ECN421

GENERAL EQUILIBRIUM AND

WELFARE

1
 Perfectly Competitive
Price System
• We will assume that all markets are
perfectly competitive
– there is some large number of homogeneous
goods in the economy
• both consumption goods and factors of
production
– each good has an equilibrium price
– there are no transaction or transportation
costs
– individuals and firms have perfect information
2
 Law of One Price
• A homogeneous good trades at the
same price no matter who buys it or
who sells it
– if one good traded at two different prices,
demanders would rush to buy the good
where it was cheaper and firms would try
to sell their output where the price was
higher
• these actions would tend to equalize the price
of the good
3
 Assumptions of Perfect
Competition
• There are a large number of people
buying any one good
– each person takes all prices as given and
seeks to maximize utility given his budget
constraint
• There are a large number of firms
producing each good
– each firm takes all prices as given and
attempts to maximize profits
4
 General Equilibrium
• Assume that there are only two goods, x
and y
• All individuals are assumed to have
identical preferences
– represented by an indifference map
• The production possibility curve can be
used to show how outputs and inputs are
related
5
 Edgeworth Box Diagram
• Construction of the production possibility
curve for x and y starts with the
assumption that the amounts of k and l
are fixed
• An Edgeworth box shows every possible
way the existing k and l might be used to
produce x and y
– any point in the box represents a fully
employed allocation of the available
resources to x and y 6
 Edgeworth Box Diagram
Labor in y production
Labor for x Labor for y Capital
Oy in y
production

Capital for y
Total Capital


A

Capital
for x
Capital
in x
production
Ox
Total Labor 7
Labor in x production
 Edgeworth Box Diagram
• Many of the allocations in the Edgeworth
box are technically inefficient
– it is possible to produce more x and more y by
shifting capital and labor around
• We will assume that competitive markets
will not exhibit inefficient input choices
• We want to find the efficient allocations
– they illustrate the actual production outcomes
8
 Edgeworth Box Diagram
• We will use isoquant maps for the two
goods
– the isoquant map for good x uses Ox as the
origin
– the isoquant map for good y uses Oy as the
origin
• The efficient allocations will occur where
the isoquants are tangent to one another
9
 Edgeworth Box Diagram
Point A is inefficient because, by moving along y1, we can increase
x from x1 to x2 while holding y constant
Oy

y1
Total Capital

y2

x2

A x1

Ox
Total Labor 10
 Edgeworth Box Diagram
We could also increase y from y1 to y2 while holding x constant
by moving along x1
Oy

y1
Total Capital

y2

x2

A x1

Ox
Total Labor 11
Edgeworth Box Diagram
At each efficient point, the RTS (of k for l) is equal in both
x and y production
Oy

y1
p4
y2
Total Capital

p3
x4
y3
p2

y4 x3
p1

x2
x1

Ox
Total Labor 12
Production Possibility Frontier
• The locus of efficient points shows the
maximum output of y that can be
produced for any level of x
– we can use this information to construct a
production possibility frontier
• shows the alternative outputs of x and y that
can be produced with the fixed capital and
labor inputs that are employed efficiently

13
Production Possibility Frontier
Quantity of y Each efficient point of production
becomes a point on the production
Ox p1
possibility frontier
y4 p2
y3
The negative of the slope of
p3
y2 the production possibility
frontier is the rate of product
transformation
p4
y1 (RPT)/Marginal Rate of
Transformation (MRT)
Quantity of x
x1 x2 x3 x4 Oy

14
Rate of Product Transformation
• The rate of product transformation (RPT)
(Alternatively referred to as the Marginal
Rate of Transformation (MRT)) between
two outputs is the negative of the slope of
the production possibility frontier
RPT (of x for y )   slope of production
possibility frontier

dy
RPT (of x for y )   (along OxOy )
dx
15
Rate of Product Transformation
• The rate of product transformation(RPT)
/Marginal Rate of Transformation (MRT)
shows how x can be technically traded for y
while continuing to keep the available
productive inputs efficiently employed

16
 Shape of the Production
Possibility Frontier
• The production possibility frontier shown
earlier exhibited an increasing RPT
– this concave shape will characterize most
production situations
• RPT is equal to the ratio of MCx to MCy

17
 Shape of the Production
Possibility Frontier
• Suppose that the costs of any output
combination are C(x,y)
– along the production possibility frontier,
C(x,y) is constant
• We can write the total differential of the
cost function as
C C
dC   dx   dy  0
x y
18
 Shape of the Production
Possibility Frontier
• Rewriting, we get
dy C / x MCx
RPT   (along OxOy )  
dx C / y MCy

