general equilibriumwalters & layard ch 21 general equilibrium general equilibrium

General EquilibriumWalters & Layard CH 2 1

GeneralGeneral Equilibrium Equilibrium GeneralGeneral Equilibrium Equilibrium


INTRODUCTION

In this chapter we will deal with positive economypositive economy theory to construct a framework for the following purposes ; First ; predicting the effect of particular cause ;First ; predicting the effect of particular cause ; Second ; detecting the cause of particular effectsSecond ; 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 1- we prove the existence and stability of general competitive equilibrium and consider whether it is competitive equilibrium and consider whether it is unique ?unique ?

2 –2 – how distribution of income is determined and how it how distribution of income is determined and how it would change with changes in factor supply ;would change with changes in factor supply ;


CONSUMPTION WITHOUT PRODUCTION CONSUMPTION WITHOUT PRODUCTION PURE EXCHANGEPURE EXCHANGE

A- BargainingA- Bargaining; ;

E = initial endowment point → MRSxyA > MRSxy

B → fruitful trade fruitful trade

is possible . As while as both are consuming in the ETis possible . As while as both are consuming in the ET00TT11 area area , both will be better off, 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.

E

T0

T1

oA

o

oBY

X

uA

uB


CONSUMPTION WITHOUT PRODUCTION PURE CONSUMPTION WITHOUT PRODUCTION PURE EXCHANGEEXCHANGE

B- Existence of equilibriumB- Existence of equilibriumWe will deal with competitive equilibriumcompetitive 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 The presence of auctioneer who will call on different prices will finally bring about the equilibrium point prices will finally bring about the equilibrium point like Tlike T. There are three question that are of interest to the auctioneer.1-First1-First , is there any price which could clear the market , is there any price which could clear the market

? Does equilibrium exist ?? Does equilibrium exist ?2-Second 2-Second , is there more than one such price ? Is , is there more than one such price ? Is

equilibrium unique ?equilibrium unique ?3-Third,3-Third, will the equilibrium be stable ? will the equilibrium be stable ?


CONSUMPTION WITHOUT PRODUCTION PURE EXCHANGE

Suppose auctioneer starts with price equal to the slope of ET0 , ( price = p0

).

cB

cA

E

oA

oB

Y

X

T0

a b

b’a’

c

d d’

c’

p0

C

P*



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=MRSxy

B,. 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.



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 + pyyo

A = value of A’s market supply = A,s income xo

A = initial endowment of x for individual A yo

A = initial endowment of y for individual A pxxA

+ pyyA = pxxoA + pyyo

A → px(xA – xoA) + py(yA – yo

A) = 0

pxxB +pyyB =pxxo

B + pyyoB → px(xB – xo

B) + py(yB – yoB) = 0

px EDxA

+ pyEDyA

=0 , px EDx B

+ pyEDy B

=0

px EDx + pyEDy =0 WALRAS LAW



WALRAS LAWWALRAS LAWThe 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 .



po

p1

EDxESx

P*

EDx=0

At p1 : excess supply

At p0 : excess demand

At p* : market is clear A

B

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 that positive excess demand should increase the price and excess supply (negative excess demand ) should decrease the price. excess supply (negative excess demand ) should decrease the price. The competitive market do respond like these (same as auctioneer)The competitive market do respond like these (same as auctioneer)



At this analysis target variable is EDtarget variable is EDxx and instrumental instrumental variable is Pvariable is Pxx. .

This is an important conclusionThis is an important conclusion , since the first response to since the first response to any disequilibrium shock will be the change in price any disequilibrium shock will be the change in price rather than quantity . This is called rather than quantity . This is called Walrasian Price Walrasian Price

adjustmentadjustment. .

Stability of equilibriumStability of equilibrium The equilibrium position is stable by the nature of the

control system.

Uniqueness of the equilibriumUniqueness of the equilibrium Curve passing through points A and B , will intersect the

vertical axis at three points : p1 , p2 ,p3 .



p4

pp33

pp22

pp11

p0

EDxESx

EDx=0

PP2 2 is unstable 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?



In the following figure the locus of all equilibrium points ( offer curve ) for consumer A is drawn.

E

A’sA’s offer curveoffer curve

C1c2

c3

yAo

xA0

u1 u2

u3

x1 x2 x3

p3p2p1

y1

y2

y3

Relative price of x (px/py) decrease from p1 to p2 to p3 .consumption of x increase from x1 to x2 to x3 . Consumption of y decrease from y1 to y2 but thenbut then

increase to increase to yy33 .

WhyWhy??



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 cAfter point c22 if D if Dxx is inelastic 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 increasey 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



y

x

E

A’s offer curve

B’s offer curveB’s offer curve

oA

oB

c1

c2

c3

p1

p2

P3

uB

uA



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 on x out of income .



