Chapter Thirty-One
Production
生产
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 …
?That’s not easy!
Contents
? A one-person economy
– Optimal outcome
– Competitive equilibrium
– Two welfare theorems
– Non-convex technologies
? Two-people economy
– Production possibility frontier
– Comparative advantage
– Pareto efficient allocation
– Competitive equilibrium
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.
Robinson Crusoe’s Technology
Production function
Labor (hours)
Coconuts
240
Robinson Crusoe’s Technology
Labor (hours)
Coconuts
Production function
240
Feasible production
plans
Robinson Crusoe’s Preferences
?RC’s preferences:
–coconut is a good
–leisure is a good
Robinson Crusoe’s Preferences
Leisure (hours)
Coconuts
More preferred
240
Robinson Crusoe’s Preferences
Leisure (hours)
Coconuts
More preferred
24 0
Robinson Crusoe’s Choice
Labor (hours)
Coconuts
Feasible production
plans
Production function
240
Robinson Crusoe’s Choice
Labor (hours)
Coconuts
Feasible production
plans
Production function
240
Leisure (hours)24 0
Robinson Crusoe’s Choice
Labor (hours)
Coconuts
Feasible production
plans
Production function
240
Leisure (hours)24 0
Robinson Crusoe’s Choice
Labor (hours)
Coconuts
Feasible production
plans
Production function
240
Leisure (hours)24 0
Robinson Crusoe’s Choice
Labor (hours)
Coconuts
Production function
240
Leisure (hours)24 0
C*
L*
Labor Leisure
Ou
tpu
t
MRS = MPL
Competitive Equilibrium
? A firm
– Produce coconuts
– Demand labor
? A consumer
– Demand coconuts
– Supply labor
? Equilibrium
– Market clears for coconuts
– Market clears for labor
Competitive Equilibrium
?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.
?Labor supply is L.
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?2
?1
0
Profit-Maximization
Labor (hours)
Coconuts
Feasible production
plans
Production function
240
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*
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 = 1? MPL = MRPL.
0
Profit-Maximization
Labor (hours)
Coconuts
Production function
24
C*
L*
Isoprofit slope = production function slope
i.e,w = MPL = 1? MPL = MRPL.
?*
? * * *? ?C wLRC gets0
Profit-Maximization
Labor (hours)
Coconuts
Production function
24
C*
L*
Isoprofit slope = production function slope
i.e,w = MPL = 1? MPL = MRPL.
? * * *? ?C wL
?* Given w,RC’s firm’s quantity
demanded of labor is L*Labor
demand
RC gets
0
Profit-Maximization
Labor (hours)
Coconuts
Production function
24
C*
L*
Isoprofit slope = production function slope
i.e,w = MPL.
?* Given w,RC’s firm’s quantity
demanded of labor is L* and
output quantity supplied is C*.Labordemand
Output
supply
? * * *? ?C wLRC gets0
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? ?? *,
Utility-Maximization
Labor (hours)
Coconuts
?*
240
C wL? ?? *,Budget constraint; slope = w
Utility-Maximization
Labor (hours)
Coconuts
More preferred
240
Utility-Maximization
Labor (hours)
Coconuts
?*
240
C wL? ?? *,Budget constraint; slope = w
Utility-Maximization
Labor (hours)
Coconuts
?*
Budget constraint; slope = w
240
C wL? ?? *,
Utility-Maximization
Labor (hours)
Coconuts
?*
240
C wL? ?? *,C*
L*
MRS = w
Budget constraint; slope = w
Utility-Maximization
Labor (hours)
Coconuts
?*
240
C wL? ?? *,C*
L*
Labor
supply
Budget constraint; slope = w
MRS = w
Given w,RC’s quantity
supplied of labor is L*
Utility-Maximization
Labor (hours)
Coconuts
?*
240
C wL? ?? *,C*
L*
Given w,RC’s quantity
supplied of labor is L* and
output quantity demanded is C*.
Labor
supply
Output
demand
Budget constraint; slope = w
MRS = w
Utility-Maximization & Profit-
Maximization
? Profit-maximization,
– 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 labor markets both clear.
