General
Equilibrium
1
Walters & Layard CH 2
General Equilibrium
INTRODUCTION
In this chapter we will deal with positive economy theory to
construct a framework for the following purposes ;
First ; predicting the effect of particular cause ;
Second ; detecting the cause of particular effects ;
A simple model is chosen ; two sector model ;
This chapter has also two independent purposes;
1- we prove the existence and stability of general
competitive equilibrium and consider whether it is
unique ?
2 – how distribution of income is determined and how it
would change with changes in factor supply ;
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Walters & Layard CH 2
General Equilibrium
CONSUMPTION WITHOUT PRODUCTION
PURE EXCHANGE
A- Bargaining ;
Y
E
o
uB
oB
T0
uA
T1
oA
X
E = initial endowment point → MRSxyA > MRSxyB → fruitful
trade is possible . As while as both are consuming in the
ET0T1 area , both will be better off .
The solution will be any point between T0 and T1 on the
contract curve . If solution would be near to T0 ,
individual A has more strength ,and vise versa .
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Walters & Layard CH 2
General Equilibrium
CONSUMPTION WITHOUT PRODUCTION PURE
EXCHANGE
B- Existence of equilibrium
We will deal with competitive equilibrium , in which there
are large number of identical consumers with identical
utility functions and identical endowments.
The presence of auctioneer who will call on different
prices will finally bring about the equilibrium point
like T. There are three question that are of interest to
the auctioneer.
1-First , is there any price which could clear the market
? Does equilibrium exist ?
2-Second , is there more than one such price ? Is
equilibrium unique ?
3-Third, will the equilibrium be stable ?
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Walters & Layard CH 2
General Equilibrium
CONSUMPTION WITHOUT PRODUCTION
PURE EXCHANGE
Suppose auctioneer starts with price equal to the slope of
ET0 , ( price = p0 ) .
a’
b’
oB
Y
E
d’
d
c’
B
c
A
c
0
T
c
p
0
C
P*
5
oA
a
b
Walters & Layard CH 2
X
General Equilibrium
CONSUMPTION WITHOUT PRODUCTION
PURE EXCHANGE
Consumer A ;
Consumer B ;
ab= excess demand for x
a’b’ =excess supply of x
cd = excess supply for y
c’d’ = excess demand for y
ab > a’b’ → aggregate excess demand for x → (px/py)↑
cd > c’d’ → aggregate excess supply for y → (px/py)↑
Price line E T0 rotates around E inward until CA and CB
coincide with each other. When CA and CB coincide with
each other at point C , excess demand for x and excess
supply of y becomes zero . The price line will be
equilibrium price ( P* ), and MRSxyA=MRSxyB,. We will be
on the contract curve .
This process could be repeated for any other price other
than the equilibrium price (p* ) , until we reach to
equilibrium price .
What is clear from this analysis is that at any price level
both consumers will be on their budget constraint.
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Walters & Layard CH 2
General Equilibrium
CONSUMPTION WITHOUT PRODUCTION
PURE EXCHANGE
It is clear from this analysis that at any price level both
consumers will be on their budget constraint. This is what we
expect from consumer utility maximization under perfect
competition;
pxxA + pyyA =value of A’s market demand = A’s expenditure
pxxoA + pyyoA = value of A’s market supply = A,s income
xoA = initial endowment of x for individual A
yoA = initial endowment of y for individual A
pxxA + pyyA = pxxoA + pyyoA → px(xA – xoA) + py(yA – yoA) = 0
pxxB +pyyB =pxxoB + pyyoB → px(xB – xoB) + py(yB – yoB) = 0
px EDxA + pyEDyA =0 ,
px EDx B + pyEDy B =0
px EDx + pyEDy =0
7
WALRAS LAW
Walters & Layard CH 2
General Equilibrium
CONSUMPTION WITHOUT PRODUCTION
PURE EXCHANGE
WALRAS LAW
The sum of price weighted excess demands summed over
all markets must be zero. So if one market has positive
excess demand , another one should have positive
excess supply or negative excess demand .
Now , consider an individual market like market for x .
Considering the above analysis, at any price level like P0
there will be either excess demand or excess supply in
the market. So it is possible to find a range of price level
from Po to P1 in such a way that excess demand convert
to excess supply .
Looking at the following figure , and assuming continuity in
the set of price level it is possible to prove (by fixed point
theorem ) that there should exist a price level at which
excess demand for x will be equal to zero.
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Walters & Layard CH 2
General Equilibrium
CONSUMPTION WITHOUT PRODUCTION
PURE EXCHANGE
B
At p1 : excess supply
p1
At p0 : excess demand
P*
At p* : market is clear
A
po
ESx
EDx
EDx=0
If AB is continuous there should be at least one point of intersection
with the vertical line representing the zero excess demand . So, there
exist an equilibrium price like P* .(fixed point theorem) .it is worth
noting that positive excess demand should increase the price and
excess supply (negative excess demand ) should decrease the price.
The competitive market do respond like these (same as auctioneer)
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Walters & Layard CH 2
General Equilibrium
CONSUMPTION WITHOUT PRODUCTION
PURE EXCHANGE
At this analysis target variable is EDx and instrumental
variable is Px .
This is an important conclusion , since the first response to
any disequilibrium shock will be the change in price
rather than quantity . This is called Walrasian Price
adjustment .
Stability of equilibrium
The equilibrium position is stable by the nature of the
control system .
Uniqueness of the equilibrium
Curve passing through points A and B , will intersect the
vertical axis at three points : p1 , p2 ,p3 .
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Walters & Layard CH 2
General Equilibrium
CONSUMPTION WITHOUT PRODUCTION
PURE EXCHANGE
p4
p3
p
2
p1
p0
ESx
EDx
EDx=0
P2 is unstable so it will rule out . Which one of the p3
or p1 is the equilibrium point ? It depends where we
start , like all dynamic systems . In the following we
will see when there will be more than one
equilibrium point ?
