이인호교수 미시경제 강의노트.pdf

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Unformatted text preview: MICROECONOMICS I N H O L EE School of Economics Seoul National University Contents 1 2 3 4 5 6 INTRODUCTION 1 1.1 Economic models . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Mathematics of Optimization . . . . . . . . . . . . . . . . . . . . 1 CONSUMER THEORY 5 2.1 Preference and Utility . . . . . . . . . . . . . . . . . . . . . . . . 5 2.2 Consumer’s Utility Maximization and Choice . . . . . . . . . . . 12 2.3 Comparative statics . . . . . . . . . . . . . . . . . . . . . . . . . 19 2.4 Application:Labor supply . . . . . . . . . . . . . . . . . . . . . . 24 PRODUCER THEORY 26 3.1 Technology and Production . . . . . . . . . . . . . . . . . . . . . 26 3.2 Cost Minimization . . . . . . . . . . . . . . . . . . . . . . . . . 29 3.3 Profit Maximization . . . . . . . . . . . . . . . . . . . . . . . . . 33 Market and Equilibrium 35 4.1 35 General Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . IMPERFECT COMPETITION 41 5.1 Monopoly . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 5.2 Duopoly . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 5.3 Game theory and Strategic behavior . . . . . . . . . . . . . . . . 45 MISSING MARKETS 51 6.1 Public Goods . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 6.2 Information and Uncertainty . . . . . . . . . . . . . . . . . . . . 52 1 INTRODUCTION 1.1 Economic models Economic models are not able to reflect the real world perfectly. And also, it is rather useless to describe the real world completely. So we need to simplify it. Through this process economic models can contain important factors in real world. Principle of Parsimony There exists a trade-off between details and explanatory power. Good economic model must be simple, systemic, and able to explain the points thoroughly. It has to remove unnecessary parts in reality. And the result of the model must be reputable. 1.2 Mathematics of Optimization Optimization is a problem of the choice. If we can express preference numerically, a choice behavior becomes the problem of minimization or maximization. 1 Unconstrained Maximization ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ max () → ... = ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ ∂ ∂1 =0 ∂ ∂2 =0 ⋅ ⋅ ⋅ ∂ ∂ =0 S.O.C.: Hessian matrix is negative semi-definite. ∙ ... is a necessary condition and ... is a sufficient condition. ∙ Minimization problem can be transformed to maximization of the negative of the objective function. min () = max − () Constrained Maximization max () s.t. ∈ We can solve this problem by Lagrangean Method. 1. Given max () s.t. () ≥ 0 ( ∈ ) formulate Lagrangian function: ℒ = () + () 2 2. Obtain the first order condition and solve for the roots of the simultaneous equation system: ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ... = ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ ∂ℒ ∂1 =0 ∂ℒ ∂2 =0 ⎛ ⋅ ⋅ ⇒ ⋅ ∂ℒ ∂ =0 ∂ℒ ∂ =0 ∂ ∂ ∂ℒ = ∂ + ⋅ ∂ =0 1 1 ⎜ ∂1 ⎜ ⎜ ⎜ ⎜ ∂ℒ ∂ ∂ ⎜ ∂2 = ∂ + ⋅ ∂ =0 2 2 ⎜ ⎜ ⎜ ⋅ ⎜ ⎜ ⎜ ⋅ ⎜ ⎜ ⎜ ⋅ ⎜ ⎜ ∂ℒ ∂ ∂ ⎜ ∂ = ∂ + ⋅ ∂ =0 ⎜ ⎜ ⎜ ⎝ ∂ℒ = () = 0 ∂ ∙ ... is a necessary condition and ... (it requires Bordered Hessian) is a sufficient condition. ∙ Minimization problem can be transformed to maximization of the negative of the objective function where the constrain set remains the same. min () s.t. () ≥ 0 = max − () s.t. () ≥ 0 ∙ Binding