Microeconomics. Please remember Spring 2018

Transcription

1 Microeconomics Please remember Spring 2018 "The time has come," the Walrus said, "To talk of many things: Of shoes - and ships - and sealing-wax - Of cabbages - and kings And why the sea is boiling hot - And whether pigs have wings" (Lewis Carroll) 1

2 memorer-micro 2 1 Prices and income Prices: p 1 and p 2 Income: m Prices always are supposed to be positive (Handling prices 0 is difficult) Budget restriction (and budget set): p 1 x 1 + p 2 x 2 m Budget line: p 1 x 1 + p 2 x 2 = m Thus x 2 = m p 2 p1 p 2 x 1 A value tax t replaces a price p by (1 + t)p A quantity tax t replaces a price p by p + t 2 Relations 3 Utility A relation R is given by a set X and for each x, y X a statement xry that is true or false A relation R is reflexive if xrx, for all x; complete if xry or yrx, for all x, y; transitive if [xry and yrz] xrz, for all x, y, z; symmetric if xry yrx, for all x, y Each complete relation is reflexive Relations in consumer theory A good bundle is a sac which contain amounts of the types of goods that are present in the economy There are infinitely many sacs A preference relation (for a consumer) is a reflexive, complete and transitive relation on the set of good bundles Each consumer may have his own specific preference relation A B means good bundle A is at least as good as B A B means A is better than B A B means A is as good as B Each consumer may have his own specific preference relation Most problematic is transitivity assumption From now on we make the unrealistic but not bad hypothesis that there are two types of goods and that good bundles have the form (x 1, x 2 ) where x i denotes the amount of good i Often one assumes monotone preferences, ie the more the better ( Principle of the swine ) and convex preferences, ie set of good bundles that is at least as good as a given good bundle is convex Remember (from mathematics): a set is convex if for any two points in the set the whole line segment between these points is in the set Also one often deals with homothetic preferences: (x 1, x 2 ) (y 1, y 2 ) λ(x 1, x 2 ) λ(y 1, y 2 ) for all λ > 0 The sets of good bundles for which the consumer is indifferent are called indifference set Remark: these sets often are, par abus de langage, called curves Indifference sets cannot intersect Under weak conditions (You do not need to know) a preference relation can be represented by a utility function u This means that (x 1, x 2 ) (y 1, y 2 ) is equivalent with u(x 1, x 2 ) u(y 1, y 2 ), (x 1, x 2 ) (y 1, y 2 ) is equivalent with u(x 1, x 2 ) > u(y 1, y 2 ), (x 1, x 2 ) (y 1, y 2 ) is equivalent with u(x 1, x 2 ) = u(y 1, y 2 ),

3 memorer-micro 3 Utility is ordinal in the sense that each strictly increasing transformation of the utility function describes the same economic situation Only cardinal utility later when we deal with uncertainty Specific (important) utility functions Cobb-douglas: u(x 1, x 2 ) = x α1 1 xα2 2 Solow: u(x 1, x 2 ) = α 1 x 1 + α 2 x 2 (Perfect substitutes) Leontief: u(x 1, x 2 ) = min (x 1 /α 1, x 2 /α 2 ) (Perfect complements) Maximum: u(x 1, x 2 ) = max (x 1 /α 1, x 2 /α 2 ) Quasi-linear: u(x 1, x 2 ) = v(x 1 ) + x 2 (Good 1 is quasi-linear) Special quasi-linear: u(x 1, x 2 ) = α x 1 + x 2 Cobb-Douglas, Solow, Leontief and many quasilinear utility functions represent convex preferences The maximum utility function does not Indifference An indifference curve x 2 (x 1 ) is such that u(x 1, x 2 (x 1 )) = constant Marginal rate of substitution at a good bundle: 1 In case x 2 (x 1 ) is an indifference curve, MRS := u x 1 / u x 2 MRS = dx 2 dx 1 For cobb-douglas: MRS = α 1 x 2 α 2 x 1 For homothetic preferences the MRS is constant along each ray through the origin For quasi-linear utility functions the indifference curves are vertical translates of each other Method of Lagrange Optimisation problem: maximise (or minimise) a function f(x 1,, x n ) under m restrictions g i (x 1,,, x n ) = 0 (1 i m) Lagrange function L = f λ 1 g 1 λ m g m L In optimum: x 1 Two laws of Gossen = 0,, L x n = 0 Gossen s first law about diminishing marginal utility is nonsense But decreasing MRS makes sense Gossen s second law: p 1 /p 2 = u x 1 / u x 2 (see below) Gossen s second law can be proved graphically or by the method of Lagrange Marshallian demand functions They are the solutions of the following utility maximisation problem: maximise u(x 1, x 2 ) under the budget restriction p 1 x 1 + p 2 x 2 m Denoted by x 1 (p 1, p 2 ; m), x 2 (p 1, p 2 ; m) For various cases (like for the cobb-douglas function) they can be determined by solving p 1 /p 2 = u x 1 / u x 2 p 1 x 1 + p 2 x 2 = m, (Gossen s second law) 1 In the book and elsewhere in economics also MRS is defined as u x 1 / u x 2, ie with a minus sign

