconuts and fish.

RC and MF are also consumers who can sell labor.

Price of coconut = pC.

Price of fish = pF.

RC’s wage rate = wRC.

MF’s wage rate = wMF.


LRC, LMF are amounts of labor purchased from RC and MF.

Firm’s profit-maximization problem is choose C, F, LRC and LMF to

max . p C p F w L w LC F RC RC MF MF


max . p C p F w L w LC F RC RC MF MFIsoprofit line equation is

constant p C p F w L w LC F RC RC MF MF



constant p C p F w L w LC F RC RC MF MFwhich rearranges to

Cw L w L

ppp

FRC RC MF MF

C

F

C

.



constant p C p F w L w LC F RC RC MF MFwhich rearranges to

Cw L w L

ppp

FRC RC MF MF

C

F

C

intercept slope 2

.


Fish

CoconutsHigher profit

Slopes = pp

F

C


Fish

Coconuts

The firm’s productionpossibility set.


Fish

Coconuts

Slopes = pp

F

C


Fish

Coconuts

Profit-max. plan

Slopes = pp

F

C


Fish

Coconuts

Profit-max. plan

Slope = pp

F

C


Fish

Coconuts

Profit-max. plan

Slope = pp

F

CCompetitive marketsand profit-maximization

MRPTpp

F

C .


So competitive markets, profit-maximization, and utility maximization all together cause

the condition necessary for a Pareto optimal economic state.

MRPTpp

MRSF

C ,


Fish

Coconuts

ORC

OMF

FRC

C

F

CRC

FMF

CMF

Competitive marketsand utility-maximization

MRSpp

F

C .


Fish

Coconuts

ORC

OMF

FRC

C

F

CRC

FMF

CMF

Competitive markets, utility-maximization and profit- maximization

MRSpp

MRPTF

C .

chapter thirty

Documents

labor production

concave production function

isoprofit slope

output markets

firmrcs firms profit

c wl c

firms technologies

input markets