• The RPT is a measure of the relative

marginal costs of the two goods

19
 Shape of the Production
Possibility Frontier
• As production of x rises and production
of y falls, the ratio of MCx to MCy rises
– this occurs if both goods are produced
under diminishing returns
• increasing the production of x raises MCx, while
reducing the production of y lowers MCy
– this could also occur if some inputs were
more suited for x production than for y
production
20
 Shape of the Production
Possibility Frontier
• But we have assumed that inputs are
homogeneous
• We need an explanation that allows
homogeneous inputs and constant
returns to scale
• The production possibility frontier will be
concave if goods x and y use inputs in
different proportions
21
 Opportunity Cost
• The production possibility frontier
demonstrates that there are many
possible efficient combinations of two
goods
• Producing more of one good
necessitates lowering the production of
the other good
– this is what economists mean by opportunity
cost
22
 Opportunity Cost
• The opportunity cost of one more unit of
x is the reduction in y that this entails
• Thus, the opportunity cost is best
measured as the RPT (of x for y) at the
prevailing point on the production
possibility frontier
– this opportunity cost rises as more x is
produced
23
Concavity of the Production
Possibility Frontier
• Suppose that the production of x and y
depends only on labor and the production
functions are
x  f (l x )  l x0.5 y  f (ly )  ly0.5
• If labor supply is fixed at 100, then
lx + ly = 100
• The production possibility frontier is
x2 + y2 = 100 for x,y  0
24
Concavity of the Production
Possibility Frontier
• The RPT can be calculated by taking the
total differential:
 dy  ( 2 x ) x
2 xdx  2ydy  0 or RPT   
dx 2y y
• The slope of the production possibility
frontier increases as x output increases
– the frontier is concave

25
 Determination of
Equilibrium Prices
• We can use the production possibility
frontier along with a set of indifference
curves to show how equilibrium prices
are determined
– the indifference curves represent
individuals’ preferences for the two goods

26
 Determination of
Equilibrium Prices
If the prices of x and y are px and py,
Quantity of y
society’s budget constraint is C
C
Output will be x1, y1
y1

Individuals will demand x1’, y1’

y1’

U3
U2 C

 px
U1 slope 
py
Quantity of x
x1 x1’ 27
 Determination of
Equilibrium Prices
There is excess demand for x and
Quantity of y
excess supply of y
C
The price of x will rise and
y1 the price of y will fall
excess
supply
y1’

U3
U2 C

 px
U1 slope 
py

x x1’
Quantity of x 28
1 excess demand
 Determination of
Equilibrium Prices
The equilibrium prices will be px* and
Quantity of y C* py*, after the price line rotates
clockwise and becomes steeper in
C response to the market addressing
the excess demand of good X and
y1 excess supply of Good Y

y1* The equilibrium output will

y1’
be x1* and y1*
U3
U2 C

 px
U1 slope 
py
C*
Quantity of x
x x1* x1’  px* 29
slope 
1 py*
Comparative Statics Analysis
• The equilibrium price ratio will tend to
persist until either preferences or
production technologies change
• If preferences were to shift toward good
x, px /py would rise and more x and less
y would be produced
– we would move in a clockwise direction
along the production possibility frontier
30
Comparative Statics Analysis
• Technical progress in the production of
good x will shift the production
possibility curve outward
– this will lower the relative price of x
– more x will be consumed
• if x is a normal good
– the effect on y is ambiguous

31
 Technical Progress in the
Production of x
Technical progress in the production
Quantity of y
of x will shift the production possibility
curve out

The relative price of x will fall

More x will be consumed

U3
U2

U1

x1* x2*
Quantity of x 32
General Equilibrium Pricing
• Suppose that the production possibility
frontier can be represented by
x 2 + y 2 = 100
• Suppose also that the community’s
preferences can be represented by
U(x,y) = x0.5y0.5