4 memorer-micro 4 4 Types of goods For cobb-douglas: x 1 (p 1, p 2 ; m) = α 1 α 1 + α 2 m p 1, x 2 (p 1, p 2 ; m) = α 2 α 1 + α 2 m p 2 For leontief: x 1 /α 1 = x 2 /α 2 holds (optimum at kink) For Solow: optimum is one boundary point or whole budget line is optimal For Maximum: optimum is one or two boundary points The marshallian demand of a quasi-linear good is independent of the income if this income is large enough (and for such incomes often can be determined with Gossens s second law) Consider a price change or income change of good i Good i is called giffen, if p i x i In a formula: xi p i > 0 ordinary, if p i x i In a formula: xi p i < 0 normal, if m x i In a formula: xi m > 0 inferior, if m x i In a formula: xi m < 0 Consider two goods i and j Good i is called a substitute for good j, if p j x i a complement for good j, if p j x i Substitution and income effect à la Slutsky 5 Elasticity Consider a price change from p 1 to p 1 for good 1 Let (x 1, x 2 ) be the old and (x 1, x 2) be the new optimal good bundle Let m be the value of (x 1, x 2 ) at prices p 1, p 2 With m := m m one has m = x 1 p 1 Let (x 1, x 2) be the optimal good bundle at p 1, p 2 and m So we have the following three optimal situations: p 1 p 2 m x 1 x 2 ; p 1 p 2 m x 1 x 2; p 1 p 2 m x 1 x 2 Define: x tot 1 := x 1 x 1 ; x subs 1 := x 1 x 1 ; x inc 1 := x 1 x 1 Slutsky equation: x tot 1 = x subs 1 + x inc 1 Law of Slutsky: The substitution effect is negative, ie the sign of x subs 1 is the opposite sign of p 1 Law of demand: each giffen good is inferior For a quasi-linear good there is no income effect if the income is large enough Setting: variable b depends on variable a, ie a function b(a) Elasticity is measure for how sensible b depends on a Elasticity should be independent of units Two types of elasticity: segment and point elasticity We only deal with point elasticities Formula for point elasticity ɛ = db a da b Interpretation: relative change of b divided by relative change of a ɛ < 1: inelastic; ɛ > 1: elastic

