Partial Equilibrium

8/3/2019 Partial Equilibrium

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1 Bene…ts from division of labour

Consider the following vision of a society. Scarce resources, that is,(i) natural resources (land, minerals, forests, water, etc.), (ii) humancapital (knowledge, skills, innovation, etc.), and (iii) physical capital(equipment, technologies etc.) are used to produce consumption goods:housing, food, entertainment, etc. People have heterogenous preferencesover consumption goods. Division of labour and specialization allow toachieve more e¢cient outcomes when people are organized in a societythan when they remain independent.

"Well then, how will our state supply these needs? It will

need a farmer, a builder, and a weaver, and also, I think, ashoemaker and one or two others to provide for our bodilyneeds. So that the minimum state would consist of four or…ve men...." (Plato, The Republic ).

Division of labor between two sexes is commonly considered as thebegging of economic specialization and exchange in human society. Aclassic example of bene…ts from division of labour by Adam Smith tellsthat the e¢ciency of production of pins increased 240 times when workersstarted to concentrate on single subtasks instead of each carrying out

the original broad task (The In The Wealth of Nations , 1776). Anothercommonly used example is e¢ciency gains from invention of the assemblyline by Henry Ford’s engineers in 1913.

Because bene…ts from specialization create mutual dependence amongeconomic agents, there is a joint decision to be made: (1) what to pro-duce; (2) how to produce, and (3) how to allocate the produced output.1

De…nition 1 (e¢cacité au sens Pareto) Economic allocation is Paretoe¢cient () it is impossible to increase well-being by one economic agent without decreasing well-being by some other agent.

1 The law of comparative advantage tells that individuals/…rms bene…t from spe-cialiasation in consumption/production in those the areas where they have a com-parative advantage. Consider the following example. Mary is an advocate. She gains400$ a day. Her Mom taught he sewing, and she can sew a pair of pants in just oneday. Instead, she can higher a professional couturier who would sew pants in twodays, and charge 100$ for this work. Despite Mary sews better than a professional,it is optimal for her to focus on law, because of 400$ opportunity costs of a day spenton sewing.

1



Figure 1: La structure générale du marche

2 General structure of competitive equilibrium model

Resources: natural, human, and physical2 are used to produce consump-tion goods.

Consumption goods are produced by pro…t-maximizing …rms (Figure1). Humans (consumers) bene…t from consumption goods. They buygoods from …rms using their budget that is composed of labour incomeand revenues from ownership in …rms.

There is a market for any good or service and information isperfect .

3 Consumer choice

3.1 Individual preferences

Fig 2 illustrates Consumption possibility sets X :

Individual preferences:

x % y «x is at least as good as y»

De…nition 2 (les préférences rationnelles) % is rational ()it is complete: x % y or x % yit is transitive: x % y and y % z ) x % z

Limitations: « just perceptible di¤erences »; « framing » (Kahne-man and Tversky 1984); social preferences; time inconsistency.

2 That is, created by humans.

2



24h

lesure

food

24h

lesure

food

food

4

3

2

1

0

food at

noon in

Montreal

food atnoon in

Quebec

pairs of

shoes

24h

lesure

food

24h

lesure

food

food

4

3

2

1

0

food at

noon in

Montreal

food atnoon in

Quebec

pairs of

shoes

Figure 2: Les possibilités de consommation

De…nition 3 x is better than y: x y ()

x % y and e y % x

De…nition 4 x is as good as y: x

y()

x % y and y % x

Verify: % is rational )(i) is irre‡exive and transitive;(ii) is re‡exive and transitive;(iii) x y % z ) x z.

Representation of preferences by utility function:

De…nition 5 (fonction d’utilité) % are representable by utility func-

tion u() : X ! R ()x % y () u(x) > u(y); 8 x 2 X; y 2 X

Two remarks:- representation of % by utility function is not unique : if u() rep-

resents % =) f (u()) also represents % where f () is monotonicallyincreasing function (Eaton, ex. 2.11, 6 page 66).

- The existence of utility function representing % is not guaranteed.

Indi¤erence curves: Eaton Fig. 2.2-2.4, ex. 7

3



Questions:

Q1. Draw indi¤erence curves for the following preferences: (i) homothe-tique (ii) quasilinear; (iii) Leontie¤ (Eaton ex 5.b)

Q2. Is it possible that two distinct indi¤erence curves intersect?

Q3. Find representation by utility function and draw indi¤erence curves

for the following preferences: (i) Mary likes two goods, and she cares only for

their total quantity; (ii) Mary likes gloves and she has two hands; (iii) Mary

likes gloves and she has only one hand.

Assumptions that are necessary for 9 u(x): % is representableby utility function u : X ! R ) % is rational.

!But: % rational ;% it is representable by a utility function (ex.:lexicographic preferences). The di¤erence between necessary and su¢-cient conditions: all women are humans, but not all humans are women.

Questions:

Q4. De…ne Pareto e¢ciency.

Q5. Explain the concept of «invisible hand».

Q6. Describe general structure of competitive equilibrium model.

Q7. Which conditions are necessary for the existence of utility function

that represents a preference relation. Are they su¢cient?

Revision

De…nition 6

x = limn!1xn , 8" > 0 9 N : jxn xj < "

Q8: …nd limn!1

xn where xn is equal to: (i) 1n

; (ii) 1; (iii) nn+1

; (iv) n; (v)

n2

n+1; (vi) (const)

1

n .

De…nition 7 f (x) is continuous function ,lim

n!1f (xn) = f (x) 8fxng1n=1 : x = lim

n!1xn

Q9: Plot the following functions: f (x) = 1 and f (x) = x; x < 1

x + 1; x > 1.

