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) human
capital (knowledge, skills, innovation, etc.), and (iii) physical capital
(equipment, technologies etc.) are used to produce consumption goods:
housing, food, entertainment, etc. People have heterogenous preferences
over consumption goods. Division of labour and specialization allow to
achieve more e¢ cient outcomes when people are organized in a society
than 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, a
shoemaker and one or two others to provide for our bodily
needs. 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 the
begging of economic specialization and exchange in human society. A
classic example of bene…ts from division of labour by Adam Smith tells
that the e¢ ciency of production of pins increased 240 times when workers
started to concentrate on single subtasks instead of each carrying out
the original broad task (The In The Wealth of Nations, 1776). Another
commonly used example is e¢ ciency gains from invention of the assembly
line by Henry Ford’s engineers in 1913.
Because bene…ts from specialization create mutual dependence among
economic agents, there is a joint decision to be made: (1) what to produce; (2) how to produce, and (3) how to allocate the produced output.1
De…nition 1 (e¢ cacité au sens Pareto) Economic allocation is Pareto
e¢ 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 specialiasation in consumption/production in those the areas where they have a comparative advantage. Consider the following example. Mary is an advocate. She gains
400$ a day. Her Mom taught he sewing, and she can sew a pair of pants in just one
day. Instead, she can higher a professional couturier who would sew pants in two
days, 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 spent
on 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 consumption goods.
Consumption goods are produced by pro…t-maximizing …rms (Figure
1). Humans (consumers) bene…t from consumption goods. They buy
goods from …rms using their budget that is composed of labour income
and revenues from ownership in …rms.
There is a market for any good or service and information is
perfect.
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 % y
it is transitive: x % y and y % z ) x % z
Limitations: « just perceptible di¤erences » ; « framing » (Kahneman and Tversky 1984); social preferences; time inconsistency.
2
That is, created by humans.
2
lesure
lesure
24h
24h
pairs of
shoes
food
food at
noon in
Montreal
4
food
3
2
1
0
food at
noon in
Quebec
food
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 function 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( ) represents % =) f (u( )) also represents % where f ( ) is monotonically
increasing 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) homothetique (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 representable
by 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 = lim xn , 8" > 0 9 N : jxn
n!1
Q8: …nd lim xn where xn is equal to: (i)
n!1
n2
n+1
1
;
n
xj < "
(ii) 1; (iii)
n
;
n+1
(iv) n; (v)
1
n
; (vi) (const) .
De…nition 7 f (x) is continuous function ,
lim f (xn ) = f (x) 8 fxn g1
n=1 : x = lim xn
n!1
n!1
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 continuos ,
8 f(xn ; yn )g1
n=1 : xn % yn 8n; x = lim xn ; y = lim xn ; =) x % y
n!1
4
n!1
Verify: Lexicographic preferences are not continuous!
Conditions that are su¢ cient for 9 u(x): If % is rational and
continuous=) % it is representable by a continuos utility function.
The model assumes that individual % are rational and continuous. Hence
consumer optimization problem can be written as:
(
maxu (x)
x2X
(1)
s:t: px 6 w
The model makes two more assumptions about %. These assumptions
are neither necessary, nor su¢ cient for the existence of utility representation, 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 > x
and 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 =)
(1
) z 2 X.
y+
De…nition 11 (Upper counter set) U CS(x) = fy 2 X j y % x g.
De…nition 12 % is: (i) convex , U CS(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 have
fun either weekend, I go skiing both weekends).
3.2
Consumer demand
Revision: Derivative of a function, the …rst- and the second-order
conditions, the method of Lagrange.
Consumer demand
(
maxu (x)
x2X
(2)
s:t: px 6 w
if p
0 and u ( ) is continuous, problem (2) has a unique solution
x (p; w).
5
X2
{(x , x ) p x
1
2
1 1
+ p2 x2 = w}
{(x , x )u(x , x )= u(x
1
2
1
2
*
1
( p, w), x* 2 ( p, w)
)}
x *2 ( p, w)
x*1 ( p, w)
X2
Figure 3: la demande de Walras.
De…nition 13 solution x (p; w) to problem (2) is called Walrasian demand.