5 memorer-micro 5 In case of an affine function, ie q = αp+β (with α < 0 and β > 0) and p on the ordinate, each point above middle of graph is elastic and each point below is inelastic In middle the elasticity is 1 Good to know: b = Aa α has ɛ = α ɛ = d ln b d ln a Consider the price elasticity of demand: ɛ = dq Revenue R(p) = p q(p) dr p dp R = 1 + ɛ If inelastic, then p R If elastic, then p R p dp q Perfect inelastic demand curve: curve is vertical Perfect elastic demand curve: curve is horzontal Consider a marshallian demand function x i Income elasticity of demand η i = xi m m x i Luxury good i: η i > 1 Necessary good i: η i < 1 6 Market demand and partial equilibrium Setting: decreasing demand function D(p) and increasing supply function S(p) Inverse demand function P (Q) Without taxes equilibrium price p is determined by Equilibrium with taxes: S(p ) = D(p ) Two prices: p D (price buyer pays) and p S (price seller gets) In case of a positive quantity tax: p D = p S + t: equilibrium prices determined by p D = p S + t, D(p D) = S(p S); result p S p p D ; p D p is part of tax passed along the consumer; it is the smaller the more elastic D and the more inelastic S p p S is part of tax passed along the producer; it is the smaller the more elastic S and the more inelastic D 7 Returns to scale for production function f(k 1, k 2 ) Constant returns to scale f(tk 1, tk 2 ) = tf(k 1, k 2 ) for all t > 1 Increasing returns to scale f(tk 1, tk 2 ) > tf(k 1, k 2 ) for all t > 1 Deceasing returns to scale f(tk 1, tk 2 ) < tf(k 1, k 2 ) for all t > 1 For cobb-douglass production function Ak β1 1 kβ2 2 : β 1 + β 2 = 1: constant returns to scale β 1 + β 2 > 1: increasing returns to scale β 1 + β 2 < 1: decreasing returns to scale 8 Cost minimisation Isoquant: set of production factor bundles with same output

6 memorer-micro 6 Technical rate of substitution: In case k 2 (k 1 ) is an isoquant, TRS := f k 1 / f k 2 TRS = dk 2 dk 1 Conditional production factor demand functions k 1 (w 1, w 2 ; q), k2 (w 1, w 2 ; q) are the solutions of the cost minimisation problem Each of the above specficic utility function has its own solution for the cost minimisation problem For example for the cobb-douglas utility function they can be determined by solving the two equations f(k 1, k 2 ) = q, w 1 /w 2 = f k 1 / f k 2 Cost function: c(q; w 1, w 2 ) = w 1 k1 (w 1, w 2 ; q) + w 2 k2 (w 1, w 2 ; q) Various cost functions: fixed costs c(0), marginal costs dc dq, variable costs c(q) c(0), average c(q) c(0), average variable costs costs c(q) q q Increasing returns to scale leads to decreasing average costs Decreasing returns to scale leads to increasing average costs Constant returns to scale leads to constant average costs Principle of the marginal leads the average : as long as the marginal is larger than the average, the average increases and as long as the marginal is less than the average, the average decreases In the short term some production factors are fixed In the long term there are no fixed costs 9 Profit maximisation Input perspective: profit function π(k 1, k 2 ) = pf(k 1, k 2 ) (w 1 k 1 + w 2 k 2 ) Output perspective: profit function Π(q) = pq c(q; w 1, w 2 ) Both perspectives give the same results; this is intuitively clear but not formally Production factor functions k 1 (p; w 1, w 2 ), k 2 (p; w 1, w 2 ) are the solutions of the profit maximisation problem For input perspective they often can be determined by solving: p f = w 1, p f = w 2 k 1 k 2 And for output perspective by solving: p = c (q) Profit maximisation is (for arbitrary prices) not compatible with constant and with increasing returns to scale (This one can easily check for the production function f k = k α ) Short term: Let p be the minimum of the average variable cost curve: this is the shutdown price Supply curve is marginal cost curve for p p For p < p supply is zero 10 Uncertainty Consider a decision maker with uncertainty about the outcomes of given choices he can made For simplicity we assume here that the outcomes are amounts of money; a negative amount means a loss

