General Equilibrium 1 Walters Layard CH 2 General

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 ; 2 Walters & Layard CH 2 General Equilibrium

CONSUMPTION WITHOUT PRODUCTION PURE EXCHANGE A- Bargaining ; Y E o u. B u. A o. B T 0 T 1 o. A X E = initial endowment point → MRSxy. A > MRSxy. B → fruitful trade is possible. As while as both are consuming in the ET 0 T 1 area , both will be better off. The solution will be any point between T 0 and T 1 on the contract curve. If solution would be near to T 0 , individual A has more strength , and vise versa. 3 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 ? 4 Walters & Layard CH 2 General Equilibrium

CONSUMPTION WITHOUT PRODUCTION PURE EXCHANGE Suppose auctioneer starts with price equal to the slope of ET 0 , ( price = p 0 ). a’ b’ o. B Y E d’ d c’ B c A c 0 T c p 0 C P* o. A a b 5 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 T 0 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 MRSxy. A=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. 6 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; pxx. A + pyy. A =value of A’s market demand = A’s expenditure pxxo. A + 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 pxx. A + pyy. A = pxxo. A + pyyo. A → px(x. A – xo. A) + py(y. A – yo. A) = 0 pxx. B +pyy. B =pxxo. B + pyyo. B → px(x. B – xo. B) + py(y. B – yo. B) = 0 px EDx. A + py. EDy. A =0 , px EDx B + py. EDy B =0 px EDx + py. EDy =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 P 0 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 P 1 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. 8 Walters & Layard CH 2 General Equilibrium

CONSUMPTION WITHOUT PRODUCTION PURE EXCHANGE B At p 1 : excess supply p 1 At p 0 : excess demand P* At p* : market is clear A po ESx EDx=0 EDx 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) 9 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 : p 1 , p 2 , p 3. 10 Walters & Layard CH 2 General Equilibrium

CONSUMPTION WITHOUT PRODUCTION PURE EXCHANGE p 4 p 3 p 2 p 1 p 0 ESx EDx=0 P 2 is unstable so it will rule out. Which one of the p 3 or p 1 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 ? 11 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 y. Ao from p 1 to p 2 to p 3 3 u. consumption of x increase from x 1 to u 1 u 2 c 3 y 3 x 2 to x 3. y 1 c 2 C 1 Consumption of y y 2 decrease from y 1 to y 2 but then x 1 x 2 x 3 increase to y 3. x. A 0 p 3 p 2 Why? p 1 12 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 c 2. We will show that demand elasticity of x will be elastic before point c 2 and inelastic after point c 2. After point c 2 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 13 Walters & Layard CH 2 General Equilibrium

CONSUMPTION WITHOUT PRODUCTION PURE EXCHANGE o. B y A’s offer curve E c 3 P 3 u. A c 2 B’s offer curve u. B c 1 p 2 p 1 x o. A 14 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 15 to spend Walters on x &out of. CHincome. Layard 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. 16 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 o. B. → point Q. A’s offer E u. A Q u. B o. A 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 u. B 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. 18 Walters & Layard CH 2 General Equilibrium

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. 19 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. 20 Walters & Layard CH 2 General Equilibrium

Production without consumption Y 2 K K/L Y 1 Y(m. K, m. L ) = 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/(L 2)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 ). 21 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-1 Y’( 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 , 22 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; 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. 23 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 24 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 d= (K/L)1 25 k/L Since area aodc is total or national income , then area abc equals to labor income Walters & Layard CH 2 General Equilibrium

Labor income 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 income of capital). bb aa d 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→ ↑=%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. 27 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. 28 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= u. K [(WK/PX) , (WK/ PY)] UL = u. L [(WL /PX ) , (WL / PY)] X WK/PX 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 ; 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 29 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. 30 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= u. K [(WK/ Px ) , (WK / PY)] = welfare of capital owners decrease UL =u. L [(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. 31 Walters & Layard CH 2 General Equilibrium