Are they continuous?

Exercise: f (x) is a continuos function; x < x. Illustrate that:

(i) f (x) < 0; f (x) > 0, 9x 2 (x; x) : f (x) = 0;

(ii) f (x) = 0; f (x) = 100, 9x 2 (x; x) : f (x) = a 8 a 2 (0; 100).

De…nition 8 (les préférences continues) Preferences % are contin-uos ,

8f(xn; yn)g1n=1 : xn % yn8n; x = limn!1

xn; y = limn!1

xn; =) x % y

4



Verify: Lexicographic preferences are not continuous!

Conditions that are su¢cient for 9 u(x): If % is rational andcontinuous=) % it is representable by a continuos utility function.

The model assumes that individual% are rational and continuous. Henceconsumer optimization problem can be written as:(

maxx2X

u (x)

s:t: px 6 w(1)

The model makes two more assumptions about %. These assumptionsare neither necessary, nor su¢cient for the existence of utility represen-tation, but they allow to use the …rst-order approach to solve consumer

optimization problem(i) “desirability”:

De…nition 9 (monotonicité ) % is: (i) strictly monotone , y > xand y 6= x =) y x; (ii) monotone , y x =) y x; (iii) LNS , 8x 2 X and 8" > 0 9 y 2 X : kx yk 6 " and y x.

Note: sometimes “enough is enough”% is strictly monotone =) monotone =) LNS;but: LNS;monotone;strictly monotone!

Revision:

De…nition 10 set U is convex , 8y 2 X and z 2 X =) y +(1 ) z 2 X .

De…nition 11 (Upper counter set) UCS (x) = fy 2 X j y % xg.

De…nition 12 % is: (i) convex , UCS (x) is convex 8x 2 X ; (ii)strictly convex , y % x; z % x =) y + (1 ) z x.

An individual with convex preferences has a taste for “diversity” (if you o¤er me two free weekends and many ways (including skiing) to havefun either weekend, I go skiing both weekends).

3.2 Consumer demandRevision: Derivative of a function, the …rst- and the second-orderconditions, the method of Lagrange.

Consumer demand (maxx2X

u (x)

s:t: px 6 w(2)

if p 0 and u () is continuous, problem (2) has a unique solutionx( p;w).

5



X 2

X 2

( ) ( ) ( ){ ),(),,(,, 2*

1*

2121w p xw p xu x xu x x =

( ){ }w x p x p x x=+

221121,

),(2*

w p x

),(1*

w p x

X 2

X 2

( ) ( ) ( ){ ),(),,(,, 2*

1*

2121w p xw p xu x xu x x =

( ){ }w x p x p x x=+

221121,

),(2*

w p x

),(1*

w p x

Figure 3: la demande de Walras.

De…nition 13 solution x( p;w) to problem (2) is called Walrasian de-mand.

Characteristics of x( p;w):

If % is LNS and u() is a continuous utility function representingpreference relation % )(i) “no money illusion”: x(p; w) = x( p;w) 8 > 0(ii) “Walras law”: px = w(iii) uniqueness: if u () is strictly convex ) x( p;w) is unique.

Q10. Which assumption on % are important for (i) and (ii)?

First order conditions for the “interior” solution of problem (2):

MRS (x) =@u=@x1

@u=@x2

=p1

p2

; (3)

p1x1 + p2x2 = w: (4)

Eaton Fig 2.7, ex. 2.8-2.10

Limitations to the …rst-order approach:(i) di¤erentiability is necessary: consider u (x1,x2) = min fx1,x2g;(ii) a “corner” solution is possible (essential goods): …nd x( p;w) for:

u (x1,x2) = ln(x1) + ln(x2 2), p1 = p2 = 1; w = 1.

6



Q11. Find consumer demand for preferences that are described in question

3 when: (i) p1 = p2 = 1; w = 2; (ii) p1 = 1; p2 = 2; w = 2; (iii) p1 = 2; p2 = 2; w = 2; (iv) p1 = 2; p2 = 2; w = 4.

Q12. Find consumer demand for preferences that are describe by util-

ity function: (i) Cobb-Douglas u (x1,x2) = ln(x1) + (1 )ln(x2); (ii)

quasilinear: u (x,m) = m + ln(x); (iii) u (x1,x2) =p

x1 +p

x2:Answer to (i):

x1( p;w) =w

p1; x2( p;w) =

(1 )w

p2:

De…nition 14 xi is: (i) “normal” good ,@x

i( p;w)

@w > 0; (ii) “inferior”good , @x

i( p;w)

@w< 0; (iii) “Gi¤en” good , @x

i( p;w)

@pi> 0

An example of an inferior good - bus to NY - substitute for airline when

get more rich;

Examples of a Gi¤en good are rare; necessary conditions: 1. the good in

question must be an inferior good, 2. there must be a lack of close substitute

goods, and 3. the good must constitute a substantial percentage of the buyer’s

income, but not such a substantial percentage of the buyer’s income that none

of the associated normal goods are consumed.

“As Mr. Gi¤en has pointed out, a rise in the price of bread makes so large

a drain on the resources of the poorer labouring families and raises so much

the marginal utility of money to them, that they are forced to curtail their

consumption of meat and the more expensive farinaceous foods: and, bread

being still the cheapest food which they can get and will take, they consume

more, and not less of it.” (Marshall, 1895 edition of Principles of Economics).

Demand function

De…nition 15 Demand elasticity is equal to @x

i ( p;w)

@pi

pix

i( p;w)

:

Value function

De…nition 16 v( p;w) = u(x( p;w)) is consumer value function.