constraints are associated with non-zero Lagrangian multiplier and non-binding ones with zero Lagrangian multiplier. This observation is formally called complementary slackness: () = 0. Example (Utility Maximization) max (, ) s.t. ⋅ + ⋅ ≤ 3 ⇒ max (, ) s.t. − ⋅ − ⋅ ≥ 0 ℒ = (, ) + ( − ⋅ − ⋅ ) = (, ) − ( ⋅ + ⋅ − ) ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ... = ⎜ ⎜ ⎜ ⎜ ⎝ ∂ℒ ∂ ⎛ =0 ∂ℒ ∂ =0 ∂ℒ ∂ =0 ⇒ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ ∂ℒ ∂ = ∂ ∂ + ⋅ = 0 ∂ℒ ∂ = ∂ ∂ + ⋅ = 0 ∂ℒ ∂ = − ⋅ − ⋅ = 0 ∂ℒ ∂ = marginal utility of income = = value of constraint 4 2 CONSUMER THEORY 2.1 Preference and Utility Preference Agent (weakly) prefers to iff ≿ Preference Relation Binary relation ≿ is a preference relation if 1. ≿ is complete. (given , ∈ , either ≿ or ≿ ) Any two bundles can be compared. That is, given any two bundles, the consumer is able to make a choice between them. 2. ≿ is transitive. (given , , ∈ , ≿ , ≿ then ≿ ) If the consumer thinks that is at least as good as and that is at least as good as , then the consumer thinks is at least as good as . Representation Theorem: Suppose ≿ is a preference relation on which is finite. Then there exists (⋅) : → such that for , ∈ , ≿ iff () ≥ () 5 ∙ (⋅) is not unique. What matters is the order, not the absolute value of the function. Any monotonic transformation of a utility function representing some preferences is a utility function representing the same preferences. ∙ As the utility function represents preference, we can use the mathematical method! ∙ For which is infinite, there may not exist utility function representing preference relation unless it is continuous. (Counterexample: lexicographic preference) Properties of Preference Since these axioms may not suffice to guarantee representation, some properties are added to facilitate the analysis using utility functions. 1. Monotonicity : The more, the better, i.e., there is no bliss point. (Local non-satiation which rules out thick indifference curve might suffice for the purpose of obtaining well-behaving demand function.) ≥ ⇒ () ≥ () 2. Continuity : The utility changes continuously as the consumption bundle changes continuously. Given a consumption bundle , the upper contour set { : ≿ } and the lower contour set { : ≿ } are closed. ♣ preference relation + continuity → continuous representation preference relation + finite consumption set → representation 6 preference relation + infinite consumption set → representation may or may NOT obtain ♣ Counter example to continuity: Lexicographic preference ≿ is lexicographic, if given (, ) ∈ and (′ , ′ ) ∈ (, ) ≿ (′ , ′ ) if > ′ or = ′ , > ′ 6 ∗ q preferred set of (∗, ∗) - ∗ Indifference Map We can draw a utility function on the space of consumption and utility. For instance, we need three dimensions if we want to describe a utility function for two commodities. This is not easy. Hence we use indifference map instead. Indifference Curve Locus of indifferent choices. Given a consumption bundle (, ), {(′ , ′ ) : (, ) ∼ (′ , ′ )} ⇔ {(′ , ′ ) : (, ) = (′ , ′ ) = ¯} 7 6 (, ) (′, ′) ¯ - Slope of indifference curve is MRS (Marginal Rate of Substitution): for a small change of , how much change of is needed to obtain the same utility? Define marginal utility as the utility change for an infinitesimal change in the consumption of a commodity: . Taking the total differentiation of the equation (, ) = ¯, ∂ ∂ ⋅ + ⋅ = 0 ∂ ∂ ⇒ ∂ ∂ = − ∂ =− ∂ Note that MRS is independent of the utility representation Proof: Let (, ) = ((, )) where (⋅) is a monotonic transformation. It follows that ∂ ∂ ∂ = − ∂ = − ∂ ∂ ∂ ∂ ⋅ ⋅ ∂ ∂ ∂ ∂ ∂ ∂ = − ∂ ∂ If two utility functions have the same MRS, then they have the same indifference curve. 