At Equilibrium …
Labor (hours)
Coconuts
24
C*
L*
?*
0
MRS = w = MPL
Given w,RC’s quantity
supplied of labor = quantity
demanded of labor = L* and
output 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 consumption
bundles.
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
–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
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.
Non-Convex Technologies
Labor (hours)
Coconuts
240
MRS = MPL The common slope ? relative
wage rate w that
implements the Pareto
efficient plan by
decentralized pricing.
Non-Convex Technologies
?Do the Welfare Theorems hold if
firms have non-convex
technologies?
?The 2nd Theorem does require that
firms’ technologies be convex.
Non-Convex Technologies
Labor (hours)
Coconuts
240
MRS = MPL,The Pareto optimal allocation
cannot be implemented by
a competitive equilibrium.
A Two-People Economy
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 (
生产可能性集 ).
?The set’s outer boundary is the production
possibility frontier（ 生产可能性边界 ）,
Production Possibilities
Fish
Coconuts
Production possibility frontier (ppf)
Production Possibilities
Fish
Coconuts
Production possibility frontier (ppf)
Production possibility set
Production Possibilities
Fish
Coconuts
Feasible but
inefficient
Production Possibilities
Fish
Coconuts
Feasible but
inefficient
Feasible and efficient
Production Possibilities
Fish
Coconuts
Feasible but
inefficient
Feasible and efficient
Infeasible
Production Possibilities
Fish
Coconuts
Ppf’s slope is the marginal rate
of product transformation
(边际生产转换率 ),or marginal rate
of transformation (边际转换率 ).
Production Possibilities
Fish
Coconuts
Ppf’s slope is the marginal rate
of product transformation.
Increasingly negative MRPT
? increasing opportunity
cost to specialization.
Production Possibilities
?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(比较优势 ).
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.
Comparative Advantage
F
C
F
C
RC
MF
20
50
30
25
MRPT = -2/3 coconuts/fish so opp,cost of one
more coconut is 3/2 foregone fish.
MRPT = -2 coconuts/fish so opp,cost of one
more coconut is 1/2 foregone fish.
MF has the comparative
opp,cost advantage in
producing coconuts.
RC has the comparative
opp,cost advantage in
producing fish.
Comparative Advantage
F
C
Economy
F
C
F
C
RC
MF
20
50
30
25
70
55
50
30
Use RC to produce
fish before using MF.
Use MF to
produce
coconuts before
using RC.
Comparative Advantage,One
Economy
F
C
Economy
F
C
F
C
RC
MF
20
50
30
25
70
55
50
30
Using low opp,cost
producers first results
in a ppf that is concave
w.r.t the origin.
Comparative Advantage,One
Economy
F
C
Economy
More producers with
different opp,costs
“smooth out” the ppf.
Comparative Advantage,
Separate Economies
F
C
F
C
RC
MF
20
50
30
25
Autarky,Produce and consume at
A.
Trade,Both countries completely
specialize,RC specializes on Fish;
MF specializes on coconuts.
Produce at B and consume at C,
Better off with trade.
A
B
C
A
B
C
Comparative Advantage,Trade
Equilibrium
F
C
F
C
RC
MF
20
50
30
25
Prices adjust to clear the markets.
At equilibrium,export of one
country = import of the other
country for each product.
o
o
Efficient Coordination of
Production & Consumption
?The ppf contains many technically
efficient output bundles.
?Which are Pareto efficient?
Fish
Coconuts
?C
?F
Output bundle is (,)? ?F C
Efficient Coordination of
Production & Consumption
Fish
Coconuts
?C
?F
Output bundle is
and is the aggregate
endowment for distribution
to consumers RC and MF.
(,)? ?F C
Efficient Coordination of
Production & Consumption
Fish
Coconuts
ORC
OMF?C
?F
Output bundle is
and is the aggregate
endowment for distribution
to consumers RC and MF.
(,)? ?F C
Efficient Coordination of
Production & Consumption
Fish
Coconuts
ORC
OMF?C
?F
Allocate efficiently;
say to RC(,)? ?F C
?CRC
?FRC
(,)? ?F CRC RC
Efficient Coordination of
Production & Consumption
Fish
Coconuts
ORC
OMF?C
?F
Allocate efficiently;
say to RC and
to MF.
