Intermediate Microeconomics (22014)

Transcription

1 Intermediate Microeconomics (22014) I. Consumer Instructor: Marc Teignier-Baqué First Semester, 2011

2 Outline Part I. Consumer 1. umer 1.1 Budget Constraints 1.2 Preferences 1.3 Utility Function Slutsky Equation under

3 Budget Constraints Preferences Utility Function Slutsky Equation TOPIC 0. CONSUMER THEORY REVIEW Budget Constraints Denitions The consumer's budget set is the set of all aordable bundles, B (p 1,..,p n ;m) = {(x 1,...,x n ) : x 1 0,...,x n 0 and p 1 x p n x n m}. The budget constraint is the upper boundary of the budget set. x 2 Budget constraint: p1x1 + p2x2 = m m /p 2 m p1 x 2 x 1 p 2 p 2 Budget set: the collection of all affordable bundles. Budget Set m /p 1 x 1

4 Budget Constraints Preferences Utility Function Slutsky Equation Preferences Denitions The set of all bundles equally preferred to a bundle x' is the indierence curve containing x'. The slope of the indierence curve at x' is the marginal rate of substitution (MRS) at x', which is the rate at which the consumer is only just willing to exchange commodity 2 for commodity 1': ( ) MRS(x x2 ) = lim = dx 2 x 1 0 x 1 dx 1 x 2 x x x 2 x x x z x y z x x 1 y x 1

5 Budget Constraints Preferences Utility Function Slutsky Equation Utility Function Denition A Utility function U(x) represents a preference relation if and only if x' x U(x') > U(x) x' x U(x') < U(x) x' x U(x') = U(x) The general equation for an indierence curve is U(x 1,x 2 ) = k, where k is a constant. Totally dierentiating this identity we obtain that the MRS is equal to the ratio of marginal utilities: U x 1 dx 1 + U x 2 dx 2 = 0 dx 2 dx 1 = U x 1 U x 2

6 Budget Constraints Preferences Utility Function Slutsky Equation Denition A decisionmaker chooses the most preferred aordable bundle, which is called the consumer's ordinary demand or gross demand. The slope of the indierence curve at ordinary demand (x 1,x 2 ) equals the slope of the budget constraint: MRS(x 1,x 2) = p 1 p 2 U/ x 1 U/ x 2 = p 1 p 2 x 2 More preferred bundles x 2 * Affordable bundles x1 * x 1

7 Budget Constraints Preferences Utility Function Slutsky Equation Slutsky Equation Changes to demand from a price change are always the sum of a pure substitution eect and an income eect. Pure substitution eect: change in demand due only to the change in relative prices. What is the change in demand when the consumer's income is adjusted so that, at the new prices, she can only just buy the original bundle? Income eect: if, at the new prices, less income is needed to buy the original bundle then real income is increased; if more income is needed, then real income is decreased.

8 Slutsky Equation graphically x 2 Budget Constraints Preferences Utility Function Slutsky Equation x 2 x 2 x 1 x 1 Pure Substitution Effect Only x 1 x 2 Adding now the income effect x 2 x 2 x 1 x 1 x 1

9 Budget Constraints Preferences Utility Function Slutsky Equation Slutsky Equation formally Let (p 1,p 2 ) be the initial price vector, m the income level and (x 1,x 2 ) the initial gross demand. Dene the Sltusky demand function x1 s as the demand of good i after the price change when income is adjusted to give the consumer just enough to consume the initial bundle (x 1,x 2 ): x s i ( p 1,p 2,x 1,x ) 2 xi p 1,p 2; p 1x 1 + p 2x 2 } {{ } m Take the derivative with respect to p 1 on both sides, x1 s = x 1 + x 1 p 1 p 1 m x 1 x 1 x1 s = x 1 p 1 p }{{} 1 m x 1 } {{ } substitution eect income eect

10 Budget Constraints Preferences Utility Function Slutsky Equation Slutsky Equation formally The sign of the susbstitution eect x s 1 p 1 is negative: the cange in demand due to the susbstitution eect is the opposite to the change in price (p 1 x s 1 and p 1 x s 1 ). If the good is normal, the sign of the income eect is also negative: an increase in a price is like a decrease in income, which leads to a decrease in demand; a price fall is like an income increase, which leads to an increase in demand. x 1 = x 1 s x 1 p 1 p }{{} 1 m x 1 } {{ } (+) ( )} {{ } ( )

11 Outline Part I. Consumer Endowments Net Demand Slutsky Equation Labor Supply 1. umer Endowments 2.2 Net Demand 2.3 Slutsky Equation 2.4 Labor Supply under

12 TOPIC 1. BUYING AND SELLING Endowments Net Demand Slutsky Equation Labor Supply So far, consumers' income taken as exogenous and independent of prices. In reality, consumers' income coming from exchange by sellers and buyers. How are incomes generated? How does the value of income depend upon commodity prices? How can we put all this together to explain better how price changes aect demands?