8 Properties of Indifference curve 1. Two indifference curve never intersect. 6 a b c - Proof: Since and are not on the same Indifference Curve and ≿ is complete, either ≻ or ≻ . However ∼ and ∼ , so that ∼ , which is a contradiction. 2. Convex to the origin: Diminishing Marginal Rate of Substitution ( ) > 0 ⇐⇒ (− ) < 0 ⇐⇒ ( ) < 0 Diminishing MRS means that the consumer weakly prefers the weighted average of two bundles to each of the two extreme bundles. Note that diminishing marginal utility does not imply diminishing marginal rate of substitution and the reverse does not hold true either. However the former is almost true in that diminishing marginal utility together with mild condition on the cross derivative implies diminishing marginal rate of substitution. 9 Examples of Indifference Curve 1. Cobb-Douglas utility function 6 - (, ) = ⋅ 1− , 0 < < 1 2. Complements 6 - (, ) = min{, } 10 3. Subsitutes 6 @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ - (, ) = + 4. Good and Bad () 6 J J J J J J J J - 11 () 2.2 Consumer’s Utility Maximization and Choice Budget constraint: Expenditure ≤ Income ⋅ + ⋅ ≤ ⇒ ≤ − ⋅ ⎛ , , : parameters ⎜ ⎜ ⎜ , : variables ⎝ : opportunity cost of consuming good Budget Constraint 6 Q Q slope = − Q Q Q Q Budget set QQ Q Q Q - ∙ If increases, the slope doesn’t change and the intercepts of and increase. 12 6 ′ Q Q Q Q slope = − Q Q Q Q Q Q  Q Q Q Q Q Q Q Q - Q ′ ∙ If increases, the slope becomes steeper and the intercept of decreases. 6 ′ Q SSQ slope = − S QQ S Q Q S slope = − Q S  Q Q S Q S Q - ′ ∙ If increases, the slope becomes flatter and the intercept of decreases. 6 Q slope = − ′ Q Q ? Q Q ′ PP slope = − Q PP PP Q PPQ PP Q Q - P Q 13 If initial endowment ( , ) is given, ⋅ + ⋅ ≤ ⋅ + ⋅ ⇒ ≤ ⋅ + ⋅ − ⋅ 6 + Q Q Q Q Q Q Q Q Q Q Q - + ∙ If increases, the slope doesn’t change. 6 Q Q Q Q slope = − Q Q Q Q Q Q  Q Q Q Q Q Q Q Q Q - ′ ∙ If increases, the slope becomes steeper. 14 6 S S ′ slope = − S Q S Q S Q S Q Q S Q SQ slope = − S Q Q S  Q S Q - ∙ If decreases, the slope becomes flatter. 6 slope = − Q Q Q XXX Q X slope = X QXX Q XXX Q XX Q Q 6 Q Q - Utility Maximization max (, ) s.t. ⋅ + ⋅ ≤ ℒ = (, ) + ( − ⋅ − ⋅ ) 15 ′ − ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ... = ⎜ ⎜ ⎜ ⎜ ⎝ ⇒ ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ ∂ℒ ∂ ∂ℒ ∂ = ∂ ∂ − ⋅ = 0 ∂ℒ ∂2 = ∂ ∂ − ⋅ = 0 = − ⋅ − ⋅ = 0 ∂ ∂ = ⋅ ⋅ ⋅ ⋅ ∗ ∂ ∂ = ⋅ ⋅ ⋅ ⋅ ∗∗ − ⋅ − ⋅ = 0 Divide * by ** side by side. ∂ ∂ ∂ ∂ = ⇒ MRS, the slope of indifference curve = Price ratio ⇔ Internal rate of substitution = External rate of substitution in market Graphical explanation If the budget is constrained and we must choose one point on the 1 indifference curve, would be an optimal solution. But in such a situation every points on the budget line between and ′ gives more utility. In result, , the contingent point of indifference curve and budget line is the utility maximization point. In this point, the slope of the budget line equals to the slope of the indifference curve. 