Characteristics of x (p; w):
If % is LNS and u( ) is a continuous utility function representing
preference 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):
M RS(x ) =
@u=@x1
p1
= ;
@u=@x2
p2
p1 x1 + p2 x2 = w:
(3)
(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 ,x2 g;
(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 utility function: (i) Cobb-Douglas u (x1 ,x2 ) = ln(x1 ) + (1
) ln(x2 ); (ii)
p
p
quasilinear: u (x,m) = m + ln(x); (iii) u (x1 ,x2 ) = x1 + x2 :
Answer to (i):
x1 (p; w) =
w
(1
)w
; x2 (p; w) =
:
p1
p2
De…nition 14 xi is: (i) “normal” good , i@w > 0; (ii) “inferior”
@x (p;w)
@x (p;w)
good , i@w < 0; (iii) “Gi¤en” good , i@pi > 0
@x (p;w)
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
@xi (p;w)
pi
:
@pi
xi (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;
@v
> 0; @v
6 0;
(ii) @w
@p
(iii) f(p; w) jv(p; w) 6 v g is convex 8 v;
(iv) v(p; w) is continuos in p and in w.
7
x2
p1
2
x * 1 (( 2 , p 2 ), w
)
x1
x2
1
x * 1 ((1 , p
x*1 ((2, p2 ), w) x*1 ((1, p2 ), w)
2
), w )
x1
x1
Figure 4: demand for "normal good".
Q13. Find v(p; w) when: (i) u (x1 ,x2 ) = x1 + x2 ; (ii) u (x1 ,x2 ) =
min fx1 ,x2 g; (iii) u (x1 ,x2 ) = x1 ; (iv) u (x1 ,x2 ) = ln(x1 ) + (1
) ln(x2 )
- answer to (iv):
w
w(1
)
+ (1
) ln
:
p1
p2
p
p
Q14. Find v(p; w) for: u (x1 ,x2 ) = x1 + x2 , (i) pV1 = pV2 = 1; w = 6;
N
(ii) pN
1 = 1; p2 = 2; w = 6.
v (p,w) =
ln
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 change
of consumer well-being? The answer is “NET”, because utility
representation is not unique!
How shall we measure the change in consumer well-being as a result
of a price change?
3.3
"Compensated" (Hicksian) demand
(
min px
x2X
s:t: u (x) > u
De…nition 17 Solution xh (p; w) to problem (5)is Hicksian demand.
8
(5)
x2
p1
SE2
IE2
IE1
x1
SE1
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):
M RS(xh ) =
@u=@x1
p1
= ;
@u=@x2
p2
(6)
u(xh1 ; xh2 ) = u:
Example: u (x1 ,x2 ) =
xh1 (p; u) = u
ln(x1 ) + (1
(1
) ln(x2 )
1
p2
)p1
(7)
; 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)
> 0; @e(p;u)
6 0 where l = 1; 2;
@u
@pl
9
X2
{(x , x )u(x , x )= u}
1
2
1
2
x h 2 ( p, u )
x h1 ( p , u )
X1
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) =
Example: u (x1 ,x2 ) =
@e(p; u) h
@e(p; u)
; x2 (p; u) =
.
@p1
@p2
ln(x1 ) + (1
e(p; u) = up1 p12
) ln(x2 )
(1
)
1
:
Duality
if % is LNS and u( ) is a continuous utility function representing
preference relation %, p
0
h
1. x (p; v(p; w)) = x (p; w) and e(p; u) = w
2. 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)) and
x (p; w); x (p; pxh (p; u)) and xh (p; u):
Slutzky equation: if % is LNS and u( ) is a continuous utility
function representing preference relation % )
@hl (p; u)
@xl (p; w) @xl (p; w)
=
+
xk (p; w) where l; k 2 f1; 2g ; u = v(p; w)
@pk
@pk
@w
Consumer surplus
10
((
)
)
((
x h1 p1 , p 2 , v nouv.
p1
x h1 p1 , p 2 , v vieux
)
)
CV
((
))
x*1 p1 , p 2 , w
x1
Figure 7: Compensated variation: "give me CV dollars so that I am not
hurt by price increase".
V
V
Suppose that price for good x1 changes from pV1 to pN
1 ; p = (p1 ; p2 ),
pN = (pN
1 ; p2 ).