7 memorer-micro 7 The Saint-Petersbourgh paradox and other examples make clear that we need some concept other than expected money value to analyse how people make decisions in risky situations Model: The choices the decision maker can make concern gambles (Simple) gamble = lottery ticket with various possible outcomes that occur with a certain probability For simplicity we further assume that there are two possible outcomes c 1, c 2 Then π 1 + π 2 = 1 Notation for such an gamble: ((c 1, π 1 ), (c 2, π 2 )) If π 1 = 1, then the gamble ((c 1, π 1 ), (c 2, π 2 )) leads to outcome c 1 for sure We denote such a gamble also by (c 1 ) or just by c 1 Assumption: the decision maker has a preference relation over the set of gambles and makes choices according to this relation Analysis of the model Under reasonable conditions (including completeness, transitivity) there exists a utility function U defined on the set of gambles representing the preference relation There even exists a utility function with the expected utility property, ie it assigns to each gamble the expected value of the utilities that might result: U((c 1, π 1 ), (c 2, π 2 )) = π 1 U(c 1 ) + π 2 U(c 2 ) Such utility function with the expected utility property is called a Von-Neumann-Morgenstern utility function Von-Neumann-Morgenstern utility functions are unique op to positive affine transformations Utility here is cardinal (and not ordinal)! Given a Von-Neumann-Morgenstern utility function U and a gamble g = ((c 1, π 1 ), (c 2, π 2 )), we have the following notions: Types of risk U(g) : expected utility of g; E(g) = π 1 c 1 + π 2 c 2 : expected money) value of g; U(E(g)) : utility of expected money value of g Risk loving: U(π 1 c 1 + π 2 c 2 ) < π 1 U(c 1 ) + π 2 U(c 2 ) (Convex U) Risk neutral: U(π 1 c 1 + π 2 c 2 ) = π 1 U(c 1 ) + π 2 U(c 2 ) (Affine U) Risk averse: U(π 1 c 1 + π 2 c 2 ) > π 1 U(c 1 ) + π 2 U(c 2 ) (Concave U) 11 Compensating variation CV and equivalent variation EV: Setting: consider a change of prices and budget For simplicity we only consider a change of the price of good 1 The price is p 1 and becomes p 1 This leads for the consumer to a utility change (new utility minus old utility) The compensating variation CV is the amount of money that when taken away from the consumer after the price change cancels the utility change The equivalent variation EV is the amount of money that when given to the consumer before the change gives the same utility effect as the the price change CV and EV have the same sign as the utility change but need not to be equal 2 Let v(p 1, p 2 ; m) be the indirect utility function ie the maximal utility the consume can reach given the prices and income It can be calculated by substituting the optimal good bundle into the utility function As p 2 is constant, we simply write v(p 1 ; m) 2 In the literature sometimes other definitions of CV and EV are used, by using in another way the above taken away and given to" These definitions only may cause an extra minus-sign in the result Do not worry about this

8 memorer-micro 8 v(p 1 ; m) = v(p 1; m CV), v(p 1 ; m + EV) = v(p 1, m) Because utility is not observable, it is a problem to determine CV and equivalent variation EV 12 Consumer s surplus Let x 1 be the (decreasing) demand function for good 1 (Net) consumer s surplus at price p 1 : S(p 1 ) := p 1 x 1 (p) dp Ie area above price under demand curve Gross consumer s surplus at price p 1 : S tot (p 1 ) = S(p 1 ) + p 1 x 1 (p 1 ) If price changes from p 1 to p 1, then change in net consumer s surplus equals S = S(p 1) S(p 1 ) = p 1 x p 1 (p)dp 1 Consumer s surplus is observable as demand is observable Under mild conditions, S is an average of CV and EV If good 1 is quasi-linear, then under mild conditions CV = EV = S holds Summing the consumer s surpli of the consumers leads to the consumers surplus 13 Producer s surplus Setting: only price of output good changes Producer s surplus at price p equals area under price p above supply curve Summing the producer s surpli of the producers leads to the producers surplus 14 Cost benefit analysis: calculation of costs and benefits of various economic policies happens in practice often by calculating the effect of the policy on the total surplus, ie on the sum of the consumers and producers surplus 15 Game theory Traditional game theory deals with mathematical models of conflict and cooperation in the real world between at least two rational intelligent players Traditional because of rationality assumption Rationality and Intelligence are completely different concepts A game can have different outcomes Each outcome has its own payoffs for each of the players Interpretation of payoff: satisfaction at end of game Nature of payoff: money, honour, activity, nothing at all, utility, real number, A game in strategic form is characterized by n players who choose simultaneously and independently a strategy Many games have already have this form or can be represented in a natural way (like chess) in this form A special case of a game in strategic form is a bi-matrix game where there are two players and each player has a finite number of strategies Strategy of a player: completely elaborated plan of play A game in strategic form is called zero-sum if the sum of the payoff functions is identical zero Strictly dominant strategy of a player: the best strategy of that player independently of strategies of the other players Strictly dominant equilibrium: strategy profile where each player has a strictly dominant strategy Nash equilibrium: strategy profile such that no player wants to deviate from it (Strongly) pareto-efficient strategy profile: a strategy profile for which there is no other strategy profile in which at least one player is better of and no player is worse off