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Walters & Layard CH 2
General Equilibrium
CONSUMPTION WITHOUT PRODUCTION
PURE EXCHANGE
In the following figure the locus of all equilibrium points
( offer curve ) for consumer A is drawn .
A’s offer curve
Relative price of x
(px/py) decrease
E
yAo
from p1 to p2 to p3
3
u
.consumption of x
increase from x1
u1 u2
c3
y3
to x2 to x3 .
y1
c2
C1
Consumption of y
y2
decrease from y1
to y2 but then
x1
x2
x3
increase to y3 .
xA0
p3
p2
Why?
12
p1
Walters & Layard CH 2
General Equilibrium
CONSUMPTION WITHOUT PRODUCTION
PURE EXCHANGE
As it is seen from the figure the consumption of y increase
with the decrease in price of x after point c2 . We will
show that demand elasticity of x will be elastic before
point c2 and inelastic after point c2 .
After point c2 if Dx is inelastic ;
px ↓(=%1) → x↑(<% 1) → px x (expenditure on x )↓ → if
total income does not change → ypy ↑ , when py is
constant then → y should increase .
So when demand for x becomes inelastic after some point ,
we will have a U shaped offer curve for individual A and
a backward-bending supple supply curve for y .
As it is shown in the following figure , when offer curves are
U shaped , we will have more than one equilibrium point
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Walters & Layard CH 2
General Equilibrium
CONSUMPTION WITHOUT PRODUCTION
PURE EXCHANGE
oB
y
A’s offer curve
E
c3
P3
uA
c2
B’s offer curve
uB
c1
p2
p1
x
oA
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Walters & Layard CH 2
General Equilibrium
CONSUMPTION WITHOUT PRODUCTION
PURE EXCHANGE
How likely a multiple equilibrium may occur ?
For unique and stable equilibrium a rise in px ( or px / py)
should bring excess supply of x through reduction in
demand for x . When px / py increase ,the change in the
demand for x will be as follows ;
1- for both A and B , there will be substitution effect away
from x → Dx will fall .
2- individual A is worse off , so the income effect leads him
to reduce demand for x ( assuming x is a normal good )
3- individual B is better off , so his income effect leads him
to increase demand for x ( assuming x is a normal good )
We will not be able to predict whether demand for x
increase or decrease? It depends on the relative
strength of the above effects . If (2) and (3) offset each
other , demand for x will decrease unambiguously. For
this to happen A and B should have similar marginal
propensity to spend Walters
on x &out
of income .
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Layard CH 2
General Equilibrium
CONSUMPTION WITHOUT PRODUCTION PURE EXCHANGE
For multiple equilibrium to occur there should be different
income effects for individuals A and B and the income
effects should be substantial .
It s important to know whether multiple equilibrium occur in
the real word . If they do , we might be able to improve
social welfare by shifting the economy from one
equilibrium to another one . We will have positive
evidence of multiple equilibrium , if we observe sudden
jumps in the economy over time .
Coalition and monopoly
The above argument does not mean , however that the
equilibrium situation which actually comes about will
necessarily be a competitive one . It merely says that if ,
if individuals act as a price taker a competitive
equilibrium will result .
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Walters & Layard CH 2
General Equilibrium
CONSUMPTION WITHOUT PRODUCTION
PURE EXCHANGE
But it is generally not in the interest of any one group of individuals to
act as a price taker , as steelworkers union leader will always tell
you .
If all members of group B ( union workers for example )organize
themselves collectively
How will they settle the equilibrium price ? They will set a price where
A’s offer curve was tangential to one of the B’s indifference curve
oB.
→ point Q .
A’s offer
A’s offer
E
uA
Q
uB
oA
17
Maximize B’s I.C.
S.T. A’s Offer curve
R
B’s offer
px/py = monopoly price
Walters & Layard CH 2
General Equilibrium
CONSUMPTION WITHOUT PRODUCTION PURE EXCHANGE
In other words he will maximize uB subject to A’s offer curve .
As it is seen , point Q is not on the contract curve , so MRSA
> MRS B . As it is seen point Q is not efficient and it may be
ethically superior to efficient point R depending upon the
needs of people ( helping workers for example) .
Clearly if groups of people can gain by forming the coalition
,we should expect to find such collusive behavior on a wide
scale . If it does not , this must be because the cost of
cooperation between members exceed the benefit they
obtain .
For the moment we simply note that perfect competition will
only be found where the transaction costs of collusion exceed
the gains from collusion , or where the law is very strong .
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Walters & Layard CH 2
General Equilibrium
Production without consumption
Production without consumption ; one sector model .
Assumptions ;
Many identical firms each owning one unit of labor , L .
Many identical capital owners each owning one unit of of
output capital , K. Output is produced by may firms with
identical production function and constant return to scale to
its equilibrium level. This is needed for considering the
aggregate production function as constant return one.
One good Y with it’s price equal to Py .
The amount of Y that a laborer can buy is WL / PY ..
the amount of Y that a capital owner can buy is WK / PY .
WL and WK are money wage of labor and capital .
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Walters & Layard CH 2
General Equilibrium
.
Production without consumption
With these assumptions we can disregard all features of
the production function, except those which describe it in
the neighborhood of equilibrium , and so we shall treat
each firm as having constant return to scale .
Dealing with constant return to scale will lead us to the
notion of homogeneous production function
Homogeneous production function
One of the most important features of the homogenous
production function is that RTSLK depends only on the
input use ratio , K/L , rather than the absolute level of
inputs . If this happens , the output expansion path will
be a straight line as while as the price ratio does not
change as it is shown in the following figure.