(,)? ?F C
?CRC ?CMF
?FMF
?FRC
(,)? ?F CRC RC
(,)? ?F CMF MF
Efficient Coordination of
Production & Consumption
Fish
Coconuts
ORC
OMF?C
?F
?CRC ?CMF
?FMF
?FRC
MRS ? MRPT
Efficient Coordination of
Production & Consumption
Fish
Coconuts
ORC
OMF?C
?F
?CRC ?CMF
?FMF
?FRC
O’MF
??C
??F
(,).?? ??F CInstead produce
Efficient Coordination of
Production & Consumption
Fish
Coconuts
ORC
OMF?C
?F
?CRC ?CMF
?FMF
?FRC
O’MF
??C
??F
(,).?? ??F CInstead produce
Efficient Coordination of
Production & Consumption
Fish
Coconuts
ORC
OMF?C
?F
?CRC ?CMF
?FMF
?FRC
??C
??F
(,).?? ??F C
O’MF
?CMF
Instead produce
Give MF same allocation
as before.?FMF
Efficient Coordination of
Production & Consumption
Fish
Coconuts
ORC
OMF?C
?F
?CRC ?CMF
?FMF
?FRC
??C
??F
(,).?? ??F C
O’MF
?CMF
Instead produce
Give MF same allocation
as before,MF’s
utility is
unchanged.
?FMF
Efficient Coordination of
Production & Consumption
Fish
Coconuts
ORC
OMF
??C
??F
(,).?? ??F C
O’MF?FMF
Instead produce
Give MF same allocation
as before,MF’s
utility is
unchanged
?CMF
Efficient Coordination of
Production & Consumption
Fish
Coconuts
ORC
OMF
??FRC
??C
??F
(,).?? ??F C
O’MF
??CRC
?FMF
?CMF
Instead produce
Give MF same allocation
as before,MF’s
utility is
unchanged
Efficient Coordination of
Production & Consumption
Fish
Coconuts
ORC
OMF
??FRC
??C
??F
(,).?? ??F C
O’MF
??CRC
?FMF
?CMF
Instead produce
Give MF same allocation
as before,MF’s
utility is
unchanged,RC’s
utility is higher
Efficient Coordination of
Production & Consumption
Fish
Coconuts
ORC
OMF
??FRC
??C
??F
(,).?? ??F C
O’MF
??CRC
?FMF
?CMF
Instead produce
Give MF same allocation
as before,MF’s
utility is
unchanged,RC’s
utility is higher;
Pareto
improvement.
Efficient Coordination of
Production & Consumption
?MRS ? MRPT ? inefficient
coordination of production and
consumption.
?Hence,MRS = MRPT is necessary for
a Pareto optimal economic state.
Efficient Coordination of
Production & Consumption
Fish
Coconuts
ORC
OMF
FRC
C
F
CRC
FMF
CMF
Efficient Coordination of
Production & Consumption
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
Decentralized Coordination of
Production & Consumption
?LRC,LMF are amounts of labor
purchased from RC and MF.
?Firm’s profit-maximization problem is
choose C,F,LRC and LMF to
m a x,? ? ? ? ?p C p F w L w LC F RC RC MF MF
A Simplification
? Suppose labor market equilibrium is
already achieved.
? Allows us to focus on the product market.
? Technically,can simultaneously look at all
the markets
– Firm,two outputs,two inputs.
– Crusoe or Friday,two consumer goods,
one leisure good.
– 4 markets clear simultaneously.
?Fish; coconuts; Crusoe’s labor;
Friday’s labor
Decentralized Coordination of
Production & Consumption
m a x,? ? ? ? ?p C p F w L w LC F RC RC MF MF
Isoprofit line equation isc o n s t a n t ? ? ? ? ?p C p F w L w L
C F RC RC MF MF
Decentralized Coordination of
Production & Consumption
m a x,? ? ? ? ?p C p F w L w LC F RC RC MF MF
Isoprofit line equation isc o n s t a n t ? ? ? ? ?p C p F w L w L
C F RC RC MF MF
which rearranges toC w L w L
p
p
p F
RC RC MF MF
C
F
C
? ? ? ??,
Decentralized Coordination of
Production & Consumption
m a x,? ? ? ? ?p C p F w L w LC F RC RC MF MF
Isoprofit line equation isc o n s t a n t ? ? ? ? ?p C p F w L w L
C F RC RC MF MF
which rearranges to
C
w L w L
p
p
p
FRC RC MF MF
C
F
C
?