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 will have positive evidence of multiple equilibrium , if we observe sudden jumps in the economy over time .we observe sudden jumps in the economy over time .

Coalition and monopoly Coalition and monopoly The above argument does not mean , however that the The above argument does not mean , however that the

equilibrium situation which actually comes about will equilibrium situation which actually comes about will necessarily be a competitive onenecessarily be a competitive one . It merely says that if , if individuals act as a price taker a competitive equilibrium will result .


CONSUMPTION WITHOUT PRODUCTION PURE EXCHANGEBut 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 . → point Q .

oA

oB

EA’s offer

B’s offer

px/py = monopoly price

A’s offer

uA

uB

QR Maximize B’s I.C.

S.T. A’s Offer curve


CONSUMPTION WITHOUT PRODUCTION PURE EXCHANGECONSUMPTION 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 Q is not efficient and it may be ethically

superior to efficient point R depending upon the needs of superior to efficient point R depending upon the needs of people ( helping workers for example) . 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 If it does not , this must be because the cost of cooperation between members exceed the benefit they obtain . between members exceed the benefit they obtain .

For the moment we simply note that perfect competition will only perfect competition will only be found where the transaction costs of collusion exceed the be found where the transaction costs of collusion exceed the gains from collusiongains from collusion , or where the law is very strong .


Production without consumptionProduction without consumption

Production without consumption ; one sector modelProduction 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 .

.



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 Dealing with constant return to scale will lead us to the notion of homogeneous production function notion of homogeneous production function

Homogeneous production functionHomogeneous 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.


Production without consumption

K

L

Y1

Y2

PL/ PK

(P(PLL/ P/ PKK )’ )’

K/L

(K/L)’Y(mK, mL ) = mα

Y(K ,L )

m=1/L

Y(K ,L )=Lα Y ( k/L , 1)

∂Y/∂K = L(α-1) Y’( K/L ,1 )

∂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 ) .


Production without consumptionProduction without consumptionTaking 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 ,


Production without consumptionTaking 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 = TRIf 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 > TRIn the increasing return caseincreasing return case; if factors of production receive their

value of marginal product, loss will occur.But in practical term , natural monopoly will emergenatural monopoly will emerge and in that

case factors of production will receive MRP=(MR)YL which is less than VMP=PYL .


Production without consumptionProduction without consumptionIf 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 distributionIncome 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



Real wage of capital =MPk =WK/py =∂Y(K,L)/∂K =YK=g(K/L )

Real wage of labor =YL=h(K/L)

k/L

Y p

er k

Yk=g(K/L)

bc

d= (K/L)1

Labor incomeLabor income

Capital Capital incomeincome

If L is total supply of labor If L is total supply of labor consist of one unit of laborconsist of one unit of labor , then area of obcd equals to capital income . Since capital income equals to capital amount (od) times price of capital (cd) .

Since area aodc is total or national income , then area abc equals to labor income

o

a

General Equilibrium26


We can similarly portray the same information with L/K as the dependent variable and provide simple answers for important

questions ;

YL

L/K

Y per L

Capital income

Labo

r in

com

e

Resident workers immigrants

ab ab

Why real wages (YL) , and standard of living (Y/L) is lower in India than Europe . The answer could simply be seen as having higher L/K ( shortage of capital).

How would immigration of workers affect the welfare of capital owner and domestic labor in a country ?As it is seen an amount bb will be transferred from domestic workers to capital owners and capital owners would gain a+ba+b and the share of domestic workers reduce to d d . . Immigrant workers receive an

amount equal to c . .

ccd


Production without consumptionSuppose that the labor supply increaselabor 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)↑>%1If (L/K) ↑=%1(L/K) ↑=%1→ YL/YK = RTSLK↓<%1→ (YL/YK)(L/K) ↑ → (Y(YLLL)/(YL)/(YKKK) ↑K) ↑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 .



Two sector modelTwo sector model ;

Suppose that there are two productive sector ; X , Y .

X is more labor intensive than Y , (K/L)(K/L)xx < (K/L) < (K/L)yy . .

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 .



Y

XWWKK/P/PXX

WWKK/P/PYY

U(x,y)U(x,y) Maximum utility of capital and labor will be a function of factor wages and commodity prices.