13 Endowments Net Demand Slutsky Equation Labor Supply Endowments In this chapter, consumers get income from endowments. This makes the budget set denition change slightly. Denition The list of resource units with which a consumer starts is her endowment, denoted by ω = (ω 1,ω 2 ). Denitions Given p 1 and p 2, the budget constraint for a consumer with an endowment ω = (ω 1,ω 2 ) is p 1 x 1 + p 2 x 2 = p 1 ω 1 + p 2 ω 2. and the budget set is formally dened as {(x 1,x 2 ) x 1 0,x 2 0 and p 1 x 1 + p 2 x 2 p 1 ω 1 + p 2 ω 2 }.

14 Endowments Net Demand Slutsky Equation Labor Supply Endowments and Budget Sets Graphically, the endowment point is always on the budget constraint. Hence, price changes pivot the constraint around the endowment point. x 2 p 1x 1 p 2x 2 p 1 1 p 2 2 Budget Set before price change Budget Set after the price change p1 ' x1 p ' 2x2 p1 ' 1 p ' 2 2 x 1

15 Endowments Net Demand Slutsky Equation Labor Supply Net demand Denition The dierence between nal consumption and initial endowment of a given good i, x i ω i, is called net demand of good i. The sum of the values of net demands is zero: x 2 p 1 x 1 + p 2 x 2 = p 1 ω 1 + p 2 ω 2 p 1 (x 1 ω 1 ) + p 2 (x 2 ω 2 ) = 0. At prices (p 1,p 2 ) the consumer sells units of good 2 to acquire more of good 1. The net demand of good 1 is, therefore, positive and the net demand dof good 2 is negative. x 2 * x 2 At prices (p 1,p 2 ) the consumer sells units of good 1 to acquire more units of good 2. The net demand of good 1 is, therefore, negative,and the net demand of good 2 is positive. ii x 2 * x 1 * 1 ' 1 ' ' 1 ' 2 2 x 1 px px p p p1( x1 1) p2( x2 2) 0 x 1 * x 1

16 Endowments Net Demand Slutsky Equation Labor Supply Price oer curve Denition Price-oer curve contains all the utility-maximizing gross demands for which the endowment is exchanged. x 2 Sell good 1, buy good 2 Buy good 1, sell good 2 x 1

17 Endowments Net Demand Slutsky Equation Labor Supply Slutsky equation revisited In an endowment economy, the overall change in demand caused by a price change is the sum of a pure substitution eect, an (ordinary) income eect, and an endowment income eect. Pure Substitution Eect: eect of relative prices change. Income Eect: eect of original bundle cost change. Endowment Income Eect: change in demand due only to the change in endowment value. x 2 Price change from (p 1,p 2 ) to (p 1, p 2 ): Pure substitution effect Ordinary income effect Endowment income effect x 2 2 x 2 x 1 1 x 1 x 1

18 Endowments Net Demand Slutsky Equation Labor Supply Slutsky Equation revisited Let x 1 (p 1,p 2 ;m (p 1,p 2 )) be the demand function of good 1 and m (p 1,p 2 ) = p 1 ω 1 + p 2 ω 2 the money income. Then, the total derivative of x 1 with respect to p 1 is dx 1 dp 1 = x 1 p 1 + x 1 m ω 1. Using that x 1 p 1 = x 1 s p 1 x 1 x m 1, we obtain dx 1 dp 1 = Rearranging, x s 1 p 1 }{{} substitution x 1 m x 1 } {{ } ord.-income + x 1 m ω 1. } {{ } end.-income dx 1 = x 1 + (ω 1 x 1 ) x 1. dp 1 p }{{} 1 }{{} m ( ) (+)

19 Endowments Net Demand Slutsky Equation Labor Supply Slutsky equation revisited Overall change in demand of normal good (demand increases with income) caused by own price change: When income is exogenous, both the substitution and (ordinary) income eects increase demand after an own-price fall; hence, a normal good's ordinary demand curve slopes down (thus, Law of Downward-Sloping Demand always applies to normal goods when income is exogenous). When income is given by initial endowments, endowment-income eect decreases demand if consumer supplies that good (negative net demand); thus, if the endowment income eect osets the substitution and the (ordinary) income eects, the demand function could be upward-sloping!