M M U = e(p0 ; v(pV ; w))
CV = e(pN ; v(pV ; w))
N
V
= e(p ; v(p ; w))
e(p0 ; v(pN ; w))
e(pN ; v(pN ; w)) = e(pN ; v(pV ; w))
V
V
e(p ; v(p ; w)) =
pRN
1
(8)
w=
xh1 (p; v(pV ; w))dp:
pV
1
EV = e(pV ; v(pV ; w))
N
N
= e(p ; v(p ; w))
e(pV ; v(pN ; w)) = w
V
N
e(p ; v(p ; w)) =
pRN
1
e(pV ; v(pN ; w)) =
xh1 (p; v(pN ; w))dp:
pV
1
De…nition 18 If preferences are quasilinear, EV = CV .3 By de…nition, this value is Consumer Marshallian Surplus.
3
Voir TP1.
11
((
)
)
((
x h1 p1 , p 2 , v nouv.
p1
x h1 p1 , p 2 , v vieux
)
)
EV
((
))
x*1 p1 , p 2 , w
x1
Figure 8: Equivalent variation: "I do not mind being hurt if you pay me
EV dollars".
4
Production
Let us divide goods in two categories: consumption goods whose consumption 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 as
an output from some composition of inputs z = (z1 ; z2 ; :::; zN ). We will
consider two ways to describe a technology.
Production function The …rst way to describe a technology is
to describe how much of an output can be produced from a given composition of inputs. Figure 9 depicts technologies that use input good
z to produce consumption good q. Shaded areas are called production
sets. The frontier of shaded areas is a production function - it describes
maximal quantity of consumption good q that can be produces out of a
given quantity of input z.
Returns to scale
Consider 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
(a)
(b)
q
q
f(z)
f(z)
f(3)
f(3)
f(1)
3
z
1
f(0)
z
q
(c)
3
f(z)
f(3)
f(3)
f(1)
z
3
1
f(1)
f(0)
q
(d)
f(z)
f(0)
1
1
z 3
f(1)= f(0)
Figure 9: Production set: (a) decreasing returns to scale; (b) increasing
returns to scale; (c) constant returns to scale; (d) constant returns to
scale with sunk setup costs.
production becomes more and more easy): f (1) f (0) < f (3) f (1). On
Figure 9(c) returns to scale are constant (as output increases, production
remains equally di¢ cult): f (1) f (0) = f (3) f (1).
Cost function The second way to describe a …rm’s technology is
to describe the minimal cost that is required to produce a given quantity
of output.5 Suppose that output q is produced out of two inputs: z1 and
z2 . Let pz1 be price of input z1 , pz2 be price of input z2 . Then, the
optimal input mix z1 (p1 ; p2 ; q), z2 (p1 ; p2 ; q) solves
(
minpz1 z1 + pz2 z2
z1 ;z2
(9)
s:t: f (z1 ; z2 ) 6 q
The cost function is equal to c(pz1 ; pz2 ; q) = pz1 z1 (pz1 ; pz2 )+ pz2 z2 (pz1 ; pz2 ; q).
Figure 10 depicts cost functions for technologies with di¤erent returns
to scale for some given prices of inputs.Compare Figures 9 and 10. On
Figure 10(a) returns to scale are decreasing (as output increases, an additional 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 output increases, an additional unit of production becomes less and less
costly): 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 will
choose to cook it in the cheapest way.
13
(b)
(a)
c(q)
c(q)
c(3)
c(3)
c(1)
c(1)
1
(c)
3
q
1
(d)
c(q)
3
q
c(q)
c(3)
c(3)
c(1)
c(1)
1
3
q
1
3
q
Figure 10: Cost …nction: (a) decreasing returns to scale; (b) increasing
returns to scale; (c) constant returns to scale; (d) constant returns to
scale 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 production 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 minimized
at 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 pays
regardless of whether its output is positive or null (registration).
c(q)
AC(q)
q
q
Figure 11: e¢ cient scale with nonsunk setup cost
14
q
f(z)
f(z*)
Slope= −
pz
p
z*
z
Figure 12: Pro…t-maximizing input mix and production
Firm’s objectives We assume that a …rm maximizes its pro…ts
taking all prices as given. In class, we have discussed limitations of this
assumption: potentially controversial objectives by di¤erent owners, and
potential con‡ict of interests between the owners and the managers to
whom the owners need to delegate decision-making.