Characteristics of v( p;w):if % is LNS and u() is a continuous utility function representing

preference relation % )(i) v(p;w) = v( p;w) 8 > 0;(ii) @v

@w> 0; @v

@p6 0;

(iii) f( p;w) jv( p;w) 6 vg is convex 8 v;(iv) v( p;w) is continuos in p and in w.

7



( )( )w p x ,,1 21*

1

2

( )( )w p x ,,2 21*

1 p

1 x

( )( )w p x ,,221

* 1 x

2 x

( )( )w p x ,,1 21* 1

x

2 x

( )( )w p x ,,1 21*

1

2

( )( )w p x ,,2 21*

1 p

1 x

( )( )w p x ,,221

* 1 x

2 x

( )( )w p x ,,1 21* 1

x

2 x

( )( )w p x ,,1 21*

11

2 2

( )( )w p x ,,2 21*

1 p

1 x

( )( )w p x ,,221

* 1 x

2 x

( )( )w p x ,,221

* 1 x

2 x

( )( )w p x ,,1 21* 1

x

2 x

( )( )w p x ,,1 21* 1

x

2 x

Figure 4: demand for "normal good".

Q13. Find v( p;w) when: (i) u (x1,x2) = x1 + x2; (ii) u (x1,x2) =min fx1,x2g; (iii) u (x1,x2) = x1; (iv) u (x1,x2) = ln(x1) + ( 1 )ln(x2)- answer to (iv):

v ( p,w) = lnw

p1 + (1 ) lnw(1

)

p2 :

Q14. Find v( p;w) for: u (x1,x2) =p

x1 +p

x2, (i) pV 1 = pV

2 = 1; w = 6;

(ii) pN 1 = 1; pN

2 = 2; w = 6.

Q15. Compare value functions in Q13(i) and Q13(ii) for: (a) p1 = p2 = 1;w = 4; (b) p1 = 1; p2 = 2; w = 4.

Can we interpret the di¤erence as a measure of the changeof consumer well-being? The answer is “NET”, because utilityrepresentation is not unique!

How shall we measure the change in consumer well-being as a resultof a price change?

3.3 "Compensated" (Hicksian) demand

(min px

x2X

s:t: u (x) > u(5)

De…nition 17 Solution xh( p;w) to problem (5)is Hicksian demand.

8



1 x

2 x

SE 1IE 1

IE 2

SE 2 1

p

1 x

2 x

SE 1IE 1

IE 2

SE 2 1

p

Figure 5: Les e¤ets de la substitution et de la revenue.

Characteristics of xh( p;u):if % is LNS and u() is a continuous utility function representing

preference relation % ) 8 p 0(i) xh(p;u) = xh( p;u) 8 > 0;

(ii) u(xh

( p;u)) = u;(iii) if % is strictly convex, xh( p;u) is unique.The …rst-order conditions that describe the “interior” solution to

problem (5):

MRS (xh) =@u=@x1

@u=@x2=

p1 p2

; (6)

u(xh1 ; xh

2) = u: (7)

Example: u (x1,x2) = ln(x1) + (1 )ln(x2)

xh1( p;u) = u

p2

(1 ) p1

1

; xh2( p;u) = u

(1 ) p1

p2

:

Expenditure function e( p;u) = pxh( p;u)Features of e( p;u):if % is LNS and u() is a continuous utility function representing

preference relation % )(i) e(p;u) = e( p;u) 8 > 0;

(ii) @e( p;u)@u

> 0; @e( p;u)@pl

6 0 where l = 1; 2;

9



X 2

X 1

( ) ( ){ }u x xu x x =2121,,

),(2 u p xh

),(1 u p xh

X 2

X 1

( ) ( ){ }u x xu x x =2121,,

),(2 u p xh

),(1 u p xh

Figure 6: la demande de Hicks.

(iii) e( p;u) is concave in p;

(iv) e( p;u) is continuos in p and u.

xh1( p;u) =

@e( p;u)

@p1 ; xh2( p;u) =

@e( p;u)

@p2 .

Example: u (x1,x2) = ln(x1) + (1 )ln(x2)

e( p;u) = up1 p

12 (1 )1 :

Dualityif % is LNS and u() is a continuous utility function representing

preference relation %, p 01. xh( p;v( p;w)) = x( p;w) and e( p;u) = w2. x( p; pxh( p;u)) = xh( p;u) and v( p; pxh( p;u)) = u

For u (x1,x2) = ln(x1) + (1 )ln(x2) compare xh( p;v( p;w)) andx( p;w); x( p;pxh( p;u)) and xh( p;u):

Slutzky equation: if % is LNS and u() is a continuous utilityfunction representing preference relation % )@hl( p;u)

@pk

=@xl( p;w)

@pk

+@xl( p;w)

@wxk( p;w) where l; k 2 f1; 2g ; u = v( p;w)

Consumer surplus

10



1

p

1 x( )( )w p p x ,, 211

*

( )( )vieuxh

v p p x ,, 211

( )( ).211 ,,

nouvhv p p x

CV

1

p

1 x( )( )w p p x ,, 211

*

( )( )vieuxh

v p p x ,, 211

( )( ).211 ,,

nouvhv p p x

CV

Figure 7: Compensated variation: "give me CV dollars so that I am nothurt by price increase".

Suppose that price for good x1 changes from pV 1 to pN

1 ; pV = ( pV 1 ; p2),

pN = ( pN 1 ; p2).