20 Endowments Net Demand Slutsky Equation Labor Supply An application: labor supply Environment description: A worker is endowed with m euros of nonlabor income and R hours of time. Consumption good's price is p c, and the wage rate is w. Worker decides amount of consumption good, denoted by C, and amount of leisure, denoted by R. Budget constraint: p c C = m + w ( R R ) p c C + wr } {{ } Expenditures value = m + wr } {{ } Endowment value

21 Endowments Net Demand Slutsky Equation Labor Supply Labor supply choice m wr p c C* m C Budget constraint equation: C w m w R R p c p c Endowment point R* R R leisure demanded labor supplied

22 Endowments Net Demand Slutsky Equation Labor Supply Labor supply curve Eect of a wage rate increase on amount labor supplied: Substitution eect: leisure relatively more expensive decrease leisure demanded / increase labor supplied. (Ordinary) income eect: cost original bundle increases decrease leisure demanded / increase labor supplied. Endowment-income eect: positive endowment income eect because worker supplies labor decrease leisure demanded / increase labor supplied. Labor supply curve may bend backwards.

23 Outline Part I. Consumer Present and Future Values Constraint Ination Valuing Securities 1. umer Present and Future Values 3.2 Budget Constraint Ination 3.5 Valuing Securities 4. under

24 TOPIC 2. INTERTEMPORAL CHOICE Present and Future Values Constraint Ination Valuing Securities So far, only static problems considered, as if consumers only alive one period or only static decisions. However, in the real world people often make intertemporal consumption decisions: Current consumption nanced by borrowing now against income to be received in the future. Extra income received now spread over the following month (saving now for consumption later). In this section, we study intertemporal choice problem using a two-period version of our consumer's choice model.

25 Problem Present and Future Values Constraint Ination Valuing Securities Notation: Let interest rate be denoted by r. Let c 1 and c 2 be consumptions in periods 1 and 2. Let m 1 and m 2 be incomes received in periods 1 and 2.. Let consumption prices be denoted by p 1 and p 2. choice problem: Given incomes m 1 and m 2, and given consumption prices p 1 and p 2, what is the most preferred intertemporal consumption bundle (c 1,c 2 )? Need to know: the intertemporal budget constraint, and intertemporal consumption preferences.

26 Present and Future Values Constraint Ination Valuing Securities Present and Future Values Denitions Given an interest rate r, the future value of M is the value next period of that amount saved now: FV = M (1 + r). The present value of M is the amount saved in the present to obtain M at the start of the next period: PV = M 1 + r. Example: Example: if r = 0.1 the future value of 100 is 100( ) = 110. if r=0.1, the present value of 1 is the amount we have 1 to pay now to obtain 1 next period: =

27 Present and Future Values Constraint Ination Valuing Securities Budget Constraint Case I: No ination, p 1 = p 2 Consumption bundle when neither saving nor borrowing: (c 1,c 2 ) = (m 1,m 2 ) If all period 1 income saved for period 2: (c 1,c 2 ) = (0,m 2 + (1 + r)m 1 ) If all period 2 income borrowed in period 1: ( (c 1,c 2 ) = m 1 + m ) r,0 ( c, c ) 0, m (1 r m c ) 1 c1, c m1, m2 m 2 2 m2 ( c1, c2 ) m1, 0 1 r 0 c 1 0 m1

28 Present and Future Values Constraint Ination Valuing Securities Budget Constraint Given a period 1 consumption of c 1, period 2 consumption is c 2 = m 2 + (1 + r)m } {{ } 1 (1 + r) } {{ } c 1 intercept slope m 2 1 r m 1 m 2 c 2 0 c 1 0 m1 m2 m1 1 r budget constraint: Future-valued form: (1 + r)c 1 + c 2 = m 2 + (1 + r)m 1 Present-valued form: c 1 + c 2 1+r = m 1 + m 2 1+r

29 Present and Future Values Constraint Ination Valuing Securities Optimal intertemporal consumption bundle given by tangency point of intertemporal indierence curves and intertemporal budget constraint: * c 2 m 2 c 2 The consumer saves. m 2 * c 2 c 2 The consumer borrows. 0 c 0 * 1 m 1 c m1 * c 1 c 1

30 Present and Future Values Constraint Ination Valuing Securities Comparative Statics: Slutsky equation re-revisited The Slutsky equation for the change in c 1 due to a change in p 1 is the same as the one seen in topic 1: dc 1 = c 1 s + (m 1 c 1 ) c 1. dp 1 p }{{} 1 }{{} m ( ) (+) Since a change in r is equivalent to a change in p 1, the Slutsky equation is exactly the same. If r, the substitution eect (the rst term in the equation above) is negative; if r the substitution eect is positive. The sign of the total income eect (the second term in the equation above) depends on whether the consumer is a saver or a borrower: If borrower (c 1 > m 1 ), total income eect is negative. If saver, (c 1 < m 1 ), total income eect is positive. Note: eects of r are the opposite as eects of r.