Firm’s supply A …rm’s whose technology is described by production function q = f (z), chooses input mix z = (z1 ; z2 ; :::; zN ) that solves
problem:
n
maxpf (z) pz z
(10)
z
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 are
equal to (z ) = pf (z ) pz z . Figure 12 illustrates pro…t-maximizing
input mix and production for strictly concave production function (decreasing returns to scale).Note, that7
p
@f (z )
6 pzi with equality if zi > 0; i = 1:::N:
@zi
A …rm’s whose technology is described by cost function c(q), chooses
to produce output q that solves problem:
maxpq
q
c(q)
(11)
For strictly convex cost function (decreasing returns to scale)
p 6 c (q ) with equality if q > 0:
0
7
(12)
Be careful not to use the …rst order approach for increasing or constant returns
to scale.
15
c(q)
Slope = -p
q
q*
Figure 13: pro…t maximizing production
That is, a …rm production, if it takes place, equalizes marginal cost
0
c (q ), that is, the cost of producing “the last”additional unit with price
that is charged for this unit, as illustrated on …gure 13.Suppose several
…rms produce the same output. For given prices, pro…t-maximizing …rm
with less e¢ cient technology, chooses to produce less.
5
Partial Equilibrium
Let us study market for one good in isolation. For illustrative
purposes, 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
function
ui (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 consumer
preferences in section 1).
Production technology by …rm j = 1; 2 is described by cost function
cj (qj ): …rm j inquires cost cj (qj ) in order to produce qj units of output.
Let us assume that cj (qj ) 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. Initially, consumer i has mi units of good m: no good x is available before
production takes place.
Equilibrium allocation Let us normalize the price of numeriare
good to be 1 (recall our discussion of “no money illusion”). Allocation
m1 ; m2 ; x1 ; x2 ; q1 ; q2 and price p of good x constitute an equilibrium if
and only if
16
1. qj = qj (p ) maximizes pro…ts by …rm j when price of the output
is equal to p , that is, it solves
maxp qj
qj
(13)
cj (qj )
(Hence, pro…ts by …rm j is j = p qj cj (qj )).
2. xi = xj (p ) solves optimization problem by consumer i when price
of consumption good is equal to p :
(
maxmi + '(xi )
xi ;mi
(14)
s:t: p xi + mi 6 mi + i1 1 + i2 2
3. Price of good x balances the market, that is, aggregate supply of
good x is equal to aggregate demand for good x:
q1 (p ) + q2 (p ) = x1 (p ) + x2 (p ):
By …rst-order approach, equilibrium allocation is characterized by:8
'i (xi ) 6 p with equality if xi > 0
(15)
p 6 cj (qj ) with equality if qj > 0
(16)
q1 + q 2 = x 1 + x 2
(17)
0
0
Pareto optimal allocation Allocation mo1 ; mo2 ; xo1 ; xo2 ; q1o ; q2o is Pareto
optimal if and only if it is impossible to …nd some other allocation
m1 ; m2 ; x1 ; x2 ; q1 ; q2 that increases utility by one consumer without decreasing utility by the other consumer.9
Suppose that perfectly informed benevolent social planner picks output in each …rm and allocates total output between the consumers so
as to maximize joint “happiness”that is measured by sum of consumer
utilities. She “solves”:
8
m1 + '1 (x1 ) + m2 + '2 (x2 )
>
< max
xi ;qj
(18)
s:t: x1 + x2 6 q1 + q2
>
:
m1 + m2 + c1 (q1 ) + c2 (q2 ) 6 m1 + m2
or, equivalently
(
max'1 (x1 ) + '2 (x2 )
xi ;qj
s:t: :
c1 (q1 )
c2 (q2 )
x 1 + x 2 6 q1 + q2
8
Recall equations (6) and (12).
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 other
allocation that increases utility by one consumer without decreasing utility by some
other consumer.
9
17
Lagrangian for this optimization problem is equal to
L = '1 (x1 ) + '2 (x2 )
c1 (q1 )
c2 (q2 ) + (q1 + q2
x1
x2 ) ;
where is Lagrangian multiplier associated with technological constraint.
Hence, the planner picks xo1 ; xo2 ; q1o ; q2o such that
6 cj (qjo ) with equality if qjo > 0
0
'i (x0i ) 6
0
with equality if xoi > 0
(q1o + q2o
xo1
xo2 ) = 0
(19)
(20)
(21)
and she allocates the remaining numeriare good in any way between the
consumers.