? ?
?
?
int e r c e pt s lo pe
? ?? ? ? ? ?? ? ? ? 2
.
Decentralized Coordination of
Production & Consumption
Fish
Coconuts Higher profit
Slopes =
? ppF
C
Decentralized Coordination of
Production & Consumption
Fish
Coconuts
The firm’s production
possibility set.
Decentralized Coordination of
Production & Consumption
Fish
Coconuts
Slopes =
? ppF
C
Decentralized Coordination of
Production & Consumption
Fish
Coconuts
Profit-max,plan
Slopes =
? ppF
C
Decentralized Coordination of
Production & Consumption
Fish
Coconuts
Profit-max,plan
Slope =
? ppF
C
Decentralized Coordination of
Production & Consumption
Fish
Coconuts
Profit-max,plan
Slope =
? ppF
CCompetitive marketsand profit-maximization
? MRPT
p
p
F
C
? ?,
Decentralized Coordination of
Production & Consumption
Fish
Coconuts
ORC
OMF
FRC
C
F
CRC
FMF
CMF
Competitive markets
and utility-maximization
? MRS
p
p
F
C
? ?,
Decentralized Coordination of
Production & Consumption
Fish
Coconuts
ORC
OMF
FRC
C
F
CRC
FMF
CMF
Competitive markets,utility-
maximization and profit-
maximization ?MR S p
p MR P T
F
C
? ? ?,
Decentralized Coordination of
Production & Consumption
?So competitive markets,profit-
maximization,and utility
maximization all together cause
the condition necessary for a Pareto
optimal economic state.
M RP T pp M RSF
C
? ? ?,
Production
生产
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 …
?That’s not easy!
Contents
? A one-person economy
– Optimal outcome
– Competitive equilibrium
– Two welfare theorems
– Non-convex technologies
? Two-people economy
– Production possibility frontier
– Comparative advantage
– Pareto efficient allocation
– Competitive equilibrium
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.
Robinson Crusoe’s Technology
Production function
Labor (hours)
Coconuts
240
Robinson Crusoe’s Technology
Labor (hours)
Coconuts
Production function
240
Feasible production
plans
Robinson Crusoe’s Preferences
?RC’s preferences:
–coconut is a good
–leisure is a good
Robinson Crusoe’s Preferences
Leisure (hours)
Coconuts
More preferred
240
Robinson Crusoe’s Preferences
Leisure (hours)
Coconuts
More preferred
24 0
Robinson Crusoe’s Choice
Labor (hours)
Coconuts
Feasible production
plans
Production function
240
Robinson Crusoe’s Choice
Labor (hours)
Coconuts
Feasible production
plans
Production function
240
Leisure (hours)24 0
Robinson Crusoe’s Choice
Labor (hours)
Coconuts
Feasible production
plans
Production function
240
Leisure (hours)24 0
Robinson Crusoe’s Choice
Labor (hours)
Coconuts
Feasible production
plans
Production function
240
Leisure (hours)24 0
Robinson Crusoe’s Choice
Labor (hours)
Coconuts
Production function
240
Leisure (hours)24 0
C*
L*
Labor Leisure
Ou
tpu
t
MRS = MPL
Competitive Equilibrium
? A firm
– Produce coconuts
– Demand labor
? A consumer
– Demand coconuts
– Supply labor
? Equilibrium
– Market clears for coconuts
– Market clears for labor
Competitive Equilibrium
?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.
?Labor supply is L.
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?2
?1
0
Profit-Maximization
Labor (hours)
Coconuts
Feasible production
plans
Production function
240
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*
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 = 1? MPL = MRPL.