Kx Production without consumption , K P 2 y P 1 P Ky Kx O S’ R’ S Ly Q 1 Q 2 WL/WK (WL/WK)’ Y=1 unit R X=1 unit Q Lx Lx , L y 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 = P x = W K K x + W L Lx cost of producing one unit of y = P y = W K K y + W L 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. 32 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 =OP 2 in terms K and OQ 2 in terms of L. Cost of producing one unit of Y =OP 1 in terms K and OQ 1 in terms of L. According to the diagram increase in the price of x (PP 2) is greater than increase in price of y (PP 1), 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. ; 33 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 ) ( WL / WK ) , and ( PX / PY ) are known and ( XL / Y K YL , XK , XL , YK are functions of (K/L)x and ) = ( WL / WK ) ( PY / PX ) (K/L)y. So two equations and two unknowns 34 [(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. Y X (WL/WK)1 (WL/WK)0 (K/L)x 1 35 (K/L)x 0 (K/L)y 1 (K/L)y 0 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)x 1 (K/L)y 0 Y=1 (K/L)x 0 X=1 (WL/WK)0 Lx , L y 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 36 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 K 0 (WL/WK) <(WL/WK)’ xo Y 0 P 1 Po Ox (K/L)Yo 37 (K/L)x 0 WL/ WK < (K/L)x 1 Y 1 x 1 (K/L)Y 1> (K/L)yo Walters & Layard CH 2 L 0 General Equilibrium

Production without consumption Consider a move from P 0 to P 1 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), L 0=Lx + Ly Qx ( at P 1 ) > Qx ( at P 0 ) Qy ( at P 1) < Qy ( at P 0) (K/L)x < (K/L)y , at P 1 and P 0 , since x is labor intensive. comparing Lx and Ly at points p 0 and p 1 ; (Ly / L 0) has reduced in P 1 , but (Lx/L 0 ) has increased in P 1. Higher weight (Lx/L 0 ) 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. 38 Walters & Layard CH 2 General Equilibrium

Production without consumption MRSx. LK = 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 Walters & Layard CH 2 X 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 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 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 K 0 (K/L)x P’ P (K/L)y ox (K/L)y L 0 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 ↓→x. L ↓, y. L↓ →x. K↑, y. K↑ →→(WL/Px)↓ , (WL/Py)↓ → (WK /Px)↑ , (WK /Py)↑ →→ UL(w. L/px , w. L/p. Y) ↓ labor worse off , UK(w. K/px , w. K/p. Y) ↑ 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 o. L Y* A 1 equilibrium UK Y 1 E A UL o. K ab = excess supply of x CK px/py X* BK x UL o. K a b c CL c x 1 A 1 x (px/py)1 Budget constraint 45 o. L UK 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 Q 2 – 1. Suppose that consumers of type A are endowed with total supply of X ( X 0 ) and consumers of type B are endowed with total supply of Y ( Y 0 ). 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. ( P x / Py ) = P S. T. ( Px / Py ) XB + YB = Y 0 , YB = 1/2 Y 0 , XB = 1/2 ( Y 0 / P ) EDX B = XB = 1/2 ( Y 0 / P ). , YB = 1/2 Y 0 , MAX UA = XA YA S. T. P ( XA - X 0 ) + YA = 0 , XA = 1/2 X 0 , YA = 1/2 X 0 P EDXA = XA - X 0 = - 1/2 X 0. EDX B + EDXA = 0 , PX / PY = Y 0 / X 0 = 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 Q 2 -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 X 0 P’ B’s offer curve X 0 Initial endowment of A YB = 1/2 Y 0 B A’s offer curve (Px / PY)=P= P. C. Price A Initial endowment of B Y 0 Solution - Infinitive. B’s offer curve is vertical at ½ Y 0 and A needs to offer barely an x in return for y in order to induce B to supply 1/2 Y 0. Therefore any positive price ( like p’ ) is sufficient to induce. General this supply. Equilibrium ( the same if type B collude ).

PROBLEMS Q 2 -3 – if Y = 100 K 1/2 L 1/2 , 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 : Country K/L 1 90000 15000 30000 2 10000 5000 10000 3 25 250 500 49 WL /PY =MPL= 50 (K/L)1/2 Walters & Layard CH 2 Y/L = 100 (K/L)1/2 General Equilibrium

Q 2 -4 , If Y = K 1/4 L 3/4 and the labor force is constant at L 0 , how does increase in capital accumulation ( from K 0 to K 1 ) 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) = Y 0 = national income Relative share of labor = ( YL L) / Y 0 [Y 0 / ( YL L) ] = 1 + (YK K)/ ( YL L) = 1 + 1/3 = 4/3 ( YL L)/ Y 0 = 3/4 , Relative share of capital =(YK K)/ Y 0 = 1/4 iii- absolute share of capital = (YK K) = 1/4 Y 0 absolute share of labor = ( YL L) = 3/4 Y 0