33
General Equilibrium Pricing
• Profit-maximizing firms will equate MRT
and the ratio of px /py
x px
MRT  
y py

• Utility maximization requires that

y px
MRS  
x py

34
General Equilibrium Pricing
• Equilibrium requires that firms and
individuals face the same price ratio
x px y
RPT     MRS
y py x

or
x* = y*

35
 Class Exercise
Given
𝑥 = 𝑙𝑥0.5 and y= 𝑙𝑦0.5
Total labour available is 100 hours
1 1
𝑈= 𝑋 2𝑌2
X and Y are goods consumed, while 𝑙𝑥
and 𝑙𝑦 are labour employed in X and Y
production.
Required:
(1) Determine the optimal allocation of
labour to producing goods X and Y 36
 Class Exercise
2) Determine the optimal production of X and Y.
3) Determine the optimal consumption of X and
Y.
4) Determine the Marginal utility at the optimal
consumption of goods x and Y in (3) above
5) Determine the Equilibrium price and
consequently Px and Py
6) If consumer preferences were to switch to favor
good y as U(x,y) = x0.1y0.9, what is the effect on
the equilibrium price?
7) Determine the effect on the price line if
consume tastes should change in favour of
good Y
37
 The Corn Laws Debate
• High tariffs on grain imports were
imposed by the British government after
the Napoleonic wars
• Economists debated the effects of these
“corn laws” between 1829 and 1845
– what effect would the elimination of these
tariffs have on factor prices?

38
 The Corn Laws Debate
Quantity of
manufactured If the corn laws completely prevented
goods (y)
trade, output would be x0 and y0

The equilibrium prices will be

px* and py*
y0

U2
U1

 px*
slope 
py*
Quantity of Grain (x)
x0

39
 The Corn Laws Debate
Quantity of
manufactured Removal of the corn laws will change
goods (y) the prices to px’ and py’
Output will be x1’ and y1’
y1’
Individuals will demand x1 and y1
y0

y1

U2
U1
 px '
slope 
py '

Quantity of Grain (x)

x1’ x0 x1

40
 The Corn Laws Debate
Quantity of
manufactured Grain imports will be x1 – x1’
goods (y)
These imports will be financed by
the export of manufactured goods
y1’
exports equal to y1’ – y1
of
y0
goods
y1

U2
U1
 px '
slope 
py '

Quantity of Grain (x)

x1’ x0 x1

41
imports of grain
 The Corn Laws Debate
• We can use an Edgeworth box diagram
to see the effects of tariff reduction on
the use of labor and capital
• If the corn laws were repealed, there
would be an increase in the production
of manufactured goods and a decline in
the production of grain

42
 The Corn Laws Debate
A repeal of the corn laws would result in a movement from p3 to
p1 where more y and less x is produced
Oy

y1
p4
y2
Total Capital

p3
x4
y3
p2

y4 x3
p1

x2
x1

Ox
Total Labor 43
 The Corn Laws Debate
• If we assume that grain production is
relatively capital intensive, the movement
from p3 to p1 causes the ratio of k to l to
rise in both industries
– the relative price of capital will fall
– the relative price of labor will rise
• The repeal of the corn laws will be
harmful to capital owners and helpful to
laborers 44
 Political Support for
Trade Policies
• Trade policies may affect the relative
incomes of various factors of production
• In the United States, exports tend to be
intensive in their use of skilled labor
whereas imports tend to be intensive in
their use of unskilled labor
– free trade policies will result in rising relative
wages for skilled workers and in falling
relative wages for unskilled workers 45
 Smith’s Invisible Hand
Hypothesis
• Adam Smith believed that the
competitive market system provided a
powerful “invisible hand” that ensured
resources would find their way to where
they were most valued
• Reliance on the economic self-interest
of individuals and firms would result in a
desirable social outcome
46
 Smith’s Invisible Hand
Hypothesis
• Smith’s insights gave rise to modern
welfare economics
• The “First Theorem of Welfare
Economics” suggests that there is an
exact correspondence between the
efficient allocation of resources and the
competitive pricing of these resources
47
 Pareto Efficiency
• An allocation of resources is Pareto
efficient if it is not possible (through
further reallocations) to make one person
better off without making someone else
worse off
• The Pareto definition identifies allocations
as being “inefficient” if unambiguous
improvements are possible
48
 Efficiency in Production
• An allocation of resources is efficient in
production (or “technically efficient”) if no
further reallocation would permit more of
one good to be produced without
necessarily reducing the output of some
other good
• Technical efficiency is a precondition for
Pareto efficiency but does not guarantee
Pareto efficiency 49
Efficient Choice of Inputs for a
Single Firm
• A single firm with fixed inputs of labor
and capital will have allocated these
resources efficiently if they are fully
employed and if the RTS between
capital and labor is the same for every
output the firm produces