9 memorer-micro 9 Weakly pareto-efficient strategy profile: a strategy profile for which there is no other strategy profile in which each player is better off Full cooperative strategy profile: a strategy profile that maximizes the total payoff Each full cooperative strategy profile is pareto efficient Prisoners dilemma: a game in strategic form where there is a strictly dominant weakly paretoinefficient nash equilibrium 16 Oligopolies Actions may be price actions or quantity actions And they may be simultanesously or sequentially This leads to four possible oligopoly models Cournot: simultaneous quantity actions Von-Stackelberg: sequential quantity actions Bertrand: simultaneous price actions Price leadership: sequential price actions Cournot model (duopoly): Setting: inverse demand function p(q) and cost functions c i (i = 1, 2) Player (oligopolist) i has profit function π i q 1, q 2 ) = p(q 1 + q 2 )q i c i (q i ) The nash equilibrium (cournot equilibrium) can (often) be determined by solving π 1 q 1 = 0, Or with R 1 and R 2 the best reply functions π 2 q 2 = 0 R 1 (q 2 ) = q 1, R 2 (q 1 ) = q 2 Collusion: the full cooperative strategy profile (q 1, q 2 ) can (often) be determined by solving c 1 (q 1 ) = c 2 (q 2 ) This strategy profile is in general not a nash equilibrium Cournot oligopoly: analysis in same way If number of oligopolists becomes larger and larger, the optimal condition becomes price = marginal costs Von-Stackelberg model (duopoly): Oligopolist 1 is the leader and oligopolist 2 is the follower Leader maximises π 1 (q 1 ) = p(q 1 + R 2 (q 1 ))q 1 c 1 (q 1 ) This gives optimal q 1 Then q 2 = R 1 (q 1) 17 Pure exchange economy Setting: 2 consumers and 2 goods There is an initial allocation: each consumer h has amount ω h i of good i O i = ωi 1 + ω2 i is the total amount of good i in the economy There is an auctioneer who determines the prices p 1, p 2 Consumers will exchange goods Each consumer h has a utility function u h Given prices, consumer h has an income m h = p 1 ω h 1 + p 2 ω h 2 Feasible allocation: each allocation that can be realized by exchange starting from the initial allocation

10 memorer-micro 10 Demand for good i of consumer h: So this demand is calculated with income m h Excess demand of good i for consumer h: Aggregate excess demand of good i: Market for good i is in equilibrium if z i = 0 Equilibrium prices are prices for which ie for which all markets are in equilibrium x h i (p 1, p 2 ; m h ) e h i = x h i ω h i z i = e 1 i + e 2 i z 1 = 0, z 2 = 0, Law of Walras: p 1 z 1 + p 2 z 2 = 0 holds for all prices This implies: if there is equilibrium in one market, then there also is equilibrium in the other market Under mild conditions (always satisfied for the situations we deal with), there exists equilibrium prices This is referred to as walrasian (or competitive) equilibrium First welfare Theorem: each equilibrium allocation is pareto efficient Second welfare theorem: each pareto efficient allocation can be supported as a walrasian equilibrium Box of Edgeworth(-Bowley) in case of two consumers A and B: this is a box with horizontal size O 1 and vertical size O 2 It represents the feasible allocations Indifference curves for A are drawn with south-west corner as origin and those for B with noth-east corner as origin Allocation is pareto efficient if MRS A = MRS B 18 Impossibilty theorem: the impossibility theorem of Arrow states that if, in the case of at least three alternatives, a social decision mechanism satisfies the completeness criterium, the reflexivity criterium, the transitivity criterium, the pareto-criterium and the independence of other alternatives crtiterium, then this mechanism is a dictatorship