16 6 @ ′ @ @ @ @ @ @ 2 @ @ @ 1 - Example1 (Cobb-Douglas utility function) (, ) = ⋅ 1− max ⋅ 1− s.t. ⋅ + ⋅ ≤ By Lagrangean method, ℒ = ⋅ 1− + ( − ⋅ − ⋅ ) By solving ... = (1 − ) ⋅ ⋅ , = Example2 (Perfect Complements) (, ) = min{, } max min{, } s.t. ⋅ + ⋅ ≤ ⇒ == + 17 6 @ @ @ ∗ ∗ @ ( , ) @ @ @ @ 2 q 1 - Example3 (Perfect Substitutes) (, ) = + max + = ¯ s.t. ⋅ + ⋅ ≤ We cannot solve perfect substitutes problem with Lagrangean method. In this case, as the slope of indifference curve is −1, we may obtain Bang-Bang solution. 6 @ @ @ @ @ @ [email protected] @@ @ HH @H @ @ @[email protected] @ @ [email protected] ∗ ∗ ) @ @ HH @ ( , q = , = 0 when <1 = 0, = 18 when >1 infinitely many solutions when 2.3 =1 Comparative statics What happens to the optimal solution when the parameter changes? Income changes 6 @ @ @ @ @ @ 2 @ @ @ 1 @ @ r r @ @ @ @ @ @ 1 2 ∙ ′ > ∙ 2 > 1 (normal good : ∂ ∂ ≥ 0) ∙ 2 > 1 (normal good) 19 2 1 - 6 @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ r r - 2 1 ∙ ′ > ∙ 2 > 1 (normal good) ∙ 2 < 1 (inferior good : ∂ ∂ < 0) Income Expansion Path This curve illustrates the bundles of goods that are demanded at different levels of income. 6 @ Income Expansion Path @ @ @ @ @ @ @ @ @ @ r @ @ @ r @ @ @ r @ @ @ @ @ @ @ @ @ @ @ - Engel Curve The engel curve is a graph of the demand for one of the goods as a function of income, with all prices being held constant. 20 demand 6 Engel curve of normal good : slope ≥ 0 Engel curve of inferior good : slope < 0 - income Definition 1 A function (1 , ⋅ ⋅ ⋅, ) is homogenous of degree if (1 , ⋅ ⋅ ⋅ , ) = (1 , ⋅ ⋅ ⋅ , ) Definition 2 Homothetic function is a positive monotonic transformation of a function which is homogenous of degree 1. Theorem 1 Suppose (, ) is homothetic. We can represent utility function as (, ) = ( (, )) where () is monotonic and (, ) is homogenous of degree 1. Then the income expansion path is straight line through origin. It means that when income increases, consumption of all commodities increases proportionally. 6 @ Income expansion path @ @ @ @ @ @ r @ @ @ @ r @ @ @ @ 21 @ @ - Price change ∙ ′ < ∙ ′ > ∙ ′ < 6 A @ [email protected] Price A @ A @ A @ A @ A @ A @ A @ A @ r a Expansion Path r b - Slutsky Decomposition Draw a new line contingent to the original indifference curve. ∙ substitution effect: 0 → 1 ∙ income effect: 1 → 2 ∂ ∂ ∂ ∂ = + ⋅ ∂ ∂ =¯ ∂ ∂ 22 6 @ A [email protected] A @ @ @A @ @A @ @ @ A @ [email protected] @ [email protected] @ A @ @ A @ r r r 0 1 2 - Properties of decomposition 1. Substitution effect is always negative because of diminishing MRS. ∂ <0 ∂ =¯ 2. Income effect may be negative or positive. ∂ > ), income effect dominates substitution ✓ In case of Giffen goods ( ∂ effect. Elasticity The price elasticity of demand, , is the percent change in quantity divided by the percent change in price. = ∙ point elasticity: ∙ arc elasticity: ∂ ∂ Δ Δ ⋅ ⋅ Δ Δ 23 2.4 Application:Labor supply 1. utility function : (, ) ∙ : leisure consumption ∙ : other consumption 2. budget constraint : + ⋅ ≤ ⋅ 24 ∙ price : = 1, = ∙ initial endowment : = 0, = 24 3. optimization ∙ max (, ) s.t. + ⋅ ≤ ⋅ 24 Wage increase makes the slope of budget line flatter and optimal solution changes from (0 , 0 ) to (1 , 1 ). (0 < 1 , 0 < 1 ) 24 1 0 ′ ′′ 6 @ A [email protected] A @ @ @A @ @A @ @ @ @ A @ @ [email protected] @ @[email protected] @ @A @ @ A @ @ @ r r r r 0 1 ∙ 0 → ′ : substitution effect ∙ ′ → ′′ : income effect 24 - ∙ ′′ → 1 : wealth effect Leisure increases when wage increases. Then is leisure an inferior good? No! Income effect and wealth effect are all positive. So leisure is not an inferior good but a normal good. Consequently it increases as the increase of wage because of the wealth effect. 