MMU = e( p0

; v( pV

; w)) e( p0

; v( pN

; w)) (8)

CV = e( pN ; v( pV ; w)) e( pN ; v( pN ; w)) = e( pN ; v( pV ; w)) w =

= e( pN ; v( pV ; w)) e( pV ; v( pV ; w)) = pN 1R

pV 1

xh1( p;v( pV ; w))dp:

EV = e( pV ; v( pV ; w)) e( pV ; v( pN ; w)) = w e( pV ; v( pN ; w)) =

= e( pN ; v( pN ; w)) e( pV ; v( pN ; w)) = pN 1R

pV 1

xh1( p;v( pN ; w))dp:

De…nition 18 If preferences are quasilinear, EV = CV .3 By de…ni-tion, this value is Consumer Marshallian Surplus.

3 Voir TP1.

11



1

p

1 x( )( )w p p x ,, 211

*

( )( )vieuxh

v p p x ,, 211

( )( ).211 ,,

nouvhv p p x

EV

1

p

1 x( )( )w p p x ,, 211

*

( )( )vieuxh

v p p x ,, 211

( )( ).211 ,,

nouvhv p p x

EV

Figure 8: Equivalent variation: "I do not mind being hurt if you pay meEV dollars".

4 Production

Let us divide goods in two categories: consumption goods whose con-sumption increases consumer utility; and inputs of production that are

used to produce consumption goods. Production takes place in …rms.A …rm has a technology that allows to produce consumption good q asan output from some composition of inputs z = (z1; z2;:::;zN ). We willconsider two ways to describe a technology.

Production function The …rst way to describe a technology isto describe how much of an output can be produced from a given com-position of inputs. Figure 9 depicts technologies that use input goodz to produce consumption good q . Shaded areas are called productionsets. The frontier of shaded areas is a production function - it describesmaximal quantity of consumption good q that can be produces out of a

given quantity of input z.Returns to scaleConsider constant > 1. Returns to scale are: (a) decreasing

, f (z) < f (z); (b) increasing , f (z) > f (z); (c) constant, f (z) = f (z).4

On Figure 9(a) returns to scale are decreasing (as output increases,production becomes more and more di¢cult): f (1) f (0) > f (3) f (1). On Figure 9(b) returns to scale are increasing (as output increases,

4 Recall that z can be a vector (z1; z2; :::;zN ) : Then, z = (z1; z2; :::;zN ).

12



q q

z

z z

z

f(z) f(z)

q q

(a) (b)

(c) (d)

f(z)f(z)

1

1

1

f(1)

f(1)

f(1)=

f(1)

f(3)

f(3)f(3)

3 3

3 3

1

f(3)

f(0) f(0)

f(0) f(0)

q q

z

z z

z

f(z) f(z)

q q

(a) (b)

(c) (d)

f(z)f(z)

1

1

1

f(1)

f(1)

f(1)=

f(1)

f(3)

f(3)f(3)

3 3

3 3

1

f(3)

f(0) f(0)

f(0) f(0)

Figure 9: Production set: (a) decreasing returns to scale; (b) increasingreturns to scale; (c) constant returns to scale; (d) constant returns toscale with sunk setup costs.

production becomes more and more easy): f (1)f (0) < f (3)f (1). OnFigure 9(c) returns to scale are constant (as output increases, productionremains equally di¢cult): f (1) f (0) = f (3) f (1).

Cost function The second way to describe a …rm’s technology isto describe the minimal cost that is required to produce a given quantityof output.5 Suppose that output q is produced out of two inputs: z1 andz2. Let pz1 be price of input z1, pz2 be price of input z2. Then, theoptimal input mix z1( p1; p2; q ), z2( p1; p2; q ) solves(

minz1;z2

pz1z1 + pz2z2

s:t: f (z1; z2) 6 q (9)

The cost function is equal to c( pz1 ; pz2 ; q ) = pz1z1( pz1; pz2)+ pz2z2( pz1 ; pz2; q ).

Figure 10 depicts cost functions for technologies with di¤erent returnsto scale for some given prices of inputs.Compare Figures 9 and 10. OnFigure 10(a) returns to scale are decreasing (as output increases, an addi-tional unit of production becomes more and more costly): c(3) c(1) >c(1) c(0). On Figure 10(b) returns to scale are increasing (as out-put increases, an additional unit of production becomes less and lesscostly): c(3) c(1) < c(1) c(0). On Figure 9(c) returns to scale

5 Think: how much at least would it cost you to bake a cake, given that you willchoose to cook it in the cheapest way.

13



c(q)(a)

1

c(1)

3

c(3)

q

c(q)

(b)

1

c(1)

3

c(3)

q

c(q)(c)

1

c(1)

3

c(3)

q

c(q)(d)

1

c(1)

3

c(3)

q

c(q)(a)

1

c(1)

3

c(3)

q

c(q)

(b)

1

c(1)

3

c(3)

q

c(q)(c)

1

c(1)

3

c(3)

q

c(q)(d)

1

c(1)

3

c(3)

q

Figure 10: Cost …nction: (a) decreasing returns to scale; (b) increasingreturns to scale; (c) constant returns to scale; (d) constant returns toscale with sunk setup costs.

are constant (as output increases, production remains equally di¢cult):c(3) c(1) = c(1) c(0).

The e¢cient scale Figure 11 depicts the e¢cient scale of produc-tion with nonsunk setup cost .6 Average cost of production AC (q ) = c(q)

q

is decreasing in region q < q , and increasing afterwards: it is minimizedat q . Level q is called the e¢cient scale .