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Walters & Layard CH 2
General Equilibrium
Production without consumption
Y2
K
Y1
K/L
Y(mK, mL ) = mα Y(K ,L )
(K/L)’
m=1/L
Y(K ,L )=Lα Y ( k/L , 1)
(PL/ PK )’
PL/ PK
∂Y/∂K = L(α-1) Y’( K/L ,1 )
L
∂Y/∂L = α L(α-1) Y ( K/L ,1 ) – K/(L2)Lα Y’ (K/L , 1 )
∂Y/∂L = α L(α-1) Y ( K/L ,1 ) – (K/L)Lα-1 Y’ (K/L ,1)
RTSLK = ( ∂Y / ∂L ) / ( ∂Y / ∂K ) =
[ αY( K/L,1 ) - ( K/L)Y’( K/L , 1 )] / Y’( K/L ,1 )=h( K/L ) .
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Walters & Layard CH 2
General Equilibrium
Production without consumption
Taking into account the above argument , for any function
like Y=Y(K, L) which is homogeneous of degree one ,
not even the MRSLK is a function of input use ration but
also the marginal product and average product of labor
and capital is also a function of input use ration.
∂Y/∂K = Lα-1Y’( K/L ,1 ) , if α=1,
YK = Y’( K/L )
∂Y/∂L = α Lα-1 Y ( K/L ,1 ) – (K/L)L Y’ (K/L ,1) , if α =1 ,
YL = ∂Y/∂L = Y ( K/L ,1 ) – (K/L)Y’ (K/L)
APL= Y(K ,L )/L = [ Lα Y ( k/L ,1)]/L, if α=1, APL= Y(k/L,1)
APK = Y(K ,L )/K = [ Lα Y ( k/L ,1)]/K , if α=1,APK=
[1/(K/L)]Y(k/L ,1)
In other words when there is constant return to scale ,
scale does not matter in terms of finding the average and
marginal products . With the expansion of output average
and marginal product does not change ,
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Walters & Layard CH 2
General Equilibrium
Production without consumption
Taking into account the Euler’s theorem for homogeneous
functions ;
Y=Y(L,K)
(YK)K + (YL)L = αY(L,K), α=degree of homogeneity
If α=1 ,constant return to scale, then
(YK)K + (YL)L = Y(L,K)
P y (YK) K +PY (YL) L =PY Y(L,K)
(WK ) K + (WL) L = TR →
TC = TR
If α > 1 , increasing return to scale ;
(YK)K + (YL)L > Y(L,K)
P y (YK) K +PY (YL) L >PY Y(L,K)
(WK ) K + (WL) L > TR →
TC > TR
In the increasing return case; if factors of production
receive their value of marginal product, loss will occur.
But in practical term , natural monopoly will emerge and in
that case factors of production will receive MRP=(MR)YL
which is less than VMP=PYL .
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Walters & Layard CH 2
General Equilibrium
Production without consumption
If α < 1 , decreasing return to scale ;
(YK)K + (YL)L = αY(L,K),
(YK)K + (YL)L < Y(L,K)
P y (YK) K +PY (YL) L <PY Y(L,K)
(WK ) K + (WL) L < TR →
TC < TR
if factors of production be paid by their VMP , an entrepreneur
excess profit will result .
Income distribution
Suppose that ; Yi = Y (Ki , Li ) , i=firm , constant return ,
WL and WK are fixed , for every firm .→ RTSLK = f(Ki /Li ) =
WL/WK = fixed → K/L is fixed for every firm
Total output of Y=Y(∑i Ki , ∑i Li ) = Y(K ,L) . If k/L is fixed for each
firm then k/L will be fixed for the aggregate production function
since each firm is alike and constant return to scale is present
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Walters & Layard CH 2
General Equilibrium
Production without consumption
Y per k
Real wage of capital =MPk =WK/py =∂Y(K,L)/∂K =YK=g(K/L )
Real wage of labor =YL=h(K/L)
If L is total supply of labor
a
consist of one unit of labor ,
then area of obcd equals to
capital income . Since
Labor income
c
capital income equals to
b
capital amount (od) times
Capital
Yk=g(K/L) price of capital (cd) .
income
o
25
d= (K/L)1
k/L
Since area aodc is total or
national income , then area
abc equals to labor income
Walters & Layard CH 2
General Equilibrium
Production without consumption
We can similarly portray the same information with L/K as the
dependent variable and provide simple answers for important
Why real wages (YL) , and standard
questions ;
of living (Y/L) is lower in India than
Y per L
Europe . The answer could simply be
seen as having higher L/K ( shortage
Capital
of capital) .
income
Labor
income
bb
d
aa
c
Resident workers
26
How would immigration of workers
affect the welfare of capital owner
YL
and domestic labor in a country ?As it
is seen an amount b will be
L/K
transferred from domestic workers to
capital owners and capital owners
immigrants
would gain a+b and the share of
domestic workers reduce to d .
Immigrant workers receive an amount
equal to c .
General Equilibrium
Production without consumption
Suppose that the labor supply increase as was mentioned in the
previous example . What will happen to the real factor income ?
Marginal product of labor ( YL) will fall and marginal product of capital
will rise .
And how does relative share change? In order to find the answer we
have to see what will happen to the relative factor share ,
[ (WLL) / (WKK) ]?
(WLL) / (WKK) = [YL Py L] / [YK Py K] =(YL/YK)(L/K)
When (L/K) increase or (K/L) decrease → ( YL/YK ) decrease . The
intensively of the substitution depends on the magnitude of the
elasticity of substitution )σ(.
σ =(%∆ [K/L] ) / (%∆ [YL/YK] )=(%∆ [K/L] ) / (%∆ [RTSLK] )
If σ>1 , RTSLK ↓=%1→(K/L)↓>%1→(L/K)↑>%1
If (L/K) ↑=%1→ YL/YK = RTSLK↓<%1→ (YL/YK)(L/K) ↑ → (YLL)/(YKK) ↑
What will happen to the total income of labor ?
When L/K increase then YK will increase too , and YKK or capital
income will increase . If σ>1 then (YLL)/(YKK) increase and (YLL)
labor income rise . If σ<1 , and [(YLL)/(YKK)] decrease, and labor
income (YLL) may still rise .
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Walters & Layard CH 2
General Equilibrium
Production without consumption
Two sector model ;
Suppose that there are two productive sector ; X , Y .