25 3 PRODUCER THEORY 3.1 Technology and Production Firm is just Production Technology. Only production function is needed. This is similar to consumer who has only preference. But producer has no budget constraint since profit of producer is public and transferable. short-term vs. long term ∙ short-term : Some input factors are fixed. =⇒ The firm could make negative profit. ∙ long-term : All input factors are variable. =⇒ The firm is always free to go out of business. The least profit is zero profit. Production technology means converting input to output. Technology Production function = (, ) And it is characterized by = ∂ ∂ , = and ∂ ∂ = , = 26 Firm’s aim Firm aims to maximize profit. Max Profit, s.t. Technology = Max (Revenue - Cost), s.t. Technology. max = ⋅ − () where () = min ⋅ + ⋅ s.t. (, ) ≥ ✓ In firm’s decision, profit maximization is achieved only after cost is minimized. If a firm is not minimizing cost, then there is a way to increase profit. It is contradiction to profit maximizer hypothesis. Isoquant Set of (.) to produce a given output ¯ where ¯ = (.). Slope of isoquant,Marginal Rate of Technical Substitution is ∂ ∂ = − ∂ ∂ K 6 Isoquant - 27 L ✓ Law of diminishing MRTS → Law of diminishing Marginal Product (X) Law of diminishing MRTS ← Law of diminishing Marginal Product (X) (See the comments on diminishing MU and diminishing MRS in the consumer theory.) Properties of Technology 1. Monotonicity (, ) > ( ′ , ) if > ′ , .., > 0 2. Convexity Production possibility set is convex. 6 - cf. Convexity is not satisfied when there is Increasing Return to Scale. 6 - 28 Return to Scale (, ) is homogenous of degree iff (, ) = ⋅ (, ). (, ) exhibits ∙ IRS if > 1 ∙ CRS if = 1 ∙ DRS if < 1 3.2 Cost Minimization Cost minimization is a problem of constrained optimization. () = min ⋅ + ⋅ s.t. (, ) ≥ Using Lagrangean method, ℒ = ⋅ + ⋅ + ( − (, )) F.O.C. ∂ℒ ∂ = − + ⋅ =0 ∂ ∂ ∂ℒ ∂ = − + ⋅ =0 ∂ ∂ ∂ℒ = (, ) − = 0 ∂ 29 ⇒ Input price ratio equals to MRTS. = ∂ ∂ ∂ ∂ = Graphical explanation Let + = be iso-cost curve. The optimal point is on the isoquant that has the lowest associated isocost curve. 6 @ @ @ @@ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ r (, ) = - Note ∙ Giffen good is impossible to exist in producer. ∙ There are neither budget constraint nor income effect in production. Only substitution effect exists. ∙ Factor demand function is always downward sloping. Exmaple (Cobb-Douglas) (, ) = ⋅ 1− () = min ⋅ + ⋅ 30 s.t. ⋅ 1− ≥ Using Lagrangean method, ℒ = −( ⋅ + ⋅ ) + ( ⋅ 1− − ) F.O.C. ∂ℒ = − + (1 − ) − = 0 ∂ ∂ℒ = − + −1 1− = 0 ∂ ∂ℒ = 1− − = 0 ∂ In result, factor demand curve is ∗ = ( ) , ∗ = ( )1− 1− ⋅ ⋅ 1− and cost function is ∗ () = ⋅ ( ) +⋅ ( ) 1− 1− ⋅ ⋅ 1− Total cost Assume that price of input factor doesn’t change. TC 6 TVC - 31 Q = = + = = + ( + ) = = = C MC AC 6 AVC - Q ✓ = when AC is minimum. Proof: at minimum point ′ () ⋅ − () 1 () () = = ⋅[ −=0 2 () () = ⇐⇒ = Average cost: short-term vs. long term ∙ SAC curve is always on or above LAC curve. ≥ ∙ The lowest point of 1 and 3 is not contingent to LAC. 32 1 3 6 2 LAC - 3.3 Profit Maximization Profit () = Revenue - Cost = ⋅ − () max () = ⋅ − () F.O.C. = − = − = 0 ∙ Under perfect competition, ∙ Otherwise, ()⋅ = = = ′ + ∴ ′ + = 33 Q Supply curve C MC AC 6 AVC ′ ∗ - Q ∙ Under perfect competition, = = . ∙ When < , firm cannot gain even variable cost. So it has no reason to stay in the market. (∗ : shutdown point) When < < , firm cannot earn more than fixed cost, but more than variable cost. So operating factory is still profitable. Therefore the upward-sloping part of MC curve above AVC curve is supply curve. Profit () = Revenue - Cost = ⋅ − () max () = ⋅ − () = ⋅ (, ) − ( + ) F.O.C. ∂ (, ) =⋅ = ∂ ∂ (, ) =⋅ = ∂ ✓ The value of the marginal product of an input factor (VMP: value of marginal product of input) is equal to the price of that factor. 34 4 Market and Equilibrium Market exists because of diversity in preference and endowment. And equilibrium price induces transactions to eliminate excess demands. When this mechanism works most well, competition market occurs. 4.1 General Equilibrium Edgeworth Box two agents : agent 1 with utility function 1 (1 , 1 ) and endowment (¯ 1 , ¯1 ), agent 2 with 2 (2 , 2 ) and (¯ 2 , ¯2 ) two commodities, and B @ @ @ @ @ r b @ @ @ r [email protected]@ @ 1 @ A 1 r @ @ @ ∙ All the points in the box represent every consumable allocation. ∙ The horizontal length is total endowment of and the vertical length is total endowment of . 35 ∙ Designate initial endowment, {(¯ 1 , ¯1 ), (¯ 2 , ¯2 )} and draw a line passing through with respect to given relative prices. This line is the same budget constraint for both of two. ∙ Under the same price ratio, find utility maximization point only for agent 1 and also do the same thing for agent 2. ∙ Prices change until excess demand and excess supply disappear. ∙ We can find Equilibrium with new price line made by invisible hand. Agents are still price takers. B @ @ @ @ @ ¯ r b @ @ @ ¯ r [email protected]@ @ r @ = − @ ¯ A ¯ @ @ excess supply of : ∣¯ − ¯ ∣ excess supply of : ∣¯ − ¯ ∣ Walrasian Equilibrium ( , ) constitute Walrasian Equilibrium, if (1) given ( , ), individual optimize. (2) optimal demand from (1) clears market for and . 36 ∗ B Q Q Q Q Q Q Q Q ∗ Q Q ∗ r eQQ Q Q Q Q r A ∗ Q slpoe = Q Q Q ′ − ′ Equiliribrium in the Edgeworth box Walras’ Law ⋅ () = 0 where is excess demand. Considering an economy with commodities, if ( − 1) markets clear, then ℎ market clears automatically. Welfare implication B M A : Mutually beneficial trade area (Pareto improvement relative to the initial endowment) 37 Pareto Optimal Allocation An allocation such that there’s no alternative allocation which makes someone better off without making anyone else worse off. ✓ While Walrasian Equilibrium is a result of voluntary movement of market, Pareto optimal allocation is a result of someone’s handling. Contract curve : Set of Pareto optimal allocation. B s s s A Pareto set, itself, does not depend on the initial endowment First Welfare Theorem Every Walrasian Equilibrium is Pareto optimal. Second Welfare Theorem Pareto optimal allocation can be obtained through Walrasian Equilibrium with lump sum transfer. Example (Cobb-Douglas function) Walrasian Equilibrium is price vector such that agents maximize utility and demand equals to supply. Find the Walrasian 38 Equilibrium. 1 2 (, ) = 3 ⋅ 3 , (¯ 1 , ¯1 ) = (10, 0) (¯ 2 , ¯2 ) = (0, 20) 1. Two agents maximize their own utility. Agent 1: 1 2 max 3 ⋅ 3 s.t. ⋅ 1 + ⋅ 1 ≤ ⋅ ¯1 + ⋅ ¯1 Let = (normalization). Then 1 2 ℒ = 13 ⋅ 13 + (¯ 1 + ¯...
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