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 equilibrium is Pareto optimal.10
A producer would increase pro…t by expanding production of the good if its price exceeded his marginal cost. Conversely, if he produced the good at all, he would contract
production if the marginal cost were to exceed the price.
This trivial result has important implications. When deciding
whether to consume one more unit of the good, a consumer
faces a price that is socially “the right one”and internalizes
the cost of producing this extra unit (Tirole (1998), “The
Theory of Industrial Organization,”The MIT Press).
Marshallian surplus Marshallian surplus is a concept of quasilinear model. It measures social welfare. It is equal to the surface that
lies below the inverse demand curve less the surface that lies below the
supply 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 equilibrium.
10
Recall, that we consider an “ideal” economy.
Recall our discussion in class: an addition unit of production increases of consumer surplus (see part 1) and imposes cost on producers.
11
18
-
+
S(p)
Augmentation de la
production/consomm
ation marginale
D(p)
Figure 14: marginal change in Marshallian surplus.
Figure 15: maximim of Marshallian surplus.
19
p
Consumer surplus
p*(t)
Tax revenues
p*
Deadweight loss
t
q,x
Producer surplus
q*(t)
q*
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, supply
of good x is characterized by equations
p (t) + t 6 cj (qj ) with equality if qj > 0;
0
(22)
where p (t) denotes new equilibrium price. Aggregate supply curve
“shifts up”, as depicted on …gure 16. As a result, equilibrium output
and consumption decrease. Taxation creates a deadweight loss: tax revenues+consumer surplus+producer surplus in equilibrium with taxes lies
below Marshallian surplus without taxes.
Note, that new equilibrium price increases as compared to that without taxation: both producers and consumers share the burden of deadweight loss. How the share that is beard by consumers depend on elasticity of demand? What happens if consumers, and not producers pay
a 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 by
cost 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 solves
maxp q
q
c(q)
2. x (p ) = N q (balance on the market);
20
(23)
3. p q
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 can
be generated by any number of …rms with any load); when returns to
scale are decreasing, there is no long-run competitive equilibrium (pro…tmaximization implies that pro…ts are positive, then, however, more …rms
want to enter the market). Indeed, in any equilibrium with determinant
number of …rms cost function must exhibit a strictly positive e¢ cient
scale (see part 2).
6
A primer in general equilibrium model: Robinson Crusoe economy
Partial equilibrium approach considers one market in isolation. Potentially, shocks on this market generate e¤ects on other markets. Partial
equilibrium approach ignores these e¤ects. A more complicated, general
equilibrium model takes these e¤ects into the account. In class, we have
considered a simple illustration of this model.
Robinson Crusoe economy Consider an economy with one consumer (Robinson Crusoe) who owns a single …rm. Robinson bene…ts
from two goods: consumption good x and leisure l. His preferences
are described by utility function u(l; x). Initially, Robinson has 24 hours
available for leisure. However, he needs to work z hours in the …rm in
order to produce f (z) units of consumption good. That is, the …rm’s
technology is described by production function f (z), where z = 24 l.
Equilibrium allocation Let p be price of good x, and w be the
wage 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 equal
to p ; w . That is, it solves12
n
maxp f (z) w z
(24)
z
(Hence, supply of good x is equal to q = f (z (p ; w )); and pro…ts by
the …rm is equal to
= p f (z (p ; w )) w z ).
2. x (p ; w ), l (p ; w ) solve consumer optimization problem:
(
max u(l; x)
x;l624
(25)
s:t: p x 6 w (24 l) +
12
Recall problem (10).
21
24h
x
q
Robinson’s indifference curve
Slope=
Supply of x
Demand for x
f(z)
−
w
p
l
z
Labour demand
Leisure demand
Labour supply
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 to
demand 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 strictly
concave, so that the …rst-order approach is valid. Then, (interior) equilibrium is characterized by:
0
p
u (l; x)
p
f (z ) =
; x0
=
; 24
w
w
ul (l; x)
0
l = z ; f (z ) = x :
(26)
On …gure 17 there is excess supply of labour and excess demand for good
x.
Consequently price ratio wp decreases, so as to balance the markets:
…gure 18.
