dvanced Microeconomics Problem Set László Sándor Central European University Pareto optima With Cobb-Douglas utilities u x ; x 2 ; x 3 = 0:4 log x 2 + 0:6 log x 3 and u x ; x 2 ; x 3 = log x 2 + log x 3, symmetric endowments! =! = (6; 0; 0), and production set Y = fy ; y 2 ; y 3 : y 0; y 2 + y 3 4y g with the rm owned by consumer, we set out to check Pareto optima and competitive equilibria.. Pareto optima In any Pareto optimal allocation the marginal rates of substitution must be equal otherwise an improvement would be available. Fortunately, with Cobb-Douglas utilities, the marginal utility of the rst in nitesimal unit of either consumption good is in nite, thus there would be no corner solutions. The technology is equally good at producing the two consumption goods (this also follows from an implicit function theorem, or the total derivative of the PPF), so MP RT 2;3 = : () lso, feasibility sets for goods 2 and 3: y 2 + y 3 4y 4!!, which implies (by the observation that allocating any remaining good from the consumers to production would be a Pareto improvement, as they attach no value to it, but some to either output) For the consumers, using (): MRS 2;3 = 0:4 x 2 MRS 2;3 = x 2 x 2 + x 3 = 48: (2) x 3 0:6 = =) 3x 2 = 2x 3 ; (3) x 3 = =) x 2 = x 3 : (4) The Pareto set is de ned by the relationships (2), (3) and (4) (which are 3 equations in four variables, as x 2 = x 2 + x 2 and x 3 = x 3 + x 3 ): as markets must clear, the Pareto e cient allocations are x ; x 2 ; x 3 ; x ; x 2 ; x 3 ; (y ; y 2 ; y 3 ) P = 0; x 2 ; 3 2 x 2 ; 0; 24 4 x 2 ; 24 4 x 2 ; 2; 24 4 x 2 ; 24 + 4 x 2 ; () where 0 x 2 96. (Note that for the limiting cases the marginal rates for substitutions are invalid. However, these are indeed Pareto-optima, as no reallocation of any of the three goods would raise the poor consumer without hurting the one better of. Note also that strictly speaking the given utilities are not identi ed for zero consumption, only their equivalent formulations with the exponentials. I assume the use of this equivalent when I allow the borderline cases.).2 The competitive equilibrium The competitive equilibrium is such a Pareto-optimum where prices balance the individual balances as well as clearing the prices. I choose good for the numeraire. The ratio of the two other prices is set by (2): p 2 p 3 = MP RT 2;3 = ;
thus let us note p 2 = p 3 = p. ut as the rm maximises pro ts, it will set this equal price to the (equal) marginal cost, p = 4. Note that the rm will not make any pro t, as we expected with such a technology exhibiting constant returns to scale, so its allocation would not change the equilibrium. Using that the equilibrium is a Pareto optimum, the budgets balance if: 4 x 2 + 3 2 x 2 = 6 + 4 24 4 x 2 + 24 + 4 x 2 2 = 6; or 2 x 2 = 24; x 2 = 48 : Substituting this into (), the Walrasian equilibrium is x ; x 2 ; x 3 ; x ; x 2 ; x 3 ; (y ; y 2 ; y 3 ) ; (p ; p 2 ; p 3 ) = = 0; 48 ; 72 ; (0; 2; 2) ; 2; 08 ; 32 ; (4p; p; p) : (6) Note also that the expenditure shares of the consumers are that were expected for this utility..3 Di erent shares s I have already noted, the rm makes no pro t in any equilibrium with such a technology, so if the economy was the same as before but with ownership 0 = 2 ; 2 instead of = (; 0), the pro t reallocation would not change anything, and the equilibrium would be the same as in (6)..4 The First Welfare Theorem The preferences are locally nonsatiated, thus the derived Walrasian equilibria are Pareto optimal.. The Second Welfare Theorem The production set is convex and the preferences are convex and locally nonsatiated (as it is known for Cobb-Douglas preferences). Therefore, every Pareto e cient allocation from () is a price quasiequilibrium with transfers for some nonzero price vector. Since the preferences are also continuous, all allocations that would allow some cheaper consumption are parts of price equilibria with transfers. Hence all Pareto optimal allocations can be equilibrium allocations apart from the limiting cases, () with 0 < x 2 < 96. However, I already noted that the utility might be invalid in those points anyway. 2 Proofs 3 Continuity X The exchange economy i ; % i I ;! i I satisfy that i= i=. X i = R L + 8i, 2. % i is continuous, strictly convex and strongly monotonic 8i, 3. and! = P I i=!i 0. Then z l (p) = P I i= zi l (p) = P I i= xi l p; p! i! i functions are continuous, since the individual demands are continuous for such preferences (by some fundamental results of utility maximisation) for those consumers with positive endowments (there is at least one), and zero for the rest. dding up continuous functions or subtracting constants does not change continuity. 4 Leontief exchange economy U = min x ; y, U = min x ; y, and x + x = x > y = y + y. 2