0
Profit-Maximization
Labor (hours)
Coconuts
Production function
24
C*
L*
Isoprofit slope = production function slope
i.e,w = MPL = 1? MPL = MRPL.
?*
? * * *? ?C wLRC gets0
Profit-Maximization
Labor (hours)
Coconuts
Production function
24
C*
L*
Isoprofit slope = production function slope
i.e,w = MPL = 1? MPL = MRPL.
? * * *? ?C wL
?* Given w,RC’s firm’s quantity
demanded of labor is L*Labor
demand
RC gets
0
Profit-Maximization
Labor (hours)
Coconuts
Production function
24
C*
L*
Isoprofit slope = production function slope
i.e,w = MPL.
?* Given w,RC’s firm’s quantity
demanded of labor is L* and
output quantity supplied is C*.Labordemand
Output
supply
? * * *? ?C wLRC gets0
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? ?? *,
Utility-Maximization
Labor (hours)
Coconuts
?*
240
C wL? ?? *,Budget constraint; slope = w
Utility-Maximization
Labor (hours)
Coconuts
More preferred
240
Utility-Maximization
Labor (hours)
Coconuts
?*
240
C wL? ?? *,Budget constraint; slope = w
Utility-Maximization
Labor (hours)
Coconuts
?*
Budget constraint; slope = w
240
C wL? ?? *,
Utility-Maximization
Labor (hours)
Coconuts
?*
240
C wL? ?? *,C*
L*
MRS = w
Budget constraint; slope = w
Utility-Maximization
Labor (hours)
Coconuts
?*
240
C wL? ?? *,C*
L*
Labor
supply
Budget constraint; slope = w
MRS = w
Given w,RC’s quantity
supplied of labor is L*
Utility-Maximization
Labor (hours)
Coconuts
?*
240
C wL? ?? *,C*
L*
Given w,RC’s quantity
supplied of labor is L* and
output quantity demanded is C*.
Labor
supply
Output
demand
Budget constraint; slope = w
MRS = w
Utility-Maximization & Profit-
Maximization
? Profit-maximization,
– 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 labor markets both clear.
At Equilibrium …
Labor (hours)
Coconuts
24
C*
L*
?*
0
MRS = w = MPL
Given w,RC’s quantity
supplied of labor = quantity
demanded of labor = L* and
output 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 consumption
bundles.
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
–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
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.
Non-Convex Technologies
Labor (hours)
Coconuts
240
MRS = MPL The common slope ? relative
wage rate w that
implements the Pareto
efficient plan by
decentralized pricing.
Non-Convex Technologies
?Do the Welfare Theorems hold if
firms have non-convex
technologies?
?The 2nd Theorem does require that
firms’ technologies be convex.
Non-Convex Technologies
Labor (hours)
Coconuts
240
MRS = MPL,The Pareto optimal allocation
cannot be implemented by
a competitive equilibrium.
A Two-People Economy
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 (
生产可能性集 ).
?The set’s outer boundary is the production
possibility frontier（ 生产可能性边界 ）,
Production Possibilities
Fish
Coconuts
Production possibility frontier (ppf)
Production Possibilities
Fish
Coconuts
Production possibility frontier (ppf)
Production possibility set
Production Possibilities
Fish
Coconuts
Feasible but
inefficient
Production Possibilities
Fish
Coconuts
Feasible but
inefficient
Feasible and efficient
Production Possibilities
Fish
Coconuts
Feasible but
inefficient
Feasible and efficient
Infeasible
Production Possibilities
Fish
Coconuts
Ppf’s slope is the marginal rate
of product transformation
(边际生产转换率 ),or marginal rate
of transformation (边际转换率 ).
Production Possibilities
Fish
Coconuts
Ppf’s slope is the marginal rate
of product transformation.
Increasingly negative MRPT
? increasing opportunity
cost to specialization.
Production Possibilities
?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(比较优势 ).
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.
Comparative Advantage
F
C
F
C
RC
MF
20
50
30
25
MRPT = -2/3 coconuts/fish so opp,cost of one
more coconut is 3/2 foregone fish.
MRPT = -2 coconuts/fish so opp,cost of one
more coconut is 1/2 foregone fish.