6 Nonsunk setup cost is the cost that a …rm pays whenever its output is positive,regardless of output level (premises). Sunk setup cost is the cost that a …rm paysregardless of whether its output is positive or null (registration).

q q

AC(q)c(q)

q q

AC(q)c(q)

Figure 11: e¢cient scale with nonsunk setup cost

14



q

z

f(z)f(z*)

Slope= p

p z

−

z*

q

z

f(z)f(z*)

Slope= p

p z

−

z*

Figure 12: Pro…t-maximizing input mix and production

Firm’s objectives We assume that a …rm maximizes its pro…tstaking all prices as given. In class, we have discussed limitations of thisassumption: potentially controversial objectives by di¤erent owners, andpotential con‡ict of interests between the owners and the managers towhom the owners need to delegate decision-making.

Firm’s supply A …rm’s whose technology is described by produc-tion function q = f (z), chooses input mix z = (z1 ; z2 ;:::;zN ) that solvesproblem:

nmaxz

pf (z)

pzz (10)

where p is the output’s price, and pz = ( pz1; pz2;:::;pzN ) is a vector of input prices. The …rm supply is equal to q = f (z), and its pro…ts areequal to (z) = pf (z) pzz. Figure 12 illustrates pro…t-maximizinginput mix and production for strictly concave production function (de-creasing returns to scale).Note, that7

p@f (z)

@zi

6 pzi with equality if zi > 0; i = 1:::N:

A …rm’s whose technology is described by cost function c(q ), chooses

to produce output q

that solves problem:max

q pq c(q ) (11)

For strictly convex cost function (decreasing returns to scale)

p 6 c0

(q ) with equality if q > 0: (12)

7 Be careful not to use the …rst order approach for increasing or constant returnsto scale.

15



c(q)

q

Slope = -p

q*

c(q)

q

Slope = -p

q*

Figure 13: pro…t maximizing production

That is, a …rm production, if it takes place, equalizes marginal cost c0

(q ), that is, the cost of producing “the last” additional unit with pricethat is charged for this unit, as illustrated on …gure 13.Suppose several…rms produce the same output. For given prices, pro…t-maximizing …rmwith less e¢cient technology, chooses to produce less.

5 Partial Equilibrium

Let us study market for one good in isolation . For illustrativepurposes, consider an economy with two consumers; two …rms and two

goods: numeriare good m and consumption good x.Preferences by consumer i = 1; 2 are described by quasilinear utility

functionui(mi; xi) = mi + '(xi);

where xi and mi denote consumption levels. Let us assume that '(xi)is a concave function (recall our discussion of convexity of consumerpreferences in section 1).

Production technology by …rm j = 1; 2 is described by cost functionc j(q j): …rm j inquires cost c j(q j) in order to produce q j units of output.Let us assume that c j(q j) is a convex function (recall our discussion of

returns to scale).Consumer i has a right to keep share ij of pro…ts in …rm j. Ini-

tially, consumer i has mi units of good m: no good x is available beforeproduction takes place.

Equilibrium allocation Let us normalize the price of numeriaregood to be 1 (recall our discussion of “no money illusion”). Allocationm

1; m

2; x1; x2; q 1; q 2 and price p of good x constitute an equilibrium if and only if

16



1. q j = q j ( p) maximizes pro…ts by …rm j when price of the output

is equal to p

, that is, it solvesmax

qj pq j c j (q j) (13)

(Hence, pro…ts by …rm j is j = pq j c j (q j )).2. xi = x j ( p) solves optimization problem by consumer i when price

of consumption good is equal to p :(maxxi;mi

mi + '(xi)

s:t: pxi + mi 6 mi + i11 + i22(14)

3. Price of good x balances the market, that is, aggregate supply of good x is equal to aggregate demand for good x:

q 1( p) + q 2( p) = x1( p) + x2( p):

By …rst-order approach, equilibrium allocation is characterized by:8

'0

i(xi ) 6 p with equality if xi > 0 (15)

p 6 c0

j (q j ) with equality if q j > 0 (16)

q 1 + q 2 = x1 + x2 (17)

Pareto optimal allocation Allocation mo1; mo

2; xo1; xo

2; q o1; q o2 is Pareto

optimal if and only if it is impossible to …nd some other allocationm1; m2; x1; x2; q 1; q 2 that increases utility by one consumer without de-creasing utility by the other consumer.9

Suppose that perfectly informed benevolent social planner picks out-put in each …rm and allocates total output between the consumers soas to maximize joint “happiness” that is measured by sum of consumerutilities. She “solves”:8><

>:maxxi;qj

m1 + '1(x1) + m2 + '2(x2)

s:t: x1 + x2 6 q 1 + q 2m1 + m2 + c1(q 1) + c2(q 2) 6 m1 + m2

(18)

or, equivalently(maxxi;qj

'1(x1) + '2(x2) c1(q 1) c2(q 2)

s:t: : x1 + x2 6 q 1 + q 2

8 Recall equations (6) and (12).9 There is only one other consumer in our economy. If there are many consumers,

an allocation is Pareto optimal if and only if it is impossible to …nd some otherallocation that increases utility by one consumer without decreasing utility by someother consumer.

17



Lagrangian for this optimization problem is equal to

L = '1(x1) + '2(x2) c1(q 1) c2(q 2) + (q 1 + q 2 x1 x2) ;

where is Lagrangian multiplier associated with technological constraint.Hence, the planner picks xo

1; xo2; q o1; q o2 such that

6 c0

j (q o j ) with equality if q o j > 0 (19)

'0

i(x0i ) 6 with equality if xo

i > 0 (20)

(q o1 + q o2 xo1 xo

2) = 0 (21)

and she allocates the remaining numeriare good in any way between theconsumers.