X is more labor intensive than Y , (K/L)x < (K/L)y .
What will determine the welfare of the factor owners ?
(Wk/px) determines the maximum amount of x an owner of
one unit of capital can buy if he spends his whole income
on x .
(WK/Py) determines the maximum amount of y an owner of
one unit of capital can buy if he spends his whole income
on y .
(Wk/px) and (WK/Py) determines the position of budget line
and thus the maximum utility he could get .
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Walters & Layard CH 2
General Equilibrium
Production without consumption
Y U(x,y)
Maximum utility of capital and
labor will be a function of factor
wages and commodity prices.
WK/PY
U K= uK [(WK/PX) , (WK/ PY)]
UL = uL [(WL /PX ) , (WL / PY)]
WK/PX
X
How are factor and commodity prices determined in a competitive
equilibrium ? We need to find out about the preference of the
individuals and the demand function for both commodities .this can
be done in two steps ;
29
First , we can establish a one to one relationship between the
relative price of products( demand for x and y ) and welfare of factor
owners. Second , we should confirm that in a closed economy the
welfare effect of changes in factor supply is the same as in one
sector model
Walters & Layard CH 2
General Equilibrium
Production without consumption
The relation between product price and factor price can be illustrated in he
following theorem
Stopler-Samuelson theorem
“ in any particular country a rise in the relative price of laborintensive good will make labor owner better off and capital
owner worse off , and vise versa, provided that some of each
good is being produced .”
In order to show the above relation we need to establish a one
to one relationship between factor prices ( the welfare of factor
owners ) and product prices .
If Px/Py increase , then production of x will increase
(p=MC) , and since x is a labor intensive commodity , demand
for labor will increase and cause the wage rate to increase
. Increase in wage rate will consequently decrease the amount
of labor demanded both for x and y .
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Walters & Layard CH 2
General Equilibrium
Production without consumption
When Lx and Ly decrease , (K/L)x and (K/L)Y will increase
and cause the labor productivity to increase and capital
productivity to decrease in the production of both X and
Y . So ;
(WK/Px =XK) and (WK/PY=YK ) decrease and
(WL /Px =XL ) and (WL /PY=YL ) increase and
UK= uK [(WK/ Px ) , (WK / PY)] = welfare of capital owners
decrease
UL =uL [(WL /PX ) , (WL / Py ) ] welfare of labor owner
increase.
The same story can be shown in the following figure. This
diagram is called “Lerner-Pierce” diagram.
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Walters & Layard CH 2
General Equilibrium
Kx
Production
without
consumption
,K
P2
y
WL/WK
(WL/WK)’
P1
S’
R’
P
Ky
S
Kx
O
Y=1 unit
R
Ly Q1 Q2
X=1 unit
Q
Lx
Lx , Ly
Suppose that there is constant return to scale in production of X and Y .
(As it was assumed earlier)
cost of producing one unit of x
= Px = WK Kx + WL Lx
cost of producing one unit of y
= Py = WK Ky + WL Ly
At the initial factor price of WL/WK → cost of producing one unit of x or y
= Px=Py=OP in terms of capital and OQ in terms of labor units .
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Walters & Layard CH 2
General Equilibrium
Production without consumption
Now suppose that factor price ratio increase to (WL/Wk)’ ;
Cost of producing one unit of x =OP2 in terms K and OQ2 in
terms of L.
Cost of producing one unit of Y =OP1 in terms K and OQ1 in
terms of L.
According to the diagram increase in the price of x (PP2) is
greater than increase in price of y (PP1), since X is labor
intensive [ (K/L)x < (K/L)y comparing S to R and S’ to R’] .
Comparing S to S’ and R to R’ , we will see that Lx and Ly has
both decreased and (K/L)x and (K/L)Y both has increased. As
discussed earlier this will cause increase in the productivity of
labor in x and Y and welfare of labor owners (UL) to increase
and welfare of capital owners (UK) to decrease.
This idea can usually be demonstrated by the following
diagram . ;
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Walters & Layard CH 2
General Equilibrium
Production without consumption
y
X
(WL/WK)0
(K/L)y
(K/L)x
O
X is labor intensive so
(K/L)x<(K/L)y
(Px/Py)0
As it is shown when Px/Py increase →( WL/WK) increase, and (K/L)x and
(K/L)y will increase too . When Px/Py is known , (K/L)x and (K/L)y could be
solved in a competitive situation .
XK=WK/Px , YK=Wk/Py , XL=WL/Px , YL=WL/Py
(YL / XK ) = ( WL / WK ) ( PX / PY )
( XL / YK ) = ( WL / WK ) ( PY / PX )
34
( WL / WK ) , and ( PX / PY ) are known and
YL , XK , XL , YK are functions of (K/L)x and
(K/L)y . So two equations and two unknowns
[(K/L)x and (K/L)y ] could be
solved
Production without consumption
Now suppose that at a low price of labor Y is indeed capital
intensive good , but at a higher values of WL/WK , X
becomes capital intensive . In these cases a given
product price is consistent with two sets of relative factor
prices , input use ratios , and welfare levels of labor and
capital owners .
X
Y
(WL/WK)1
(WL/WK)0
(K/L)x0
(K/L)x1
(K/L)y1
35
(K/L)y0
Walters & Layard CH 2
(Px/Py)0
General Equilibrium
Production without consumption
The Lerner- Pearse diagram could show us why this happen
Kx
Ky
(k/L)x1
(K/L)y1
(K/L)y0
(K/L)x0
Y=1
X=1
(WL/WK)1
36
(WL/WK)0
Lx, Ly
As it is shown when WL/WK
increases Px will increase but
still Px/Py =1 . As it is shown ,
factor intensity reversal
occurred . That is ; when factor
price ratio changes , input use
ratio responds much more in x
industry than y . It means that
elasticity of substitution is
higher in industry x than
industry y .