MF has the comparative
opp,cost advantage in
producing coconuts.
RC has the comparative
opp,cost advantage in
producing fish.
Comparative Advantage
F
C
Economy
F
C
F
C
RC
MF
20
50
30
25
70
55
50
30
Use RC to produce
fish before using MF.
Use MF to
produce
coconuts before
using RC.
Comparative Advantage,One
Economy
F
C
Economy
F
C
F
C
RC
MF
20
50
30
25
70
55
50
30
Using low opp,cost
producers first results
in a ppf that is concave
w.r.t the origin.
Comparative Advantage,One
Economy
F
C
Economy
More producers with
different opp,costs
“smooth out” the ppf.
Comparative Advantage,
Separate Economies
F
C
F
C
RC
MF
20
50
30
25
Autarky,Produce and consume at
A.
Trade,Both countries completely
specialize,RC specializes on Fish;
MF specializes on coconuts.
Produce at B and consume at C,
Better off with trade.
A
B
C
A
B
C
Comparative Advantage,Trade
Equilibrium
F
C
F
C
RC
MF
20
50
30
25
Prices adjust to clear the markets.
At equilibrium,export of one
country = import of the other
country for each product.
o
o
Efficient Coordination of
Production & Consumption
?The ppf contains many technically
efficient output bundles.
?Which are Pareto efficient?
Fish
Coconuts
?C
?F
Output bundle is (,)? ?F C
Efficient Coordination of
Production & Consumption
Fish
Coconuts
?C
?F
Output bundle is
and is the aggregate
endowment for distribution
to consumers RC and MF.
(,)? ?F C
Efficient Coordination of
Production & Consumption
Fish
Coconuts
ORC
OMF?C
?F
Output bundle is
and is the aggregate
endowment for distribution
to consumers RC and MF.
(,)? ?F C
Efficient Coordination of
Production & Consumption
Fish
Coconuts
ORC
OMF?C
?F
Allocate efficiently;
say to RC(,)? ?F C
?CRC
?FRC
(,)? ?F CRC RC
Efficient Coordination of
Production & Consumption
Fish
Coconuts
ORC
OMF?C
?F
Allocate efficiently;
say to RC and
to MF.
(,)? ?F C
?CRC ?CMF
?FMF
?FRC
(,)? ?F CRC RC
(,)? ?F CMF MF
Efficient Coordination of
Production & Consumption
Fish
Coconuts
ORC
OMF?C
?F
?CRC ?CMF
?FMF
?FRC
MRS ? MRPT
Efficient Coordination of
Production & Consumption
Fish
Coconuts
ORC
OMF?C
?F
?CRC ?CMF
?FMF
?FRC
O’MF
??C
??F
(,).?? ??F CInstead produce
Efficient Coordination of
Production & Consumption
Fish
Coconuts
ORC
OMF?C
?F
?CRC ?CMF
?FMF
?FRC
O’MF
??C
??F
(,).?? ??F CInstead produce
Efficient Coordination of
Production & Consumption
Fish
Coconuts
ORC
OMF?C
?F
?CRC ?CMF
?FMF
?FRC
??C
??F
(,).?? ??F C
O’MF
?CMF
Instead produce
Give MF same allocation
as before.?FMF
Efficient Coordination of
Production & Consumption
Fish
Coconuts
ORC
OMF?C
?F
?CRC ?CMF
?FMF
?FRC
??C
??F
(,).?? ??F C
O’MF
?CMF
Instead produce
Give MF same allocation
as before,MF’s
utility is
unchanged.
?FMF
Efficient Coordination of
Production & Consumption
Fish
Coconuts
ORC
OMF
??C
??F
(,).?? ??F C
O’MF?FMF
Instead produce
Give MF same allocation
as before,MF’s
utility is
unchanged
?CMF
Efficient Coordination of
Production & Consumption
Fish
Coconuts
ORC
OMF
??FRC
??C
??F
(,).?? ??F C
O’MF
??CRC
?FMF
?CMF
Instead produce
Give MF same allocation
as before,MF’s
utility is
unchanged
Efficient Coordination of
Production & Consumption
Fish
Coconuts
ORC
OMF
??FRC
??C
??F
(,).?? ??F C
O’MF
??CRC
?FMF
?CMF
Instead produce
Give MF same allocation
as before,MF’s
utility is
unchanged,RC’s
utility is higher
Efficient Coordination of
Production & Consumption
Fish
Coconuts
ORC
OMF
??FRC
??C
??F
(,).?? ??F C
O’MF
??CRC
?FMF
?CMF
Instead produce
Give MF same allocation
as before,MF’s
utility is
unchanged,RC’s
utility is higher;
Pareto
improvement.