Notice, that if we take = p, the systems of equations (29)-(31) and(15)-(17) are equivalent. Therefore,

The First Fundamental Welfare Theorem: any competitive equi-librium is Pareto optimal.10

A producer would increase pro…t by expanding produc-tion of the good if its price exceeded his marginal cost. Con-versely, if he produced the good at all, he would contractproduction if the marginal cost were to exceed the price.This trivial result has important implications. When decidingwhether to consume one more unit of the good, a consumerfaces a price that is socially “the right one” and internalizesthe cost of producing this extra unit (Tirole (1998), “TheTheory of Industrial Organization,” The MIT Press ).

Marshallian surplus Marshallian surplus is a concept of quasi-linear model. It measures social welfare. It is equal to the surface thatlies below the inverse demand curve less the surface that lies below thesupply curve (see …gure 14).11 Its share above line p = p is consumer

surplus , the rest is producer surplus .Figure 15 illustrates that Marshallian surplus is maximized in equi-librium.

10 Recall, that we consider an “ideal” economy.11 Recall our discussion in class: an addition unit of production increases of con-

sumer surplus (see part 1) and imposes cost on producers.

18



D(p)

S(p)Augmentation de la

production/consomm

ation marginale

+-

D(p)

S(p)Augmentation de la

production/consomm

ation marginale

+-

Figure 14: marginal change in Marshallian surplus.

Figure 15: maximim of Marshallian surplus.

19



q,x

t

p

Producer surplus

Consumer surplus

Tax revenuesDeadweight lossp*

p*(t)

q* q*(t)

q,x

t

p

Producer surplus

Consumer surplus

Tax revenuesDeadweight lossp*

p*(t)

q* q*(t)

Figure 16: Deadweight loss from commodity taxation.

Deadweight loss from commodity taxation Suppose that a…rm is taxed at rate t for each unit of good that it sells. Then, supplyof good x is characterized by equations

p(t) + t 6 c0

j (q j ) with equality if q j > 0; (22)

where p

(t) denotes new equilibrium price. Aggregate supply curve“shifts up”, as depicted on …gure 16. As a result, equilibrium outputand consumption decrease. Taxation creates a deadweight loss: tax rev-enues+consumer surplus+producer surplus in equilibrium with taxes liesbelow Marshallian surplus without taxes.

Note, that new equilibrium price increases as compared to that with-out taxation: both producers and consumers share the burden of dead-weight loss. How the share that is beard by consumers depend on elas-ticity of demand? What happens if consumers, and not producers paya tax?

Number of …rms in the market Consider market for good x.Demand is given by x( p). There is an in…nite number of …rms. Each…rm has an access to production technology that is characterized bycost function c(q ): c(0) = 0, and it can enter or exit the market. A triple( p,q ; N ) is a long-run competitive equilibrium if

1. a …rm’s optimization: q solvesmax

q pq c(q ) (23)

2. x( p) = N q (balance on the market);

20



3. pq c(q ) = 0 (no pro…ts).

As we have discussed in class: when returns to scale are constant, q

and N cannot be determined (quantity of good x that is demanded canbe generated by any number of …rms with any load); when returns toscale are decreasing, there is no long-run competitive equilibrium (pro…t-maximization implies that pro…ts are positive, then, however, more …rmswant to enter the market). Indeed, in any equilibrium with determinantnumber of …rms cost function must exhibit a strictly positive e¢cientscale (see part 2).

6 A primer in general equilibrium model: Robin-son Crusoe economy

Partial equilibrium approach considers one market in isolation. Poten-tially, shocks on this market generate e¤ects on other markets. Partialequilibrium approach ignores these e¤ects. A more complicated, generalequilibrium model takes these e¤ects into the account. In class, we haveconsidered a simple illustration of this model.

Robinson Crusoe economy Consider an economy with one con-sumer (Robinson Crusoe) who owns a single …rm. Robinson bene…tsfrom two goods: consumption good x and leisure l. His preferencesare described by utility function u(l; x). Initially, Robinson has 24 hoursavailable for leisure. However, he needs to work z hours in the …rm inorder to produce f (z) units of consumption good. That is, the …rm’stechnology is described by production function f (z), where z = 24 l.

Equilibrium allocation Let p be price of good x, and w be thewage rate (price of time). x; z and p; w constitute an equilibrium, if and only if

1. z = z( p; w) maximizes pro…ts by …rm j, when prices are equalto p; w. That is, it solves12

nmax

z pf (z) wz (24)

(Hence, supply of good x is equal to q = f (z( p; w)); and pro…ts bythe …rm is equal to = pf (z( p; w)) wz).

2. x( p; w), l( p; w) solve consumer optimization problem:(max

x;l624u(l; x)

s:t: px 6 w(24 l) + (25)

12 Recall problem (10).

21



24h

q

z

f(z)

x

l

Robinson’s indifference curve

Leisure demand Labour supp ly

Labour demand

Slope= p

w−

D e m a n d f o r x

S u p p l y o f x

24h

q

z

f(z)

x

l


Leisure demand Labour supp ly

Labour demand

Slope= p

w−

D e m a n d f o r x

S u p p l y o f x

Figure 17: de…cit of consumption good and excess supply of labour.

3. Prices balance the markets,13 that is, supply of good x is equal todemand for good x:

f (z( p; w)) = x( p; w);

and labour supply is equal to labour demand

24 l( p; w) = z( p; w):

Suppose that both utility function and production function are strictlyconcave, so that the …rst-order approach is valid. Then, (interior) equi-librium is characterized by:

f 0

(z) =p

w;

u0

x(l; x)

u0

l(l; x)=

p

w; 24 l = z; f (z) = x: (26)

On …gure 17 there is excess supply of labour and excess demand for goodx.