This problem matters if we wish
to compare countries engaged in
international trade . Within a
country this may not be a
problem .
Cost of producing one unit of x or y is the same for both of
these factor price ratios px /py=1
Walters & Layard CH 2
General Equilibrium
Production without consumption
The relation of output-mix to real factor prices;
How the pattern of output is changing as prices changes,
and why transformation curve is concave ?
OY
K0
(WL/WK) <(WL/WK)’
xo
Y0
Po
Ox
(K/L)Yo
37
(K/L)x1
P1
(K/L)x0
WL/ WK
<
Y1
x1
(K/L)Y1> (K/L)yo
Walters & Layard CH 2
L0
General Equilibrium
Production without consumption
Consider a move from P0 to P1 on the contract curve . How does
relative factor prices will change ?
When factor are fully employed ;
(K/L)o = (K/L)x (Lx/Lo) + (K/L)Y (LY/Lo),
L0=Lx + Ly
Qx ( at P1 ) > Qx ( at P0 )
Qy ( at P1) < Qy ( at P0)
(K/L)x < (K/L)y , at P1 and P0 ,since x is labor intensive.
comparing Lx and Ly at points p0 and p1 ;
(Ly / L0) has reduced in P1 , but (Lx/L0 ) has increased in P1 .
Higher weight (Lx/L0 ) is being attached to lower K/L {=(K/L)x}
The right hand side of the following relation will reduce. To
maintain the full employment of factors of production either
(K/L)x or (K/L)y or both should increase to maintain the equality .
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Walters & Layard CH 2
General Equilibrium
Production without consumption
MRSxLK = f(K/L)x =WL/WK
MRSY LK = f(K/L)Y =WL/WK
factor prices are the same for the production of both goods.
So both (K/L)x and (K/L)Y should rise with together,
because if one of them increase, the other one will increase
too . This will lead to increase in WL/WK .because MRS
has increased. this will cause an increase in Px/Py , since x
is labor intensive. And also an increase in MCx/MCy in the
context of perfect competition which is consistent with
increase in the production of X and decrease in the
production of Y . This means moving on the production
possibility frontier from
point A to point B .that is
Y
producing more from x
A
and less from y .
B
39
X
Walters & Layard CH 2
General Equilibrium
Production without consumption
When X is labor intensive PPF is concave to the origin and
contract curve is below the diagonal .
Contract curves (or production possibility frontier) can not
intersect each other . If there is one common point of
intersection , all points should be common.( why? )
If the two industries have the same K/L , the contract curve
and PPF or transformation curve will be a straight line. But
if K/L differs between the two industries , contract curve will
be convex or concave towards diagonal (why?).
When (Px/Py) permanently increase along with the
production of x , then (WL/WK) will increase which will
cause (Px/Py) to increase again . Production of x will
increase again . This process may continue until all factors
of production engaged in the production of x . When this
happen . (K/L)x will be fixed , and (WL/WK) will be fixed too.
But Px/Py has increased , which is the violation of the
Stopler-Samuelson theorem. So in order for the theorem to
work , some of each good should be produced .
40
Walters & Layard CH 2
General Equilibrium
Production without consumption
Effect a of changes in factor supply on income
distribution in a closed economy
suppose that in a closed economy labor supply increase as
a result of migration . But the increase is such that factor
intensity reversal does not occur. We would like to see
what will be the effect of this migration on the income
distribution between factors of production .
In order to analyze the effect of labor migration on income
distribution , in the beginning we will keep the price
levels constant (Since (WL/WK) is constant and see what
will happen to the demand and supply of factors .
Since at the beginning (WL/WK) is fixed , (K/L)x and (K/L)y
are constant and does not change , since MRS =
MRS(K/L) = (WL/WK) .
41
Walters & Layard CH 2
General Equilibrium
Production without consumption
oy
o’y
K0
(K/L)x
P’
P
(K/L)y
ox
(K/L)y
L0
L’
Since (K/L) remains constant production occur along The oxp line .
Production point shift from point P to P’ . As a result production of x
will increase and y will decrease to maintain full employment .
When supply of labor increase with wages remain unchanged , total
labor income will increase and cause their demand for x and y to
increase too.
42
Walters & Layard CH 2
General Equilibrium
Production without consumption
But production of x increase and for y decrease and cause
the relative price of x (px/py) decrease in order to
maintain equilibrium . As a result WL/WK will fall based
on Stopler-Samuelson theorem .
(WL/WK)↓→Lx↑, Ly↑→(K/L)x ↓, (K/L)y ↓→xL ↓,yL↓ →xK↑, yK↑
→→(WL/Px)↓ , (WL/Py)↓ → (WK /Px)↑ , (WK /Py)↑ →→
UL(wL/px , wL/pY) ↓ labor worse off ,
UK(wK/px , wK/pY) ↑ capital better off .
This the same result when we were considering one
sector analysis.
For further and exact analysis we need to know the
elasticity of substitution between industry x and y (when x
increase and y decrease ) to find out the degree of
relative price decrease .
43
Walters & Layard CH 2
General Equilibrium
Production and consumption
In the final step we have to take into account production
and consumption altogether and see if there is an
equilibrium set of prices and if they are unique ?
Existence of equilibrium could be brought about by using
fixed point theorem .
We could imagine a very low px which Qx = 0 , and
QY = max , and a very high price of x in which Qy=0 , and
Qx = max . In the first case excess demand for x is very
high and in the second case excess demand for x is equal
to zero . So there should be an equilibrium level for px/py
in which there is no excess demand for x .
For uniqueness of the equilibrium we have to see whether
excess demand for x (for both workers and capital owners)
decrease with increase in the relative price of x (px/py).
44
Walters & Layard CH 2
General Equilibrium
Production and consumption
Y
Y
bc = excess demand for y
A
BL
ab = excess supply of x
oL
Y*
A1
equilibrium
UK
Y1
E
a
oK
45
px/py
X*
Budget
constraint
BK
b
CK
A
UL
oL
UK
x
UL
oK
CL
c
x1
A1
x
(px/py)1
Walters & Layard CH 2
General Equilibrium
Production and consumption
As it is shown in the figure , Px has increased and production of x
increase and production of y decrease
As it was discussed earlier , welfare of labor owners increased and for
capital owner decreased as a result of change of the budget line .