PROBLEMS Q 2 -5 How would you rank the welfare of the workers in the following table WL PY State 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 /PY 1 1 1 2 2/3 3 2 1/2 51 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 /2 Px , Y = WL / 2 Py U = XY = 1/4 (WL /2 Px ) (WL / 2 Py ) U 1 = (1/4)(1)(1)= 1/4 , U 2 = 1/9 , U 3 = 1/4 Q 2 -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. Q 2 -7 - Suppose that X = Kx 2/3 Lx 1/3 , Y = Ky 1/3 Ly 2/3 , and the economy is endowed with K 0 and L 0 measured in units such that K 0 = L 0 =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

PROBLEMS Evaluate the following at points (a) and (b). WK / P x , WK /Py , WL /Px , WL /Py , W L / WK , P x / Py , At which point labor is better off. Solution ; being on the transformation curve we have RTSLK x =RTSLKy XL / X K = Y L / Y K , 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 (K 0 - Kx )/ (L 0 – Lx ) → 1/2 ( K 0 – Ky )/ ( L 0 – 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 / W K XL / X K Px / P y XK YK XL YL 2/3 (Kx / Lx)- 1/3 (K y/ Ly ) 1/3 (Kx / Lx) 2/3 (K y/ Ly ) a 2/3 (0. 4)1/3 (1. 6)2/3 1/3 (2. 5)2/3 (0. 6)1/3 1/0. 8 1/2 (6. 4)1/3 b 2/3 (1/1. 6)1/3 (1/0. 4)2/3 1/3 (1. 6)2/3 (0. 4)1/3 0. 8 1/2 (10)1/3 2/3 YK /XK 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

Q 2 -8 Same as question 2 -7 , but with Y = 2 Ky 2/3 Ly 1/3 and every thing else as before. Solution ; X = Kx 2/3 Lx 1/3 , Y = 2 Ky 2/3 Ly 1/3 , both X and Y are equally capital intensive so , X L / 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

Q 2 -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 ) K x + Ky = 1 Kx = 1/3 Ky = 2/3 L x + 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. Q 2 -10 Suppose X= K x + Lx Y = 2 K y + Ly K 0 = L 0 =1 What are the following parameters in general equilibrium ; WK / P x , WK /Py , WL /Px , WL /Py , W L / WK , P x / Py 56 Walters & Layard CH 2 x

PROBLEMS i- if ii- if Uk = XK 3/4 yk , UL = XL 3/4 y. L Uk = XK yk 3/4 , UL = XL y. L 3/4 X=2 K 0 Y =1 3 P correspond to 0 P 1 P correspond to xo. All K in Y. L transferred from ‘ X to Y PP 2 correspond to yo. All L in X. K transferred from Y to X Y -dy/dx = 1 p 1 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 Y=3 L 0 =1 o 1 2 p 2 X 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 MRSxy. L = 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 = Mp. L 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 yk 3/4 , UL = XL y. L 3/4 , 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 PP 2 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 Q 2 -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 K 0 = 1. 8 , and L 0 = 1. ii- in a less reach country with K 0 = 1. 4 , and L 0 = 1. iii- in a poor country with K 0 = 0. 5 , and L 0 = 1. 5 9

PROBLEMS Y K 0 = 2 X + Y Slope = -2 K 0 ≥ 2 X + Y A B’ L 0 ≥ X + Y • B Production may occur on AB Feasible region or BC region or at point B. At point B there is full employment , since both constraints are binding. L 0 = X + Y slope = -1 On AB region K is unemployed ( only L is binding), on BC region L X in unemployed. C O , C i. K= 1. 8 , L=1 , 1. 8 = 2 X + 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 + 2 Kx , Px = WL + 2 WK 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 + 2 Kx , 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 / P x = W L / P y = 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 = P x / 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 Walters & Layard CH 2 General Equilibrium

PROBLEMS iii- at B , X is negative , , 0. 5 = 2 X + Y , 1 = X + Y , X = -0. 5 , Y = 1. 5 X = Lx + 2 Kx , 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 , 2 K=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 P y /Px = Wk / Px =1/2 0. 5 x and some of the capital goes for the production of Y K=0. 5=2 X+Y 1 -0. 5 0. 25 X Q 2 -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 ).

PROBLEMS Rx P” Oy P P’ Increase in (K/L)x Ox 63 Walters & Layard CH 2 K 0 L 0 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