50
Efficient Choice of Inputs for a
Single Firm
• Assume that the firm produces two
goods (x and y) and that the available
levels of capital and labor are k’ and l’
• The production function for x is given by
x = f (kx, lx)
• If we assume full employment, the
production function for y is
y = g (ky, ly) = g (k’ - kx, l’ - lx)
51
Efficient Choice of Inputs for a
Single Firm
• Technical efficiency requires that x
output be as large as possible for any
value of y (y’)
• Setting up the Lagrangian and solving for
the first-order conditions:
L = f (kx, lx) + [y’ – g (k’ - kx, l’ - lx)]
L/kx = fk + gk = 0
L/lx = fl + gl = 0
L/ = y’ – g (k’ - kx, l’ - lx) = 0 52
Efficient Choice of Inputs for a
Single Firm
• From the first two conditions, we can see
that
fk g k

fl gl
• This implies that
RTSx (k for l) = RTSy (k for l)

53
 Efficient Allocation of
Resources among Firms
• Resources should be allocated to those
firms where they can be most efficiently
used
– the marginal physical product of any
resource in the production of a particular
good should be the same across all firms
that produce the good

54
 Efficient Allocation of
Resources among Firms
• Suppose that there are two firms
producing x and their production
functions are
f1(l1, k1)
f2(l2, k2)
• Assume that the total supplies of capital
and labor are k’ and l’
55
 Efficient Allocation of
Resources among Firms
• The allocational problem is to maximize
x = f1(k1, l1) + f2(k2, l2)
subject to the constraints
l1 + l2 = l’
k1 + k2 = k’
• Substituting, the maximization problem
becomes
x = f1(k1, l1) + f2(l’ - l1 , k’ - k1,)
56
 Efficient Allocation of
Resources among Firms
• First-order conditions for a maximum
are
x f1 f2 f1 f2
    0
l1 l1 l1 l1 l2
x f1 f2 f1 f2
    0
k1 k1 k1 k1 k 2

57
 Efficient Allocation of
Resources among Firms
• These first-order conditions can be
rewritten as
f1 f2 f1 f2
 
l1 l2 k1 k 2

• The marginal physical product of each

input should be equal across the two
firms 58
 Efficient Choice of Output
by Firms
• Suppose that there are two outputs (x
and y) each produced by two firms
• The production possibility frontiers for
these two firms are
yi = fi (xi ) for i=1,2
• The overall optimization problem is to
produce the maximum amount of x for
any given level of y (y*)
59
 Efficient Choice of Output
by Firms
• The Lagrangian for this problem is
L = x1 + x2 + [y* - f1(x1) - f2(x2)]
and yields the first-order condition:
f1/x1 = f2/x2
• The rate of product transformation
(RPT) should be the same for all firms
producing these goods
60
 Efficient Choice of Output
by Firms
Firm A is relatively efficient at producing cars, while Firm B
is relatively efficient at producing trucks
Cars Cars 1
2 RPT 
RPT  1
1
100 100

50 Trucks 50 Trucks
Firm A Firm B 61
 Efficient Choice of Output
by Firms
If each firm was to specialize in its efficient product, total
output could be increased
Cars Cars 1
2 RPT 
RPT  1
1
100 100

50 Trucks 50 Trucks
Firm A Firm B 62
 Theory of Comparative
Advantage
• The theory of comparative advantage
was first proposed by Ricardo
– countries should specialize in producing
those goods of which they are relatively
more efficient producers
• these countries should then trade with the rest
of the world to obtain needed commodities
– if countries do specialize this way, total
world production will be greater
63
 Efficiency in Product Mix
• Technical efficiency is not a sufficient
condition for Pareto efficiency
– demand must also be brought into the
picture
• In order to ensure Pareto efficiency, we
must be able to tie individual’s
preferences and production possibilities
together
64
 Efficiency in Product Mix
• The condition necessary to ensure that
the right goods are produced is
MRS = RPT
– the psychological rate of trade-off between
the two goods in people’s preferences must
be equal to the rate at which they can be
traded off in production

65
 Efficiency in Product Mix
Output of y Suppose that we have a one-person (Robinson
Crusoe) economy and PP represents the
combinations of x and y that can be produced
P

Any point on PP represents a

point of technical efficiency

Output of x
P

66
 Efficiency in Product Mix
Output of y Only one point on PP will maximize
Crusoe’s utility

P At the point of
tangency, Crusoe’s
MRS will be equal to
the technical RPT
U3

U2

U1

Output of x
P

67
 Efficiency in Product Mix
• Assume that there are only two goods
(x and y) and one individual in society
(Robinson Crusoe)
• Crusoe’s utility function is
U = U(x,y)
• The production possibility frontier is
T(x,y) = 0
68
 Efficiency in Product Mix
• Crusoe’s problem is to maximize his
utility subject to the production
constraint