Equilibrium point convert to a non-equilibrium one . We will expect
three effect ;
1- there will be substitution effect away from x and towards
consumption of y . Demand for x decrease and for y increase as a
result of increase in the price of x .
2- capital owners become worse off , so there will be income effect
away from consumption of x ( reduction in the demand for x) .
3- labor owners become better off . So there will be income effect
for the increase in the demand for x .
If 2 and 3 offset each other , the final effect will be the decrease in
the demand for x . So with increase in the price of x , excess supply
of x will emerge . So the equilibrium will be unique .
46
Walters & Layard CH 2
General Equilibrium
PROBLEMS
Q2 – 1 . Suppose that consumers of type A are endowed with total
supply of X ( X0 ) and consumers of type B are endowed with total
supply of Y ( Y0 ) . UA = XA YA and UB = XB YB . In a competitive
market what is an equilibrium relative price of X ? Is this equilibrium
unique and stable ?
Solution ; in the equilibrium total excess demand shoud be equal to
zero .
MAX UB = XB YB .
( Px / Py ) = P
S.T. ( Px / Py ) XB + YB = Y0 ,
YB = 1/2 Y0 , XB = 1/2 ( Y0 / P )
EDX B = XB = 1/2 ( Y0 / P ) . , YB = 1/2 Y0 ,
MAX UA = XA YA
S.T.
P ( XA - X0 ) + YA = 0 , XA = 1/2 X0 , YA = 1/2 X0 P
EDXA = XA - X0 = - 1/2 X0 .
EDX B + EDXA = 0 , PX / PY = Y0 / X0 = equilibrium price
if equilibrium is unique and stable , the excess demand for X
decreases with increase in P . EDXA is fixed and EDX B decrease
with increase in relative price of X .
47
Walters & Layard CH 2
General Equilibrium
PROBLEMS
Q2-2 Suppose that in the above problem consumers of type A could
agree among themselves on a price at which they would sell x ( but
consumers of type B could not collude ) . What price would they set
?.
XA
=1/2
X0
P’
B’s offer curve
X0
YB = 1/2 Y0
Initial endowment of A
B
A’s offer
curve
(Px / PY)=P=
P.C. Price
A
Initial endowment of B
Y0
Solution - Infinitive . B’s offer curve is vertical at ½ Y0 and A needs to offer
barely an x in return for y in order to induce B to supply 1/2 Y0 .
Therefore any positive price ( like p’ ) is sufficient to induceGeneral
this supply
.
Equilibrium
( the same if type B collude ).
PROBLEMS
Q2-3 – if Y = 100 K1/2 L1/2 , where Y is output per head , K is capital
stock and L is man-year . What is the real wage and output per
worker in the following countries .
Country
K
L
1
9000000
100
Us
2
200000
20
UK
3
5000
200
INDIA
Solution :
49
WL /PY =MPL=
50 (K/L)1/2
Y/L = 100 (K/L)1/2
Country
K/L
1
90000
15000
30000
2
10000
5000
10000
3
25
250
500
Walters & Layard CH 2
General Equilibrium
Q2-4 , If Y = K1/4 L3/4 and the labor force is constant at L0 , how does
increase in capital accumulation ( from K0 to K1 ) affect
i- the real wage and real capital rental
ii- The relative shares of national income
iii- the absolute share of capital
Solution ;
i- we know that increase in ( K/L) increase the real wage ( YL ) of labor
and decrease the real capital rental ( YK ) .
ii- Relative real share of factor income are equal to ( YL L) / (YK K) .
( YL L) = (3/4 y L-1 ) ( L ) ,
(YK K) = ( 1/4 Y K-1 ) K
( YL L) / (YK K) = 3 , this holds independently of K/L .
( YL L) + (YK K) = Y0 = national income
Relative share of labor = ( YL L) / Y0
[Y0 / ( YL L) ] = 1 + (YK K)/ ( YL L) = 1 + 1/3 = 4/3
( YL L)/ Y0 = 3/4 , Relative share of capital =(YK K)/ Y0 = 1/4
iii- absolute share of capital = (YK K) = 1/4 Y0
absolute share of labor = ( YL L) = 3/4 Y0
50
PROBLEMS
Q2-5 How would you rank the welfare of the workers in the following
table
WL
51
PY
State 1
1
1
1
State 2
2
3
3
State 3
2
1
4
Assume ;
i- worker’s utility is not known
ii- worker’s utility is U= XY.
Solution ;
State
PX
WL /PX
WL /PY
1
1
1
2
2/3
2/3
3
2
1/2
i- state 1 is preferred to 2 , but we
can not say anything about the
other states.
Walters & Layard CH 2
General Equilibrium
PROBLEMS
Max U = XY
S.T. Px X + Py Y = WL
Demand functions for Labor for the production of X and Y :
X = WL /2Px , Y = WL / 2Py
U = XY = 1/4 (WL /2Px ) (WL / 2Py )
U1 = (1/4)(1)(1)= 1/4 ,
U2 = 1/9
, U3 = 1/4
Q2-6 , are workers rational to lobby for tariffs on labor-intensive imports.
Yes , the tariff on labor intensive commodity will increase the price of
labor intensive good and raise the real wage of labor .
Stopler-Samuelson theorem .
Q2-7 - Suppose that X = Kx 2/3 Lx1/3 , Y = Ky1/3 Ly2/3 , and the economy is
endowed with K0 and L0 measured in units such that K0 = L0 =1.
i- What are the values of x and y on the transformation curve
corresponding to first (a) and then (b) ;
(a) Kx = Ky
52
(b) Lx = Ly
PROBLEMS
Evaluate the following at points (a) and (b) .