31 Present and Future Values Constraint Ination Valuing Securities Comparative Statics: Interest rate decrease Graphically, since slope budget constraint curve is (1 + r), r attening budget constraint. m 2 c 2 0 c 0 m1 1 Eects of r on optimal intertemporal consumption bundle: Substitution eect: increase in cost future consumption relative to present consumption. Total income eect: If saver, total income eect is negative. If borrower, total income eect is positive.

32 Comparative Statics: Interest rate decrease Total eect: Present and Future Values Constraint Ination Valuing Securities c 2 If saver, c 1?, c 2 If borrower, c 1, c 2? The consumer saves. c 2 The consumer borrows. * c 2 ** c 2 m 2 0 c * ** 0 m1 1 c 1 c 1 m 2 * c 2 ** c m 1 * c 1 ** c 1 c 1

33 Present and Future Values Constraint Ination Valuing Securities Ination Denitions The ination rate is the rate at which the level of prices for goods increases. It is equal to π = p 2 p 1 p π = p 2 p 1 The real-interest rate, ρ, is an interest rate adjusted to remove the eects of ination. It is equal to 1 + ρ = 1 + r 1 + π ρ = r π 1 + π and, if the ination rate is small, it can be approximated by the dierence between the interest rate and the ination rate: ρ r π.

34 Present and Future Values Constraint Ination Valuing Securities Budget Constraint Case II: Ination, p 2 = (1 + π) p 1 budget constraint with ination: p 1 c 1 + p 2c 2 (1 + r) = p 1m 1 + p 2m 2 (1 + r) c 1 + c ρ = m 1 + m ρ budget constraint curve: c 2 = (1 + ρ)m 1 + m } {{ } 2 (1 + ρ) } {{ } intercept slope 1 r m1 m2 1 c 2 same effects as r c 1 m 2 p m1 p1 m2 m1 1 r c 1

35 Valuing Financial Securities Present and Future Values Constraint Ination Valuing Securities Denition A nancial security is a nancial instrument that promises to deliver an income stream. Example: Consider a security that pays m 1 at the end of period 1, m 2 at the end of period 2, and m 3 at the end of period 3. What is the most that should be paid now for this security? The present value of this security! PV = m 1 (1 + r) + m 2 (1 + r) 2 + m 3 (1 + r) 3.

36 Present and Future Values Constraint Ination Valuing Securities Valuing Bonds Denition A bond is a type of security that pays a xed amount x for T years (its maturity date) and then pays its face value F. Denition Year T 1 T Income paid x x... x F Present Value x (1+r) x x... (1+r) 2 (1+r) T 1 The value of the bond is its present value: F (1+r) T PV = x (1 + r) + x (1 + r) x (1 + r) T 1 + F (1 + r) T

37 Outline Part I. Consumer Contingent BC Preferences Insurance Diversication 1. umer under 4.1 State-contingent budget constraints 4.2 Preferences under uncertainty 4.3 Insurance 4.4 Risk spreading

38 Contingent BC Preferences Insurance Diversication TOPIC 3. CHOICE UNDER UNCERTAINTY So far, dynamic problems considered had no uncertainty. However, in the real world people often make decisions with uncertainty about future prices, future wealth, or other agents' decisions. In this section, we study the choice problem under uncertainty using a two-state version of our consumer's choice model. Optimal responses: insurance purchase, risk diversication. Example: 2 possible states of nature: car accident (loss of L ), no car accident. Probabilities for each state: π a, π na. Insurance: get K if accident by paying γk as insurance premium.