U K= uK [(WK/PX) , (WK/ PY)]

UL = uL [(WL /PX ) , (WL / PY)]

How are factor and commodity prices determined in a competitive How are factor and commodity prices determined in a competitive equilibrium ?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;

First First , we can establish a one to one relationship between the a one to one relationship between the relative price of products( demand for x and y ) and welfare of factor relative price of products( demand for x and y ) and welfare of factor ownersowners. SecondSecond , we should confirm that in a closed economy the welfare effect of changes in factor supply is the same as in one sector model


Production without consumptionThe relation between product price and factor price can be illustrated in he The relation between product price and factor price can be illustrated in he

following theoremfollowing theorem

Stopler-Samuelson theoremStopler-Samuelson theorem

“ in any particular country a rise in the relative price of labor-rise in the relative price of labor-intensive good will make labor owner better off and capital intensive good will make labor owner better off and capital owner worse offowner 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 PPxx/P/Pyy increase increase , then production of x will increase (p=MC)production of x will increase (p=MC) , and since x is a labor intensive commodity , demand for demand for labor will increase and cause the wage rate to increase .labor will increase and cause the wage rate to increase . Increase in wage rate will consequently decreasedecrease the amount of labor demandedlabor demanded both for x and y .



When Lx and Ly decrease , (K/L)x and (K/L)Y will increase and cause the labor productivity to increaselabor productivity to increase and capital capital productivity to decreaseproductivity 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”“Lerner-Pierce” diagram.



Lx , Ly

Kx , Ky

PP

QQ

PP11

QQ11

PP22

QQ22

Y=1 unitY=1 unit

X=1 unitX=1 unitRR

SS

SS’’ RR’’

KKxx

LLxxLLyy

KKyy

WWLL/W/WKK))WWLL/W/WKK’(’(

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 .

O


Production without consumptionNow 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 . ;

34


O

XXyy

(K/L)x)K/L(y (Px/Py)0

(WL/WK)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 )

X is labor intensive so (K/L)x<(K/L)y

) 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



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 .

(Px/Py)0

XXYY

))WWLL/W/WKK((00

(WL/WK)1(WL/WK)1

(K/L)(K/L)x0x0

(K/L)(K/L)y0y0))K/LK/L((y1y1

))K/LK/L((x1x1



The Lerner- Pearse diagram could show us why this happen

LLxx, L, Lyy

KKxx

KKyy

)WL/WK(0(W(WLL/W/WKK))11

)K/L(x0

X=1X=1

Y=1Y=1

(K/L)(K/L)y0y0

(K/L)(K/L)y1y1

(k/L)x1

Cost of producing one unit of x or y is the same for both of these factor price ratios px /py=1

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 .



The relation of output-mix to real factor prices;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 ?

K0

L0

)K/L(Yo )K/L(Y1> (K/L)yo

Po

P1

xo

Y0

WL/ WK

Ox

OY

x1

Y1

)K

/L(

x0

) >

K

/L(

x1

)WL/WK()> WL/WK’(


Production without consumptionConsider 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 .



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 producing more from xand less from y .

Y

X

AA

BB


Production without consumptionProduction without consumptionWhen X is labor intensive PPF is concave to the origin and contract curve is below the When X is labor intensive PPF is concave to the origin and contract curve is below the diagonal .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 (PWhen (Pxx/P/Pyy) permanently increase along with the production of x , then (W) permanently increase along with the production of x , then (WLL/W/WKK) will increase ) will increase which will cause (Pwhich will cause (Pxx/P/Pyy) to increase again . Production of x will increase again . This process ) 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 . may continue until all factors of production engaged in the production of x . When this happen . (K/L)(K/L)x x will be fixed , and (W will be fixed , and (WLL/W/WKK) will be fixed too. But P) will be fixed too. But Pxx/P/Py y has increased , which is the has increased , which is the violation of the Stopler-Samuelson theorem. So in order for the theorem to work , some of violation of the Stopler-Samuelson theorem. So in order for the theorem to work , some of each good should be produced .each good should be produced .


Production without consumptionEffect 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) .



ox

o’yK0

L0 L’

P

P’

oy

)K/L(x

)K/L(y )K/L(y

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.


Production without consumptionBut 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 theelasticity of substitution between industry x and y (when xincrease and y decrease ) to find out the degree of relative price decrease .


Production and consumptionProduction 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 , andQx = max . In the first case excess demand for x is veryhigh and in the second case excess demand for x is equalto 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 whetherexcess demand for x (for both workers and capital owners)decrease with increase in the relative price of x (px/py).


Production and consumption

px/py

oL

A

A

BL

BK

Y

x

Y*

X*

E

UK

UL

Y

x

A1

A1

(px/py)1

oL

oK oK

x1

Y1

CK

CL

UK

UL

aa bb

equilibrium

Budget constraint

abab = excess supply of x

bcbc = excess demand for y

cc


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 .

PROBLEMSQ2 – 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 . EDX

A is fixed and EDX B decrease with increase in relative price of X .


PROBLEMSQ2-2 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 ? .