WK / Px ,
WK /Py ,
WL /Px ,
WL /Py ,
WL / WK ,
Px / Py ,
At which point labor is better off .
Solution ; being on the transformation curve we have RTSLK x =RTSLKy
XL / XK = YL / YK ,
1/3 ( X / Lx ) / 2/3 ( X / Kx ) = 2/3 ( Y / Ly ) / 1/3 ( Y / Ky )
1/2 ( Kx / Lx ) = 2 (Ky / Ly ) → 1/2 ( Kx / Lx ) = 2 (K0 - Kx )/ (L0 – Lx )
→ 1/2 ( K0 – Ky )/ ( L0 – Ly )= 2 (Ky / Ly )
a- Kx = 0.5
Ky = 0.5
Lx = 0.2
Ly = 0.8
x = (0.05)1/3 Y = (0.32)1/3
b- Lx = 0.5
L Y = 0.5
Kx = 0.8
Ky = 0.2
X = (0.32)1/3 Y = (0.05)1/3
53
Walters & Layard CH 2
General Equilibrium
PROBLEMS
WK / P x
,
WK /Py
WL /Px
WL /Py
WL / WK
XL / XK
YK /XK
XK
YK
XL
YL
2/3 (Kx / Lx)-
1/3 (K y/ Ly )
1/3 (Kx / Lx)
2/3 (K y/ Ly )
1/3
2/3
2/3
1/3
Px / Py
a
2/3 (0.4)1/3
1/3 (1.6)2/3
1/3 (2.5)2/3
2/3 (0.6)1/3
1/0.8
1/2 (6.4)1/3
b
2/3 (1/1.6)1/3
1/3 (1/0.4)2/3
1/3 (1.6)2/3
2/3 (0.4)1/3
0.8
1/2 (10)1/3
Y is labor intensive good ((Ky / Ly ) < ( Kx / Lx ) )
Labor is better off in a in which the production of Y is higher
54
Walters & Layard CH 2
General Equilibrium
Q2-8 Same as question 2-7 , but with Y = 2 Ky2/3 Ly1/3 and every thing
else as before .
Solution ;
X = Kx 2/3 Lx1/3 , Y = 2 Ky2/3 Ly1/3 , both X and Y are equally capital
intensive so ,
XL / XK = YL / YK ,
1/3 ( X / Lx ) / 2/3 ( X / Kx ) = 1/3 ( Y / Ly ) / 2/3 ( Y / Ky )
1/2 ( Kx / Lx ) = 1/2 (Ky / Ly ) → Kx / Lx = Ky / Ly
a- Kx = 0.5 , Ky = 0.5 ,
Lx = 0.5 , Ly = 0.5 , X= 1/2 , Y= 1
b- Kx = 0.5 , Ky = 0.5
Lx = 0.5 , Ly = 0.5
X= ½ , Y = 1
The contract curve and transformation curve are straight lines .
Contact curve is straight line since ( K/L) is the equal to 1 for both
X and Y . National output will be measured by the line X + 1/2 Y .
The output mix will not affect the relative prices of goods and
factors. Since contract curve is straight line and relative prices
remain constant when we move on the contract curve.
55
Walters & Layard CH 2
General Equilibrium
Q2-9 Suppose that with the production function as in Question 2-8 we
evaluate x and Y such that Kx = 1/2 Ky , Lx = 1/2 Ly . How
does welfare of workers and capital owners compare with that found
in Question 2-8 .
Solution both in ( a ) and ( b )
Kx + Ky = 1
Kx = 1/3 Ky = 2/3
Lx + L y = 1
Lx =1/3
Ly = 2/3
X = 1/3
Y = 4/3
The contract curve and transformation curve are both straight lines . So
change in the output mix does not affect the relative prices and
welfare of workers.
Q2-10
Suppose
X= Kx + Lx
Y = 2 Ky + Ly
K0 = L0 =1
What are the following parameters in general equilibrium ;
WK / Px ,
WK /Py ,
WL /Px ,
WL /Py ,
WL / WK ,
Px / Py
56
Walters & Layard CH 2
x
PROBLEMS
i- if
ii- if
Uk = XK3/4 yk , UL = XL3/4 yL
Uk = XK yk3/4 , UL = XL yL3/4
X=2
K0
Y
Y
=1
X=1.5
P correspond to 0
P1P correspond to xo .All K in Y . L transferred
‘ X to Y
from
PP2 correspond to yo . All L in X. K transferred
from Y to X
-dy/dx = 1
3
Y=0.5
X=1
Y=1
2
Y=2
1
P
All L in X
All K in Y
X=1/2
-dy/dx = 2
Y= 2.5
X
L0 =1
p2 X
o
1
2
Y=3
Contract
curve
57
Walters & Layard CH 2
General Equilibrium
If the slope equilibrium price line lies between the slope of 1 and 2 ,
then point P will be the equilibrium point , otherwise we should find
the equilibrium point by maximizing the community indifference
curve subject to one of the linear segments of the transformation
curve ( MRS = MRT ) . 4
i - At P , we will have MRSxyL = MRSxyK = 3/4 ( Y/X)L = 3/4 ( Y/X)K =
(3/4)(2/1) = 3/2
Since at point P , MRS = Px / Py , Px / Py = 3/2
1 ≤ ( Px / Py ) = 3/2 ≤ 2 . Point P will be the equilibrium point . Then
, X=1 , Y =2
Wk/Py = Mpky =2 , since all K is employed in Y production so real
wage is equal to marginal productivity.