4. The exchange demand functions For any nonnegative price vector p, the consumers demand equal amount of both goods, thus for both, excess demand is z x (p) = p xx + p y y x + p xx + p y y x = p xx + p y y x; z y (p) = p xx + p y y y: These are continuous, homogenous of degree zero, satisfy Walras law and are bounded. However, if we take a generic sequence of prices p n = p ; p2 n! p = (p ; 0) 6= 0, max fz x (p n ) ; z y (p n p x )g! max p x; p x p We can conclude that the existence of the equilibrium is not guaranteed. 4.2 Pareto optimal allocations y < : In this case, Pareto optimality is not about marginal substitution. Instead, x = y and x = y are required individually. Note that the sum of these would imply equality for all the allocated consumption goods. Thus, even if such an allocation exhausts the resources in y, there will be x rest, that can be arbitrarily allocated, it does not change either utility. The Pareto optimal allocations are: y + a; y ; x y a; y y : y 2 [0; y] ; a 2 [0; x y] : (7) 4.3 First welfare theorem The preferences are locally nonsatiated still, so if an equilibrium exists, it is Pareto optimal. 4.4 Second welfare theorem The preference relation is convex and locally nonsatiated, thus every Pareto optimal allocation can be a price quasiequilibrium with transfers. Thus apart from the points where one consumer would end up with zero amount of at least one good (some excess x is small consolation), any allocation of (7) can be upheld by some nonzero price vector (with zero price for x). This exempli es the limitations of our result on existence. We just listed some su cient conditions. This does not rule out existence in some other cases as well. The 22 production model. The factor intensities o -diagonal In an Edgeworth box, the diagonal is the set of points where K L = K L. elow that, the ray form O X is less steep, i.e. K X L X < K L. However, the ray from O Y is steeper, in a reverse coordinate system, thus K L < K Y Obviously, K X L X < K Y..2 Intersections with rays With constant returns to scale, the rays exhibit the same marginal rate of technical substitution all along. If a ray intersected the Pareto set twice, but both o -diagonal, two separate rays would connect those intersections to the other origin. Thus for the two points, the MRTS of, say, X would be the same, but that of Y would be di erent. Therefore the MRTS of X and Y cannot be equal for both points, thus cannot both be in the Pareto set. The slopes of rays to a continuous curve, like the Pareto set in this model, is continuous. If the slopes cannot be the same for two points, the slopes must change monotonically. ut those slopes are just the factor intensities. nd they uniquely determine the relative factor prices.. 3
6 The Stolper-Samuelson theorem ecause of the constant returns to scale, we had The total derivatives are dp X = @c X (w; r) @w dp Y = @c Y (w; r) @w c X (w; r) = p X ; c Y (w; r) = p Y : dw + @c X (w; r) dr = L X (w; r) dw + K X (w; r) dr; @r dw + @c Y (w; r) dr = (w; r) dw + K Y (w; r) dr; @r since we know L X K Y K X > 0 as the factor intensity assumption, we can invert the matrix implicit in the expression, to get dw KY K = X dpx : dr L X K Y K X L X dp Y So, if dp X > 0, dp Y = 0, dw = K Y L X K Y K X dp X > 0; dr = L X K Y K X dp X < 0; which proves the Stolper-Samuelson theorem for this case. 7 The Rybcszynski theorem s no output prices or technologies change when some extra endowment is found, the new equilibrium must be on the same rays with the same factor intensities. ctually, if the extra factor is what X is intensive in (below the diagonal, horizontally added endowment), the new equilibrium will be on the very same ray, while the ray from O Y shifts as the point itself shifts. s the invariant ray slopes upward, it follows that more will be produced of X, and less of Y. 8 strengthened Rybcszynski theorem The graphical proof is the following. If the rise in the production of X would be proportional to the rise in the factor endowment, if the gure, the two extra lines (rays from the lower-right corners of the respective Edgeworth-boxes) were parallel. Then we could quote the intercept theorem. However, the noted angles should be equal then. ut the outer one is larger (thus the new equilibrium is away than 4
the parallels would imply in the proportional case, the rise is more proportional), since it faces the same section (of length K), but has a smaller height, as decreases.