General Equilibrium

A

B X0

Y0

XA =1/2 X0

YB = 1/2 Y0

A’s offer curve B’s

offe

r cu

rve

(Px / PY)=P= P.C. Price

of A Initial endowment

Initial endowment of B

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 induce this supply . ( the same if type B collude ).

P’

PROBLEMSQ2-3 – 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 .

Solution :


CountryKL

19000000100Us

220000020UK

35000200INDIA

CountryK/LWL /PY =MPL=50 (K/L)1/2

Y/L = 100 (K/L)1/2

1900001500030000

210000500010000

325250500

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 Y050

PROBLEMSQ2-5 How would you rank the welfare of the workers in the following

table

Assume ;

i- worker’s utility is not known

ii- worker’s utility is U= XY.

Solution ;


WL PX PY

State 1111

State 2233

State 3214

State WL /PX WL /PY

111

22/32/3

321/2

i- state 1 is preferred to 2 , but we can not say anything about the other states.

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 Lx

1/3 , Y = Ky1/3 Ly

2/3 , 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

(b) Lx = Ly 52

PROBLEMSEvaluate 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


PROBLEMS

WK / Px ,WK /Py WL /Px WL /Py WL / WK Px / Py

XKYK XLYLXL / XK YK /XK

2/3) Kx / Lx(-1/3

1/3 (K y/ Ly )

2/3

1/3 (Kx / Lx) 2/3

2/3 (K y/ Ly )

1/3

2/3) 0.4(1/31/3 (1.6)2/31/3 (2.5)2/3

2/3) 0.6(1/3 1/0.81/2 (6.4)1/3

2/3 (1/1.6)1/31/3 (1/0.4)2/31/3 (1.6)2/3 2/3 (0.4)1/30.81/2 (10)1/3


a

b

Y is labor intensive good ((Ky / Ly ) < ( Kx / Lx ) )Labor is better off in a in which the production of Y is higher

Q2-8 Same as question 2-7 , but with Y = 2 Ky2/3 Ly

1/3 and every thing else as before .

Solution ;

X = Kx 2/3 Lx

1/3 , Y = 2 Ky2/3 Ly

1/3 , 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.


Q2-9 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 + Ly = 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 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 xWalters & Layard CH 2 56

PROBLEMSi- if Uk = XK

3/4 yk , UL = XL3/4 yL

ii- if Uk = XK yk3/4

, UL = XL yL3/4 ‘


K0 =1

X=1

X=1/2

X=1.5

X

Y

Y=1

Y=2

Y

X

1

2

3

1 2

Contract curve

-dy/dx = 1

-dy/dx = 2

All L in XAll K in Y

Y=3

X=2

L0 =1

P

p1

p2

o

P correspond to 0P correspond to 0 PP11P correspond to xo .All K in Y . L transferred P correspond to xo .All K in Y . L transferred

from X to Yfrom X to YPPPP22 correspond to yo . All L in X. K transferred correspond to yo . All L in X. K transferred

from Y to X from Y to X

Y= 2.5

Y=0.5

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 = MRSxy

K = 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


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 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

• K0 ≥ 2X + Y , L0 ≥ X + Y


Y

X

K0 = 2X + Y Slope = -2

L0 = X + Y slope = -1

A

CO

Feasible region Production may occur on AB region or BC region or at point B. At point B there is full employment , since both constraints are binding. On AB region K is unemployed ( only L is binding), on BC region L in unemployed .

i-i- K= 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.

BB’

C

at point B’ the relative price is equal to Px / Py = 1 ,

We should apply exhaustion theorem ;

X = MPLx Lx + MPK

x Kx , MPLx =1 , MPK

x =2 , X = Lx + 2Kx , Px = WL + 2WK

Y = MPLY LY + MPK

Y KY , MPLY =1 , MPK

Y =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 W L / 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


PROBLEMSiii- at B , X is negative ,

, 0.5 = 2X + Y , 1 = X + Y , X = -0.5 , Y = 1.5

X = Lx + 2Kx , Y = Ly + Ky

capital constraint highly dominates the labor constraint . There are too 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

2-all the capital goes for y production k=0.5 , L=0.5 , Y = 0.5

→ → MRS =Y/X=2=Px /Py= WK/Py

Py /Px = Wk / Px =1/2

Q2-12Q2-12 – 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

L=1=X+Y

Y

X

K=0.5=2X+Y

1

1

0.5

0.25-0.5

1.5

3 -some of the capital goes for the production of x and some of the capital goes for the production of Y

PROBLEMS


Ox

Oy

L0

K0

P

P’

P”

Rx

Increase in (K/L)x

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

general equilibriumwalters & layard ch 21 general equilibrium general equilibrium

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