Y = 2 KY
WL/Py =( WL / Px ) / (Px /Py) = 1 (3/2) = 3/2
WL/Px = MpL x= 1 , since all L is employed in X production so real
wage is equal to marginal productivity. X= Lx
Wk/Px =(Wk / Py ) ( Px /Py ) =2(2/3)=4/3
58
Walters & Layard CH 2
General Equilibrium
ii- Uk = XK yk3/4 , UL = XL yL3/4 , at P we will have MRS =
(4/3)(Y/X)=(4/3)(2/1)= 8/3, therefore , P is not equilibrium point ,
since Px /Py is grater than 2 so equilibrium point should lie on the
section PP2 with an slope (Px /Py = 2 = MRS), where all L is
employed in the production of X . So X production should increase
Wk/Py = Mpky =2 , since
Y = 2 KY
WL/Py =( WL / Px ) / (Px /Py) = 1 (2) = 2
WL / PX = MPLX = 1 , since
X= Kx + Lx
Wk/Px =MPK X =1 since
X= Kx + Lx
Q2-11
To produce 1 uint of x requires 1 unit of L and 2 units of K . To
produce 1 unit of Y requires 1 unit of L and 1 unit of K .
Suppose U = XY . Will there will be full employment of labor , and
what is the structure of the prices
i- in a rich country with K0 = 1.8 , and L0 = 1 .
ii- in a less reach country with K0 = 1.4 , and L0 = 1 .
iii- in a poor country with K0 = 0.5 , and L0 = 1 .
59
PROBLEMS
Y
K0 = 2X + Y
Slope = -2
K0 ≥ 2X + Y
A
B’
L0 ≥ X + Y •
B
C
O
,
C
Production may occur on AB
Feasible region
region or BC region or at point B.
At point B there is full employment
, since both constraints are
L0 = X + Y slope = -1 binding. On AB region K is
unemployed ( only L is binding),
X
on BC region L in unemployed .
iK= 1.8 ,
L=1 ,
1.8 = 2X + Y , 1 = X + Y
MRSxy = Y/X = 1/4 < 1=slope of AB . As it is clear from the figure
production at point B brings production of X more than what is needed
. So production of X will decrease and the production point move to
point B’ . A point in which labor constraint is satisfied and some capital
is unemployed.
60
Walters & Layard CH 2
General Equilibrium
at point B’ the relative price is equal to Px / Py = 1 ,
We should apply exhaustion theorem ;
X = MPLx Lx + MPKx Kx , MPLx =1 , MPKx =2 , X = Lx + 2Kx , Px = WL + 2WK
Y = MPLY LY + MPKY KY , MPLY =1 , MPKY =1 , Y = Ly + Ky , PY = WL + WK
Px /Py = price of one X in terms of Y
Wk / Py = marginal product of K in production of X in terms of y ( price of one K in
terms of y in the production of X)
WL / Py = marginal product of L in production of X in terms of y ( price of one L in
terms of y in the production of X)
X = Lx + 2Kx , Px /Py =2 Wk / Py + WL / Py
, Px /Py = 1
Y = Ly + Ky ,
Py /Px = Wk / Px + WL / Px
,
WK / Py = WK / Px =0 , capital is not binded .
WL / Px = WL / Py = 1 .
ii- at
B , when K= 1.4 , L=1 , with the same procedure we will find that
X = 0.4 , Y = 0.6 , MRS = Y/X = 3/2 = Px / Py , 1 < 3/2 < 2 , so point B is
equilibrium point
WK / Py = 1/2 =marginal productivity of capital in production of X in terms of Y
WL / Py = 1/2 =marginal productivity of labor in production of X in terms of Y
WL / Px = 1/3= marginal productivity of labor in production of Yin terms of X
WK / Px = 1/3 = marginal productivity of capital in production of Yin terms of X 61
Walters & Layard CH 2
General Equilibrium
PROBLEMS
iii- at B , X is negative ,
,
0.5 = 2X + Y , 1 = X + Y , X = -0.5 , Y = 1.5
X = Lx + 2Kx , Y = Ly + Ky
Y capital constraint highly dominates the labor constraint . There are too
1.5
much labor and small amount of capital. Labor is not bonded . Only capital is
bounded . WL / Py =0 , WL / Px =0 three cases may happen :
1- all the capital goes for x production , 2K=0.5 , K=0.25, L=0.25, X =0.25
1
2-all the capital goes for y production k=0.5 , L=0.5 , Y = 0.5
3- some of the capital
→ MRS =Y/X=2=Px /Py= WK/Py
L=1=X+Y
goes for the production of
Py /Px = Wk / Px =1/2
0.5
x and some of the capital
goes for the production of
Y
K=0.5=2X+Y
1
-0.5
0.25
X
Q2-12 – Suppose a minimum wage is imposed in one industry (X) , the wage in y
being uncontrolled . The minimum is expressed in terms of WL / Px and is above
the equilibrium level. Will such a wage necessarily make workers who can not get
jobs in the X industry worse off ? ( the X industry may be capital intensive or labor
intensive ).
62
PROBLEMS
Rx
P”
Oy
P
P’
Increase
in (K/L)x
Ox
63
Walters & Layard CH 2
K0
L0
General Equilibrium
Starting from point P , increase in real wage will increase
MPLx = WL / Px so , (K/L)x will increase and OP will change to ORx
Since real wage is fixed at the new level , so MPLx is fixed at the new level ,
and (K/L)x is fixed at the new level . So the new equilibrium point should lie
on the Ox Rx line . The new equilibrium point can not lie on the Ox P’ section
of the contract curve. Because , if it lies on the Ox P’ section , any point on
the contract curve between Ox and P’ will have a higher (K/L)x and higher
wage level than the minimum wage . So the equilibrium point could not be
located on this section of contract curve and wage level can not logically rise
above the minimum wage. The equilibrium point could not be located on
P’ P or P’OY section either , since the wage level will be less than the
minimum wage
So the equilibrium should lie on the P’Rx section .
A- if it lies on P”Rx section . (K/L)y would fall and YL would fall and labor in Y is
worse off . Since (K/L)x and XL would rise we don’t know the direction of the
change in UL ( WL / Px , WL / Py ) .
B – if it lies on P’P” section , (K/L)y would rise and YL = WL / Py would rise and
labor in Y is better off .
So UL ( WL / Px , WL / Py ) will rise , since both WL / Px , and WL / Py has risen
64