Consequently price ratio

w

p decreases, so as to balance the markets:…gure 18.

Pareto optimal allocation Suppose that Robinson gives up withtrading with himself and simply decides how much to work and howmuch of good x to consume. That is, he solves:(

maxx;l624

u(l; x)

s:t: x = f (24 l)

13 Indeed, if one market is balaced, the other is also balanced.

22



q

z

f(z)

24h

x

l


Leisure demand Labour supply

Labour demand

Slope=*

*

p

w−

S u p p l y o f x

D e m a n d f o r x

q

z

f(z)

24h

x

l


Leisure demand Labour supply

Labour demand

Slope=*

*

p

w−

S u p p l y o f x

D e m a n d f o r x

Figure 18: equilibrium allocation and price ratio

Because we have assumed that utility and production function are bothstrictly concave, there is the unique Pareto optimal allocation xo, lo:

u0

l(lo; f (24 lo)) = u0

x(lo; 24 lo)f 0

(24 lo); xo = f (24 lo):

It is the same as equilibrium allocation (The First Fundamental Welfare

Theorem): compare …gures 19 and 18.

7 A scope for public intervention: an illustrativeexample

“An ideal family” Marie and Pierre live together. An individual’sutility depends on consumption of food x, and on the amount of moneym:

U M = mM + ln(1 + xM ); (27)

U P = mP + 2 ln(1 + xP ): (28)

Hence, (i) Mary and Pierre like money equally; (ii) Pierre likes foodtwice more than Mary:

There are two ways to get food out of money. The …rst way is tosend Mary to the market. She is able to bring home q units of food inexchange for q2

2 dollars. If Peter goes shopping, he has to spend twicemore money to bring home the same amount of food. That is, thereare two technologies that allow to produce consumption good out of thenumeriare, with cost functions:

cM (q ) =q 2

2; cP (q ) = q 2:

23



q

z

f(z)

24h

x

l


Leisure Labour

q

z

f(z)

24h

x

l


Leisure Labour

Figure 19: optimal allocation.

Figure 20: Tastes for food

24



Figure 21: Production technologies

Mary and Peter have 100 dollars in the pocket each. The optimalshopping/consumption plan (xM ; xP ; q M ; q P ) solves14

(max

xM ;xP ;qa;qbln(1 + xM ) + 2 ln(1 + xP ) q2M

2 q 2P

s:t: : xM + xP = q M + q P

Lagrangian for this optimization problem is equal to

L = ln(1 + xM ) + 2 ln(1 + xP ) q 2M

2 q 2P + (q M + q P xM xP ) ;

where is the Lagrangian multiplier associated with technological con-straint.

= c0M (q M ) = q M = c0P (q P ) = 2q P = (29)

= (ln(1 + xM ))0 =1

1 + xM

= (2ln(1 + xP ))0 =2

1 + xP

(30)

(q M + q P xM xP ) = 0 (31)

Solving (29)-(31), we …nd:15

xM =

p 10 2

3 0:39; (32)

14 We have to verify later that the solution satis…es endownment constraint q i 6100, i = M; P .

15 Verify.

25



xP = 1 + 2xM =2p

10 1

3 1:77; (33)

q P =1

1 + xM

=3

1 +p

10

22 0:72; (34)

q M = 2q P =3

1 +p

10

11 1:44: (35)

Hence, perfectly informed benevolent “family planner” (i) would askMary to buy more food than Peter: q M > q P ; and she (ii) would letPeter eat more than Mary: xM < xP .

Suppose now that the planner knows that Peter’s preferences are de-scribed by equation (28). She also knows that (i) with probability 1

2Mary has the same preferences as Peter, (ii) with probability 12 Mary’s

preferences are described by equation (27). However, the planner doesnot know Mary’s preferences exactly. Because Mary’s utility is monoton-ically increasing in her consumption of food, she claims that she likesfood as much as Peter (recall the failure of planned economies!).16 Away to discipline Mary is to make her pay for each additional unit of

food that she demands .

Perfect Market Equilibrium Suppose that Peter and Mary tradefood at home at price p. They (i) know everything about each other pref-erences and production technologies, and (ii) behave as price-takers both

when they sell food to each other, and when they buy food from eachother.

SupplyFor a given price p, Mary’s supply of food on the home market solves

maxq

pq q 2

2

Hence, it is equal toq M ( p) = p (36)

Mary’s supply of food on the home market solves

maxqa

pq q 2

q P ( p) =p

2(37)

Therefore, at a given price p, aggregate supply of food on the homemarket is equal to

q M ( p) + q P ( p) =3 p

2: (38)

16 In this case it is optimal to allocatep 512

units of food to either family member.

26



Demand

For a given price p, Mary’s demand for food on the home marketsolves8<:

maxxM ;qM

ln(1 + xM ) + mM

s:t: : pxM + mM 6 1 + pq M q2M

2

Hence, her demand for food xM ( p) satis…es equation

1

1 + xM ( p)= p; (39)

or, equivalentlyxM ( p) =

1

p 1: (40)

Peter’s demand for food on the home market solves(maxxP ;qb

2 ln(1 + xP ) + mP

s:t: : pxP + mP 6 1 + ( pq P q 2P )

Hence,2

1 + xP ( p)= p; (41)

which is equivalent to

xP ( p) =2

p 1: (42)

Consequently, at a given price p, aggregate demand for food on the homemarket is equal to

xM ( p) + xP ( p) =3

p 2: (43)

Balance on the MarketIn equilibrium, price balances the market:17

3

p 2 =3 p

2 : (44)

Notice, that if we pick = p, the systems of equations (29)-(31) and (36),(37), (39), and (41) are equivalent. Therefore, in equilibrium individualproduction and consumption are e¢cient: that is, they are given byequations (32)-(35): recall the First Fundamental Welfare Theorem.

