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lecture09_slides - Econ 121. Intermediate Microeconomics....

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Unformatted text preview: Econ 121. Intermediate Microeconomics. Eduardo Faingold Yale University Lecture 9 Outline of the course I. Introduction II. Individual choice III. Competitive markets IV. Market failure Outline of the course I. Introduction II. Individual choice             Budget constraint (Ch. 2) Preferences (Ch. 3) Utility (Ch. 4) Consumer problem (Ch. 5) Revealed preference (Ch. 7) Slutsky equation (Ch. 8) Endowment income effect (Ch. 9) Intertemporal choice (Ch. 10) Choice under uncertainty (Ch. 12) [First-midterm material ends here!] Consumer surplus (Ch. 14) Aggregate demand (Ch. 15) Partial equilibrium (Ch. 16) III. Competitive markets IV. Market failure Consumer’s surplus Want a measure of how much a person is willing to pay for something. How much a person is willing to sacrifice of one thing to get something else. Consider a discrete-good setting with quasi-linear utility: u.x1; x2 / D v.x1 / C x2; x1 D 0; 1; 2; : : : ; x2 2 RC: Normalizing the price of good 2 to p2 D 1, the consumer’s maximization problem (given income m and price p1) is to choose x1 to maximize v.x1 / over all x1 D 0; 1; 2; : : : . p1x1 C m Consumer’s surplus The solution is given by reservation prices: r.1/ D v.1/ v .0/; r.2/ D v.2/ v .1/; : : : ; r.k/ D v.k/ v .k 1/; so that  x1 .p1/ D k if and only if r.k C 1/ < p1  r.k/: The reservation price for the k th unit measures consumer’s marginal willingness to pay for an extra unit when he has k 1 units already. Natural to assume r.1/ > r.2/ > r.2/ >    That is, consumer is willing to pay more for an extra unit when he has fewer units. Consumer’s surplus Add up over all different outputs to get total willingness to pay for k units. Total willingness to pay for k units D r.1/ C r.2/ C : : : C r.k/ This is a measure of the consumer’s benefit from consuming k units, since r.k/ D v.k/ v .k 1/ and therefore, r.1/ C    C r.k/ D .v.1/ D v.k/ v .0// C .v.2/ v .1// C    .v.k/ v .k v .0/ The sum r.1/ C    C r.k/ is called the consumer’s (gross) surplus. To get the consumer’s net surplus must subtract the total amount that the consumer has to spend to get the benefit.. 1// Continuous demand Suppose utility has form u.x; y/ D v.x/ C y (quasilinear) Thus, the inverse demand curve has form p .x/ D v 0.x/. By the Fundamental Theorem of Calculus, v.x/ v .0/ D Z x 0 v .t/ dt D 0 Z x p .t/ dt : 0 This is the generalization of the discrete-goods argument. If utility is not quasilinear, things are a bit complicated. We’ll not see this in this course. In general, we are interested in the change in consumer’s surplus due to some policy (e.g., imposing a tax). Market demand  To get market demand, just add up individual demands – add inverse demands horizontally, – properly account for zero demands  Example: linear demand Market demand  Often think of market behaving like a single individual.  This is the so-called representative consumer model.  Not valid in general, but reasonable assumption sometimes. (Common assumption in macroeconomics.)  Inverse of aggregate demand curve measures the MRS of the representative individual. Elasticity  measures responsiveness of demand to price in percentage terms  If p.q/ is the inverse demand function, then elasticity is given by p1 p dq D D : 0 .q/ qp q dp  Linear demand: q D a bp . Then, bp p dq D D q dp a bp  Note that in the linear demand case,  D the demand curve 1 when we are halfway down Elasticity  Now consider q D Ap b .Then, D p bAp b Ap b 1 D b  thus elasticity is constant along this demand curve  note that log q D log A b log p , i.e., log-linear demand  what does elasticity depend on? In general, how many and how close substitutes a good has. How does revenue change when you change price?  Consider a downward sloping demand q.p/.  R D pq.p/, so R0.p/ D q C pq 0.p/ D q.1 j.p/j/  If price increases, two effects: lower q and higher p . Who wins? Depends on elasticity.  R0.p/ > 0 if and only if j j < 1 (inelastic).  Increase in p dominates decrease in q for inelastic demand.  This is a local phenomenon! Consider linear demand case Partial equilibrium  We will maintain the setup in which preferences are quasi-linear: u.q1; q2/ D v.q1/ C q2  He will normalize the price of good 2 to 1 and will focus on the demand for good 1 as a function of its price p . The first-order condition yields: p D v 0.q/ where q D q1 hereafter. So, the marginal utility v 0.q/, as a function of the quantity q , is the inverse demand.  In general, we will be given the inverse demand function Pd .q/ directly, or the demand function D.p/. Supply function  A supply function S.p/ measures the amount the supplier is willing to supply at each price.  We let Ps .q/ designate the inverse supply function.  Supply functions come from the firm’s profit maximization. We have not seen that yet, so we will take the supply functions as primitives. Equilibrium  Competitive market - each agent takes prices as given (outside of their control)  Foundations for competitive behavior: large markets (many small agents)  Equilibrium price - the price where desired demand equals desired suply D.p/ D S.p/  Special cases – vertical supply - quantity determined by supply, price determined by demand – horizontal supply - price determined by supply, quantity determined by demand Equilibrium  equivalent definition of equilibrium: where inverse demand crosses inverse supply Pd .q/ D Ps .q/  Linear model: a b q  D.q/ D S.q/  c C d q ∴ qD ac bCd Comparative Statics  For example, in linear model can increase or decrease the intercept of Pd .q/.  In general, can change a parameter of the functions Pd .q/ and/or Ps .q/ and study the effect in the equilibrium price and quantity.  Changing a preference parameter, for instance, affects demand because it affects the marginal utility v 0.q/, which is the inverse demand function. Taxes  Consider quantity taxes.  If payed by consumer, then posted price is ps and consumer effectively pays pd D ps C t  If payed by supplier, then posted price is pd and supplier effectively receives ps D pd t  equilibrium happens when D.pd / D S.ps / or equivalently, D.ps C t/ D S.ps / Passing along a tax  horizontal supply function  vertical supply function Deadweight loss of a tax  change in consumer’s surplus  change in producer’s surplus  tax revenue  deadweight loss Price
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