MICROECONOMICS - cheatsheets and fundamentals of theory, 1ST PARTIAL, Dispense di Microeconomia

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Microeconomics Midterm BESS Contents 1 Budget constraint 3 2 Preferences 3 2.1 Utility functions: . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2.2 Marginal utility . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 3 Choice and demand functions 4 3.1 Constrained optimal choice . . . . . . . . . . . . . . . . . . . . . 4 3.2 Direct demand functions (Marshallian) . . . . . . . . . . . . . . 4 3.2.1 Optimality conditions with interior solutions: . . . . . . . 5 3.2.2 Optimality conditions with corner solutions: . . . . . . . . 5 3.2.3 Optimality conditions with mixed solutions: . . . . . . . . 6 3.2.4 Own price changes: . . . . . . . . . . . . . . . . . . . . . . 6 3.2.5 Cross price changes: . . . . . . . . . . . . . . . . . . . . . 6 3.2.6 Income changes and Engel curve . . . . . . . . . . . . . . 6 4 Elasticity 7 4.1 Own price elasticity of demand . . . . . . . . . . . . . . . . . . . 7 4.2 Cross price elasticity of demand . . . . . . . . . . . . . . . . . . . 7 4.3 Income of elasticity of demand . . . . . . . . . . . . . . . . . . . 8 5 Effects of price changes 8 5.1 Substitution and income effect . . . . . . . . . . . . . . . . . . . 8 5.2 Uncompensated (Marshallian) and compensated (Hicksian) de- mand functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 5.3 Expenditure function . . . . . . . . . . . . . . . . . . . . . . . . . 10 5.4 Calculating analitically SE and IE . . . . . . . . . . . . . . . . . 11 1 6 The rational choice model of the labor supply 11 6.1 Labor and leisure . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 6.1.1 The budget constraint: . . . . . . . . . . . . . . . . . . . . 12 6.1.2 Constrained optimal choice . . . . . . . . . . . . . . . . . 12 6.2 Effects of nonlabor income change on labor supply . . . . . . . . 13 6.3 Effects of wage change . . . . . . . . . . . . . . . . . . . . . . . . 13 6.4 Reservation wage . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 7 Intertemporal choice 14 7.1 Intertemporal budget constraint . . . . . . . . . . . . . . . . . . . 14 7.2 Costrained optimal choice . . . . . . . . . . . . . . . . . . . . . . 15 7.3 Increase in i from i∗ to i∗∗ . . . . . . . . . . . . . . . . . . . . . . 15 8 Choice under uncertainty 15 8.1 Lotteries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 8.1.1 Expected value(EV) . . . . . . . . . . . . . . . . . . . . . 16 8.1.2 Expected utility (EU) . . . . . . . . . . . . . . . . . . . . 16 8.1.3 Utility of the expected value u(EV) . . . . . . . . . . . . 16 8.1.4 Attitudes towards risk/uncertainty . . . . . . . . . . . . . 16 8.1.5 Certainty equivalent (CE) . . . . . . . . . . . . . . . . . . 16 8.1.6 Risk premium . . . . . . . . . . . . . . . . . . . . . . . . . 17 8.1.7 Insurance . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 2 3.2.1 Optimality conditions with interior solutions: - Cobb-Douglas: tangency condition for optimality always holds. Demand function:{ p1x ∗ 1 + p2x ∗ 2 = m feasibility condition |MRS(x∗)| = p1 p2 optimality condition (1) { p1x ∗ 1 + p2x ∗ 2 = m feasibility condition x2 x1 = p1 p2 optimality condition (2) |MRS(x∗)| = p1 p2 |MRS| ≡ MUx1 MUx2 = p1 p2 MUx1 p1 = MUx2 p2 (3) - Perfect complements:{ p1x ∗ 1 + p2x ∗ 2 = m feasibility condition x1 = x2 optimality condition (4) 3.2.2 Optimality conditions with corner solutions: { p1x ∗ 1 + p2x ∗ 2 = m |MRS(x∗)| ≤ p1 p2 x∗ 1 > 0 x∗ 2 = 0 (5) { p1x ∗ 1 + p2x ∗ 2 = m |MRS(x∗)| ≥ p1 p2 x∗ 1 = 0 x∗ 2 > 0 (6) - Neutral goods: 1. Vertical IC: if MRS(x∗) = ∞ then x∗ 1 = m p1 , x2 = 0 2. Horizontal IC: if MRS(x∗) = 0 then x∗ 1 = 0, x2 = m p2 - Concave preferences: 1. If —MRS(x∗)| > p1 p2 then x∗ 1 = m p1 , x2 = 0 2. If —MRS(x∗)| < p1 p2 then x∗ 1 = 0, x2 = m p2 5 3.2.3 Optimality conditions with mixed solutions: - Quasi-linear: { p1x1 + p2x2 = m |MRS| = p1 p2 (7) - Interior: if x1 > 0 - Corner: if x1 < 0 → x1(x1, x2,m) = 0 and x2(x1, x2,m) = m p2 - Perfect substitutes: Corner solutions: 1. If |MRS| > p1 p2 then x∗ 1 = m p1 , x2 = 0 |MRS| > p1 p2 ⇐⇒ MUx1 MUx2 > p1 p2 ⇐⇒ MUx1 p1 > MUx2 p2 2. If |MRS| < p1 p2 then x∗ 1 = 0, x2 = m p2 |MRS| < p1 p2 ⇐⇒ MUx1 MUx2 < p1 p2 ⇐⇒ MUx1 p1 < MUx2 p2 Interior solution: If |MRS| = p1 p2 then p1x1 + p2x2 = m 3.2.4 Own price changes: - Ordinary: the quantity demanded of it always increases as its own price decreases - Giffen: if, for some values of its own price, the quantity demanded rises as its own prices increases 3.2.5 Cross price changes: - Gross substitute: if an increase in p2 increases the demand for com- modity 1 - Gross complement: if an increase in p2 decreases the demand for com- modity 1 3.2.6 Income changes and Engel curve Income consumption curve: the income-consumption curve (ICC) repre- sents the set of the optimal consumption bundles as income varies. Engel curve: a plot of quantity demanded against income - Normal good: a good for which quantity demanded rises with income (Engel curve is positively sloped) - Inferior goods: a good for which quantity demanded decreases with income (Engel curve is negatively sloped) Engel curve for Cobb-Douglas, u(x1, x2) = xa 1x b 2: 6 Ordinary demand equations and Engel curves: x∗ 1 = am (a+ b)p1 m = (a+ b)p1 a x∗ 1 (8) x∗ 2 = bm (a+ b)p2 m = (a+ b)p2 a x∗ 2 (9) 4 Elasticity Elasticity: key concept that tells us how sensitive an individual’s demand func- tions are to changes in any of the exogenous variables, i.e. the prices of the commosities p1 and p2 and the consumer’s income, m. - Own price elasticity: the price of the good changes - Cross-price elasticity: the price of another good changes - Income elasticity: the consumer’s income changes 4.1 Own price elasticity of demand ϵ ≡ %changeinx1demanded %changeinp1 ϵ ≡ ∆x1 x1 ∆p1 p1 = ∆x1 ∆p1 x1 p1 = ∆x1 ∆p1 p1 x1 =⇒ ϵ = ∆x1 ∆p1 p1 x1 Definition Values Elastic ϵ > 1 Inelastic 0 < ϵ < 1 Infinitely elastic |ϵ| = ∞ Infinitely inelastic |ϵ| = 0 Unitary elastic |ϵ| = 1 For ordinary goods, elasticity is a negative number because the slope of the demand function is negative (there is a negative relationship between the price and the quantity demanded of a good). 4.2 Cross price elasticity of demand ϵ1,2 ≡ %changeinx1demanded %changeinp2 ϵ ≡ ∆x1 x1 ∆p2 p2 = ∆x1 ∆p2 x1 p2 = ∆x1 ∆p2 p2 x1 =⇒ ϵ = ∆x1 ∆p2 p2 x1 7 5.2 Uncompensated (Marshallian) and compensated (Hick- sian) demand functions - The Marshallian demand functions (x1(p1, p2,m), x2(p1, p2,m)) are un- compensated : the consumer’s utility is allowed to vary with the price of the good and both SE and IE are considered. - TheHicksian demand functions are compensated (h1(p1, p2, Ū), h2(p1, p2, Ū)): they show how much the quantity demandeed changes when price changes, holding utility constant. Only the SE is considered, so the individual must be compensated with extra income as the price rises in order to hold his/her utility constant (viceversa when price decreases) 5.3 Expenditure function Expenditure function: smallest expenditure p1x1 + p2x2 that enables the indi- vidual to achieve a given level of utility Ū based on given market prices p1 and p2. E = E(p1, p2, Ū) Two methods: 1. Expenditure minimization problem: min x1,x2 E(p1h1, p2h2) s.t. U(h1, h2) = Ū We get: h1(p1, p2, Ū) h2(p1, p2, Ū) 2. Shortcut: δE(p1, p2, Ū) δp1 = h1(p1, p2, Ū) δE(p1, p2, Ū) δp2 = h2(p1, p2, Ū) (11) 10 5.4 Calculating analitically SE and IE We must consider the 3 optimal bundles: - x∗ initial optimal bundle - x∗∗ new optimal bundle after price change - h∗ compensated optimal bundle (calculated using one of the two methods and plugging the new prices Then: - SE = h∗ − x∗ - IE = x∗∗ − h∗ - TE = SE+ IE = x∗∗ − x∗ 6 The rational choice model of the labor supply - Preferences over leisure and consumption using indifference curves: Slope of the indifference curve: MRS ≡ −∆c ∆l - Preferences over leisure and consumption using utility functions: MRS ≡ −MUl MUc 6.1 Labor and leisure l + h = T 11 6.1.1 The budget constraint: w · l+ p · c = w ·T+ m̄ p · c = w · T − w · l + m̄ = w(T − l) + m̄ = w · h+ m̄ =⇒ p · c = w · h+ m̄ w = hourly wage l = number of hours of leisure p = price of consumption c = consumption T = total time endowment m̄= non-labor income Solving for c the budget costraint is: c = w · T + m̄ p − w p · l - Vertical intercept: w·T+m̄ p The vertical intercept represents the value of c when l = 0. - Horizontal intercept: T The horizontal intercept represents the value of l when c = 0, and it is equal to T, the maximum number of hours available. - Slope : −w p It is the opportunity cost of leisure in terms of consumption, i.e. the amount of consumption (the individual has to give up if he/she wants to consume one extra hour of leisure. 6.1.2 Constrained optimal choice max l,c U(l, c) s.t. w · l + p · c = w · T + m̄ Labor supply function: h∗(w, p, T, m̄) = T − l∗(w, p, T, m̄) Solution of the optimization problem: 12 7.2 Costrained optimal choice - Neither saves nor borrows: Optimal bundle: c∗ = ω that is c∗0 = m0, c ∗ 1 = m1. In c∗:MRSc0,c1 = 1 + i - The individual saves: saving S0 = m0 − c∗0 c0 < m0 Optimal bundle: c∗ = (c∗0, c ∗ 1) In c∗:MRSc0,cl = 1 + i In ω: MRSc0,cl < 1 + i - The individual borrows: borrowing B = c∗0 −m0 c0 > m0 Optimal bundle: c∗ = (c∗0, c ∗ 1) In c∗: MRSc0,c1 = 1 + i In ω: MRSc0,c1 > 1 + i 7.3 Increase in i from i∗ to i∗∗ c0 becomes relatively more expensive than c1 - saver: interest rate earnings increase (a saver is richer because the value of his/her endowment bundle is bigger) - borrower: the interest rate payments on the loan increase (a borrower is poorer because the present discounted value of his endowment bundle is now smaller) 8 Choice under uncertainty 8.1 Lotteries A lottery L is a probability distribution: p1, p2, ...pn with p1 ≥ 0 and ∑n i=1 pi = 1 over the possible monetary values (payoffs, outcomes) that may occur v1, v2, ..., vn A lottery is: L = (v1p1; v2p2; ...; vnpn) 15 8.1.1 Expected value(EV) The expected value EV is the weighted average of the lottery L (weights=probabilities): EV ≡ EV(L) = n∑ i=1 pivi = p1v1 + p2v2 + ...+ pnvn 8.1.2 Expected utility (EU) The expected utility ranks lotteries through the weighted average of the utility values: EU(L)= n∑ i=1 pi · u(vi) = p1 · u(v1) + p2 · u(v2) + ...+ pn · u(vn) 8.1.3 Utility of the expected value u(EV) The utility of the expected value u(EV) of a lottery is simply the utility that the individual obtains from receiving the expected value EV of the lottery 8.1.4 Attitudes towards risk/uncertainty - risk averse: his / her utility function u(v) is concave u(UV ) > EU(L) - risk loving: his / her utility function u(v) is convex u(EV ) < EU(L) - risk neutral: his / her utility function u(v) is linear u(EV ) = EU(L) 8.1.5 Certainty equivalent (CE) The certainty equivalent CE(L) of a lottery L is the risk-free (certain) monetary value that gives the individual a utility level u which is equal to the expected utility, EU, of the lottery. u(CE)=EU(L) CE = u−1(EU) 16 8.1.6 Risk premium RP=EV-CE - RP for risk averse > 0 - RP for risk loving < 0 - RP for risk neutral = 0 · We need to pay RP to a risk averse individual for making him/her willing to participate to the lottery · Vice versa, a risk loving individual is willing to pay RP to participate to the lottery 8.1.7 Insurance - Full coverage: EU(L with insurance z) = ∑n i=1 p1 · (u(vi − z)) - Partial coverage L = {v1 − aδ, 1− p; v2 − aδ, 1− p} a: units of policy δ : price of the unitary policy 17