17 Recall, that when all-but-one markets are balanced, all markets are balanced.Use equations (38), (43), and (44) to …nd equilibrium price and quantities. Drawaggregate demand and aggregate supply curves.

27



Missing markets Suppose that Peter has a powerful stereo sys-

tem, and he likes to listen to pop-music (denote by h 6 24 his consump-tion of music per day):

U P = mP + 2 ln(1 + xP ) + ln(1 + h):

Mary, instead, su¤ers when music is playing:

U M = mM + ln(1 + xM ) h

2:

By listening to music Peter generates a negative externality on Mary.18

When Peter and Mary do not try to reach an agreement for how long

shall the music play on, Peter listens to music all the time h = 24. Thisoutcome is suboptimal. It would be e¢cient to take into the accountMary’s preferences, and let the music play so as to

maxh624

ln(1 + h) h

2;

that is, forh = 1

hour a day.

Private bargaining over externality A benevolent social plan-ner could restore the e¢ciency (i) by requiring that the music cannot beplayed more than an hour (that is, by imposing a quota), or else (ii) bycharging Peter t = 1

2dollar for each hour that he listens to the music

(that is, by imposing a tax). However, these forms of intervention arenot necessary.19

Suppose that the government enforces Peter’s property rights to lis-ten to the music as long as Peter wants. Suppose furthermore thatthe government allows Mary to make Peter a take-it-or-leave-it o¤er of a monetary transfer T in exchange for playing a music for h hours aday. If Peter agrees, this agreement is enforced by the government. If

no agreement is reached, the outcome is as described in the previoussection: Peter listens to music non-stop.

Peter agrees to Mary’s o¤er if and only if

ln(1 + h) T > ln(25) (45)

18 See page 352 in Mas-Colell, Whinston, and Green 1995 for de…nition of an ex-ternality.

19 Because government intervention is costly (recall that taxation is associated witha deadweightloss), the general presumption would be to keep it on the lowest neces-sary level.

28



Hence, the Mary’s best o¤er solves(max

T 6100; h624T h

2

s:t: : (45)

Indeed, Mary o¤ers to Peter to pay him T = ln 25 ln 2 if the musicplays for one hour, and he agrees.

Assignment of property rights a¤ects only the …nal distribution of wealth between Peter and Mary, and not the number of hours of musicplayed. Indeed, suppose that the government guarantees Mary that shecan lock the stereo system in a wardrobe for h hours. Mary o¤ers toPeter to pay him T = ln 25

h

ln 2 if the music plays for one hour,

and he agrees: the higher h, more wealth is left to Mary.Furthermore, when Mary and Peter can bargain, the resulting num-

ber of hours of music on does not depend on the form of bargaining.Suppose that it is not Mary, but Peter who can make a “take-it-or-leave-it” o¤er. Then, he proposes Mary to pay him T = 11 h

2 if themusic plays an hour, and Mary agrees.

The following three insights are general in bargaining games: a player’spayo¤ is higher, (i) the better his “outside option”; (ii) the higher hisbargaining power; and (iii) the more patient he is (in dynamic games).

Coase Theorem (1960):20 when trade of the externality can occur,

bargaining leads to an e¢cient outcome, no matter how property rights are allocated.21

Public goods Peter and Mary rent an apartment together andnone of them can limit the other’s access to the place. They can hire acleaning lady who increases the level of order in the apartment by y ate¤ort costs y2. Peter cares for order less than Mary:

U P = mP + 2 ln(1 + xP ) + 3 ln(1 + yP + yM );

U M = mM + ln(1 + xM ) + ln(1 + yP + yM ):

Order in the apartment is a public good . Socially optimal level of ordersolves

maxy

4 ln(1 + y) y2

It is equal toy = 1:

20 See page 357 in Mas-Colell, Whinston, and Green 1995.21 Hence, the givernment should simply enforce private agreements.

29



y

1 Pierre’s demand

for public good

Mary’s demand

for public good

p y

y=1Social optimum

3

4

Equilibrium

y=0.82

y

1 Pierre’s demand

for public good

Mary’s demand

for public good

p y

y=1Social optimum

3

4

Equilibrium

y=0.82

Figure 22: free-riding.

In equilibrium, cleaning lady o¤ers

y =py

2

units of order, and only Mary is ready to pay for it - Peter bene…ts fromthe order that is paid by Mary: he would not like to pay so as to maketheir apartment even cleaner:

1

1 + yP + yM

6 py with an equality when yP > 0

3

1 + yP + yM

6 py with an equality when yM > 0.

As a result, y is suboptimal (see Fig 22):Verify that if Mary’s consump-tion of order is subsidized at rate sM = 1

2 , the e¢ciency is restored.22

An alternative way to restore the e¢ciency is to impose a quota

y > 1:

However, if the government is uncertain about Mary’s taste for order(unlike in our model), the task to …nd an optimal quota becomes non-

trivial.

References

[1] Mas-Colell, A., Whinston, D.,M., and J.,R. Green. (1995), “Micro-economic Theory,” Oxford University Press .

[2] Tirole, J. (1988), “The Theory of Industrial Organization,” The MIT Press .

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Partial Equilibrium