• Setting up the Lagrangian yields

L = U(x,y) + [T(x,y)]

69
 Efficiency in Product Mix
• First-order conditions for an interior
maximum are
L U T
  0
x x x
L U T
  0
y y y
L
 T ( x, y )  0
 70
 Efficiency in Product Mix
• Combining the first two, we get
U / x T / x

U / y T / y

or
dy
MRS ( x for y )   (along T )  RPT ( x for y )
dx

71
 Competitive Prices and
Efficiency
• Attaining a Pareto efficient allocation of
resources requires that the rate of
trade-off between any two goods be the
same for all economic agents
• In a perfectly competitive economy, the
ratio of the prices of the two goods
provides the common rate of trade-off to
which all agents will adjust
72
 Competitive Prices and
Efficiency
• Because all agents face the same
prices, all trade-off rates will be
equalized and an efficient allocation will
be achieved
• This is the “First Theorem of Welfare
Economics”

73
 Efficiency in Production
• In minimizing costs, a firm will equate
the RTS between any two inputs (k and
l) to the ratio of their competitive prices
(w/v)
– this is true for all outputs the firm produces
– RTS will be equal across all outputs

74
 Efficiency in Production
• A profit-maximizing firm will hire
additional units of an input (l) up to the
point at which its marginal contribution
to revenues is equal to the marginal
cost of hiring the input (w)
pxfl = w

75
 Efficiency in Production
• If this is true for every firm, then with a
competitive labor market
pxfl1 = w = pxfl2
fl1 = fl2
• Every firm that produces x has identical
marginal productivities of every input in
the production of x

76
 Efficiency in Production
• Recall that the RPT (of x for y) is equal
to MCx /MCy
• In perfect competition, each profit-
maximizing firm will produce the output
level for which marginal cost is equal to
price
• Since px = MCx and py = MCy for every
firm, RTS = MCx /MCy = px /py
77
 Efficiency in Production
• Thus, the profit-maximizing decisions
of many firms can achieve technical
efficiency in production without any
central direction
• Competitive market prices act as
signals to unify the multitude of
decisions that firms make into one
coherent, efficient pattern
78
 Efficiency in Product Mix
• The price ratios quoted to consumers
are the same ratios the market presents
to firms
• This implies that the MRS shared by all
individuals will be equal to the RPT
shared by all the firms
• An efficient mix of goods will therefore
be produced
79
 Efficiency in Product Mix
Output of y x* and y* represent the efficient output mix
px*
slope  
py*
P
Only with a price ratio of
px*/py* will supply and
y* demand be in equilibrium

U0

Output of x
x* P

80
 Laissez-Faire Policies
• The correspondence between
competitive equilibrium and Pareto
efficiency provides some support for the
laissez-faire position taken by many
economists
– government intervention may only result in
a loss of Pareto efficiency

81
 Departing from the
Competitive Assumptions
• The ability of competitive markets to
achieve efficiency may be impaired
because of
– imperfect competition
– externalities
– public goods
– imperfect information

82
 Imperfect Competition
• Imperfect competition includes all
situations in which economic agents
exert some market power in determining
market prices
– these agents will take these effects into
account in their decisions
• Market prices no longer carry the
informational content required to achieve
Pareto efficiency
83
 Externalities
• An externality occurs when there are
interactions among firms and individuals
that are not adequately reflected in
market prices
• With externalities, market prices no
longer reflect all of a good’s costs of
production
– there is a divergence between private and
social marginal cost
84
 Public Goods
• Public goods have two properties that
make them unsuitable for production in
markets
– they are nonrival
• additional people can consume the benefits of
these goods at zero cost
– they are nonexclusive
• extra individuals cannot be precluded from
consuming the good

85
 Imperfect Information
• If economic actors are uncertain about
prices or if markets cannot reach
equilibrium, there is no reason to expect
that the efficiency property of
competitive pricing will be retained

86
 End of Topic Practise
Exercises
1) Work on exercises in practice exercise
2.
2) Work on the below:

87
 End of Topic Practise
Exercises
3) Suppose two rice farms have production
functions of the simple form

But that one rice farm is more mechanized than

the other. If capital for the first farm is given by
K1=16 and for the second farm by K2=625, show
why an equal allocation of Labour for the
production of q by both firms will be inefficient for
the industry. Also show each firms optimum output
88
 End of Topic Practise
Exercises
4) With the aid of appropriate diagram(s)
that are well labelled, explain how in a
competitive market where there exists
excess demand for good Y and excess
supply of good X, disequilibrium is
corrected.

89