Pareto optimal allocation Suppose that Robinson gives up with
trading with himself and simply decides how much to work and how
much of good x to consume. That is, he solves:
(
max u(l; x)
x;l624
s:t: x = f (24
13
l)
Indeed, if one market is balaced, the other is also balanced.
22
24h
x
q
Robinson’s indifference curve
f(z)
w*
p*
Demand for x
Supply of x
Slope= −
l
z
Labour demand
Leisure demand
Labour supply
Figure 18: equilibrium allocation and price ratio
Because we have assumed that utility and production function are both
strictly concave, there is the unique Pareto optimal allocation xo , lo :
0
ul (lo ; f (24
0
lo )) = ux (lo ; 24
0
lo )f (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 illustrative
example
“An ideal family” Marie and Pierre live together. An individual’s
utility depends on consumption of food x, and on the amount of money
m:
UM = mM + ln(1 + xM );
(27)
UP = mP + 2 ln(1 + xP ):
(28)
Hence, (i) Mary and Pierre like money equally; (ii) Pierre likes food
twice more than Mary:
There are two ways to get food out of money. The …rst way is to
send Mary to the market. She is able to bring home q units of food in
2
exchange for q2 dollars. If Peter goes shopping, he has to spend twice
more money to bring home the same amount of food. That is, there
are two technologies that allow to produce consumption good out of the
numeriare, with cost functions:
cM (q) =
q2
; cP (q) = q 2 :
2
23
24h
x
q
Robinson’s indifference curve
f(z)
l
z
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 optimal
shopping/consumption plan (xM ; xP ; qM ; qP ) solves14
(
2
qM
max ln(1 + xM ) + 2 ln(1 + xP )
qP2
2
xM ;xP ;qa ;qb
s:t: :
x M + x P = qM + qP
Lagrangian for this optimization problem is equal to
L = ln(1 + xM ) + 2 ln(1 + xP )
2
qM
2
qP2 + (qM + qP
xM
xP ) ;
where is the Lagrangian multiplier associated with technological constraint.
= c0M (qM ) = qM = c0P (qP ) = 2qP =
(29)
= (ln(1 + xM ))0 =
1
2
= (2 ln(1 + xP ))0 =
1 + xM
1 + xP
(qM + qP
Solving (29)-(31), we …nd:15
p
xM =
xM
10
3
2
(30)
xP ) = 0
(31)
0:39;
(32)
We have to verify later that the solution satis…es endownment constraint qi 6
100, i = M; P .
15
Verify.
14
25
p
2 10 1
xP = 1 + 2xM =
1:77;
(33)
3
p
3 1 + 10
1
0:72;
(34)
qP =
=
22
1 + xM
p
3 1 + 10
qM = 2qP =
1:44:
(35)
11
Hence, perfectly informed benevolent “family planner” (i) would ask
Mary to buy more food than Peter: qM > qP ; and she (ii) would let
Peter eat more than Mary: xM < xP .
Suppose now that the planner knows that Peter’s preferences are described by equation (28). She also knows that (i) with probability 12
Mary has the same preferences as Peter, (ii) with probability 12 Mary’s
preferences are described by equation (27). However, the planner does
not know Mary’s preferences exactly. Because Mary’s utility is monotonically increasing in her consumption of food, she claims that she likes
food as much as Peter (recall the failure of planned economies!).16 A
way 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 trade
food at home at price p. They (i) know everything about each other preferences and production technologies, and (ii) behave as price-takers both
when they sell food to each other, and when they buy food from each
other.
Supply
For a given price p, Mary’s supply of food on the home market solves
maxpq
q
q2
2
Hence, it is equal to
qM (p) = p
(36)
Mary’s supply of food on the home market solves
maxpq
qa
q2
p
(37)
2
Therefore, at a given price p, aggregate supply of food on the home
market is equal to
3p
qM (p) + qP (p) = :
(38)
2
qP (p) =
16
In this case it is optimal to allocate
p
5 1
2
26
units of food to either family member.