39 Contingent BC Preferences Insurance Diversication State-contingent budget constraints Denitions A contract is state contingent if it is implemented only when a particular state of Nature occurs. A state-contingent consumption plan species the consumption to be implemented when each state of Nature occurs. Example: Consumption if no accident: c na = M γk Consumption if accident: c a = M L γk + K. Hence, K = c a M+L 1 γ and ( ) ca M + L c na = M γ = M γl γ 1 γ 1 γ 1 γ c a

40 State-contingent budget constraints Car Insurance example State-contingent budget constraint in car insurance example: c na = m γl γ c a 1 γ 1 γ } {{ } } {{ } intercept slope Contingent BC Preferences Insurance Diversication C na Bundle with no consumption if accident: M Endowment bundle L M K 1 Full insurance bundle (C a =C na ) M L Bundle with no consumption if no accident: M K C C a

41 Contingent BC Preferences Insurance Diversication Preferences under To know what is the agents' choice, we need to know their preferences about the dierent state-contingent consumption plans. Utility across state-contingent consumption plans is a function of the consumption levels and probabilities at each state,u (c 1,c 2,π 1,π 2 ). Denition A utility function U (c 1,c 2,π 1,π 2 ) satises the expected utility or von NeumannMorgenstern property if it can be written as the weighted sum of the utility at each state, where the weights are the probabilities of each state: U (c 1,c 2,π 1,π 2 ) = π 1 v (c 1 ) + π 2 v (c 2 ) It satises the independence property, which means that the utility in a given state is independent of the utility in other states.

42 Contingent BC Preferences Insurance Diversication Risk aversion Denition We say an agent is risk averse if the expected utility of wealth is lower than the utility of expected wealth, risk lover if it is higher, and risk neutral if it is equal. Example: Lottery: 90 with probability 1/2, 0 with prob 1/2. Utility levels: U($90) = 12, U($0) = 2. Expected utility: EU=1/2*12+1/2*2=7. Expected money value: EM=1/2*90+1/2*0=45. Risk averse consumer Risk lover consumer Risk neutral consumer 12 U(45) EU=7 EU=7 U(45) U(45)=EU= Wealth Wealth Wea

43 Contingent BC Preferences Insurance Diversication Indierence Curves State-contingent consumption plans that give equal expected utility are equally preferred and on the same indierence curve. Slope of indierence curves: EU = π 1 U (c 1 )+π 2 U (c 2 ) deu = π 1 U (c 1 ) c 1 dc 1 +π 1 U (c 2 ) c 2 dc 2 π 1 U (c 1 ) c 1 dc 1 + π 2 U (c 2 ) c 2 dc 2 = 0 dc 2 dc 1 = π 1 U (c 1 )/ c 1 π 2 U (c 2 )/ c 2 C 2 Indifference curves EU 1 < EU 2 < EU 3 EU 3 EU 2 EU 1 C 1

44 Contingent BC Preferences Insurance Diversication under uncertainty The optimal choice under uncertainty is the most preferred aordable state-contingent consumption plan. In the car insurance example, the optimal consumption plan is where the slope of indierence curves is tangent to the budget constraint: C na m π a U (c a )/ c a = γ π na U (c na )/ c na 1 γ Most preferred affordable plan Affordable plans m L m L C a

45 Contingent BC Preferences Insurance Diversication Insurance Fair Insurance Denition We say an insurance is fair or competitive if the expected economic prot of the insurer is zero, or, equivalently, if the price of a 1 insurance is the probability of the insured state. Car insurance example: γk π }{{} a K (1 π a )0 = 0 γ = π } {{ } a revenues expected expenditures If the insurance is fair, the optimal choice of risk-averse consumers is full insurance: π a π na U (c a )/ c a U (c na )/ c na = π a U (c a) = U (c na) 1 π a c a c na Hence, for risk averse consumers, c a = c na.

46 Contingent BC Preferences Insurance Diversication Insurance Unfair Insurance Denition We say an insurance is unfair if the insurer makes positive expected economic prots. If the insurance is unfair, the optimal choice of risk-averse consumers is less than full insurance: γk π }{{} a K (1 π a )0 > 0 γ } {{ } 1 γ > π a π na revenues expected expenditures Hence, π a U(c a )/ c a π na U(c na )/ c na = γ 1 γ U (c a ) c a implies that > U (c na) c na so, for risk averse consumers,c a < c na.

47 Contingent BC Preferences Insurance Diversication Diversication Asset diversication typically lowers (or keeps) expected earnings in exchange for lowered risk. This is going to be the case as long as the asset prices are not perfectly correlated across states. Example: two rms, two states (prob. 1/2), agent with 100 to spend in rms' share. Firm A: shares' cost 10, prots per share in state 1 100, in state Firm B: shares' cost 10, prots per share in state 1 20, in state shares of A 10 shares of B 5 of A, 5 of B Prots in Prots in Expected prots