Efficient Coordination of
Production & Consumption
?MRS ? MRPT ? inefficient
coordination of production and
consumption.
?Hence,MRS = MRPT is necessary for
a Pareto optimal economic state.
Efficient Coordination of
Production & Consumption
Fish
Coconuts
ORC
OMF
FRC
C
F
CRC
FMF
CMF
Efficient Coordination of
Production & Consumption
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
Decentralized Coordination of
Production & Consumption
?LRC,LMF are amounts of labor
purchased from RC and MF.
?Firm’s profit-maximization problem is
choose C,F,LRC and LMF to
m a x,? ? ? ? ?p C p F w L w LC F RC RC MF MF
A Simplification
? Suppose labor market equilibrium is
already achieved.
? Allows us to focus on the product market.
? Technically,can simultaneously look at all
the markets
– Firm,two outputs,two inputs.
– Crusoe or Friday,two consumer goods,
one leisure good.
– 4 markets clear simultaneously.
?Fish; coconuts; Crusoe’s labor;
Friday’s labor
Decentralized Coordination of
Production & Consumption
m a x,? ? ? ? ?p C p F w L w LC F RC RC MF MF
Isoprofit line equation isc o n s t a n t ? ? ? ? ?p C p F w L w L
C F RC RC MF MF
Decentralized Coordination of
Production & Consumption
m a x,? ? ? ? ?p C p F w L w LC F RC RC MF MF
Isoprofit line equation isc o n s t a n t ? ? ? ? ?p C p F w L w L
C F RC RC MF MF
which rearranges toC w L w L
p
p
p F
RC RC MF MF
C
F
C
? ? ? ??,
Decentralized Coordination of
Production & Consumption
m a x,? ? ? ? ?p C p F w L w LC F RC RC MF MF
Isoprofit line equation isc o n s t a n t ? ? ? ? ?p C p F w L w L
C F RC RC MF MF
which rearranges to
C
w L w L
p
p
p
FRC RC MF MF
C
F
C
?
? ?
?
?
int e r c e pt s lo pe
? ?? ? ? ? ?? ? ? ? 2
.
Decentralized Coordination of
Production & Consumption
Fish
Coconuts Higher profit
Slopes =
? ppF
C
Decentralized Coordination of
Production & Consumption
Fish
Coconuts
The firm’s production
possibility set.
Decentralized Coordination of
Production & Consumption
Fish
Coconuts
Slopes =
? ppF
C
Decentralized Coordination of
Production & Consumption
Fish
Coconuts
Profit-max,plan
Slopes =
? ppF
C
Decentralized Coordination of
Production & Consumption
Fish
Coconuts
Profit-max,plan
Slope =
? ppF
C
Decentralized Coordination of
Production & Consumption
Fish
Coconuts
Profit-max,plan
Slope =
? ppF
CCompetitive marketsand profit-maximization
? MRPT
p
p
F
C
? ?,
Decentralized Coordination of
Production & Consumption
Fish
Coconuts
ORC
OMF
FRC
C
F
CRC
FMF
CMF
Competitive markets
and utility-maximization
? MRS
p
p
F
C
? ?,
Decentralized Coordination of
Production & Consumption
Fish
Coconuts
ORC
OMF
FRC
C
F
CRC
FMF
CMF
Competitive markets,utility-
maximization and profit-
maximization ?MR S p
p MR P T
F
C
? ? ?,
Decentralized Coordination of
Production & Consumption
?So competitive markets,profit-
maximization,and utility
maximization all together cause
the condition necessary for a Pareto
optimal economic state.
M RP T pp M RSF
C
? ? ?,