Demand
For a given price p, Mary’s demand for food on the home market
solves
8
<
max ln(1 + xM ) + mM
xM ;qM
: s:t: :
pxM + mM 6 1 + pqM
2
qM
2
Hence, her demand for food xM (p) satis…es equation
1
= p;
1 + xM (p)
(39)
or, equivalently
1
1:
p
Peter’s demand for food on the home market solves
(
max2 ln(1 + xP ) + mP
xM (p) =
(40)
xP ;qb
s:t: :
pxP + mP 6 1 + (pqP
qP2 )
Hence,
2
= p;
1 + xP (p)
(41)
which is equivalent to
2
1:
(42)
p
Consequently, at a given price p, aggregate demand for food on the home
market is equal to
3
xM (p) + xP (p) =
2:
(43)
p
Balance on the Market
In equilibrium, price balances the market:17
xP (p) =
3
p
2=
3p
:
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 individual
production and consumption are e¢ cient: that is, they are given by
equations (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. Draw
aggregate demand and aggregate supply curves.
27
Missing markets Suppose that Peter has a powerful stereo system, and he likes to listen to pop-music (denote by h 6 24 his consumption of music per day):
UP = mP + 2 ln(1 + xP ) + ln(1 + h):
Mary, instead, su¤ers when music is playing:
UM = 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. This
outcome is suboptimal. It would be e¢ cient to take into the account
Mary’s preferences, and let the music play so as to
max ln(1 + h)
h624
h
;
2
that is, for
h =1
hour a day.
Private bargaining over externality A benevolent social planner could restore the e¢ ciency (i) by requiring that the music cannot be
played more than an hour (that is, by imposing a quota), or else (ii) by
charging Peter t = 12 dollar for each hour that he listens to the music
(that is, by imposing a tax). However, these forms of intervention are
not necessary.19
Suppose that the government enforces Peter’s property rights to listen to the music as long as Peter wants. Suppose furthermore that
the 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 a
day. If Peter agrees, this agreement is enforced by the government. If
no agreement is reached, the outcome is as described in the previous
section: Peter listens to music non-stop.
Peter agrees to Mary’s o¤er if and only if
ln(1 + h)
T > ln(25)
18
(45)
See page 352 in Mas-Colell, Whinston, and Green 1995 for de…nition of an externality.
19
Because government intervention is costly (recall that taxation is associated with
a deadweightloss), the general presumption would be to keep it on the lowest necessary level.
28
Hence, the Mary’s best o¤er solves
(
max T h2
T 6100; h624
s:t: :
(45)
Indeed, Mary o¤ers to Peter to pay him T = ln 25 ln 2 if the music
plays 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 music
played. Indeed, suppose that the government guarantees Mary that she
can lock the stereo system in a wardrobe for h hours. Mary o¤ers to
Peter 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 number 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-orleave-it” o¤er. Then, he proposes Mary to pay him T = 11 h2 if the
music plays an hour, and Mary agrees.
The following three insights are general in bargaining games: a player’s
payo¤ is higher, (i) the better his “outside option”; (ii) the higher his
bargaining 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 and
none of them can limit the other’s access to the place. They can hire a
cleaning lady who increases the level of order in the apartment by y at
e¤ort costs y 2 . Peter cares for order less than Mary:
UP = mP + 2 ln(1 + xP ) + 3 ln(1 + yP + yM );
UM = mM + ln(1 + xM ) + ln(1 + yP + yM ):
Order in the apartment is a public good. Socially optimal level of order
solves
max4 ln(1 + y) y 2
y
It is equal to
y = 1:
20
21
See page 357 in Mas-Colell, Whinston, and Green 1995.
Hence, the givernment should simply enforce private agreements.
29
py
4
3
Mary’s demand
for public good
1
Equilibrium
y=0.82
Pierre’s demand
for public good
y
Social optimum
y=1
Figure 22: free-riding.
In equilibrium, cleaning lady o¤ers
py
y=
2
units of order, and only Mary is ready to pay for it - Peter bene…ts from
the order that is paid by Mary: he would not like to pay so as to make
their apartment even cleaner:
1
1 + yP + yM
6 py with an equality when yP > 0
3
6 py with an equality when yM > 0.
1 + yP + yM
As a result, y is suboptimal (see Fig 22):Verify that if Mary’s consumption of order is subsidized at rate sM = 12 , 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 nontrivial.
References
[1] Mas-Colell, A., Whinston, D.,M., and J.,R. Green. (1995), “Microeconomic Theory,”Oxford University Press.
[2] Tirole, J. (1988), “The Theory of Industrial Organization,”The MIT
Press.
22
Note that the government plays a more e¢ cient role than just law enforcement.
30