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Advanced Microeconomics The household optimum Harald Wiese University of Leipzig Harald Wiese (University of Leipzig) Advanced Microeconomics 1 / 72 Part B. Household theory and theory of the …rm 1 The household optimum 2 Comparative statics and duality theory 3 Production theory 4 Cost minimization and pro…t maximization Harald Wiese (University of Leipzig) Advanced Microeconomics 2 / 72 Nobel price 2015 In 2015, the Sveriges Riksbank Prize in Economic Sciences in Memory of Alfred Nobel was awarded to the economist Angus Deaton (Princeton University, NJ, USA) for his analysis of consumption, poverty, and welfare By linking detailed individual choices and aggregate outcomes, his research has helped transform the …elds of microeconomics, macroeconomics, and development economics. How do consumers distribute their spending among di¤erent goods? How much of society’s income is spent and how much is saved? How do we best measure and analyze welfare and poverty? Harald Wiese (University of Leipzig) Advanced Microeconomics 3 / 72 The household optimum Overview 1 Budget 2 The household optimum 3 Comparative statics and vocabulary 4 Applying the Lagrange method (recipe) 5 Indirect utility function 6 Consumer’s rent and Marshallian demand Harald Wiese (University of Leipzig) Advanced Microeconomics 4 / 72 Budget Money budget and budget line De…nition The expenditure for a bundle of goods x = (x1 , x2 , ..., x` ) at a vector of prices p = (p1 , p2 , ..., p` ) is the dot product (or the scalar product): ` p x := ∑ pg xg . g =1 De…nition The money budget: n B (p, m ) := x 2 R`+ : p x The budget line: Harald Wiese (University of Leipzig) n o m , p 2 R` , m 2 R+ x 2 R`+ : p x = m Advanced Microeconomics o 5 / 72 Budget Money budget: A two goods case x2 Problem m p2 Assume that the household consumes bundle A. Identify the “left-over” in terms of good 1, in terms of good 2 and in money terms. C B A m p1 x1 Problem What happens to the budget line if price p1 doubles; if both prices double? Harald Wiese (University of Leipzig) Advanced Microeconomics 6 / 72 Budget Money budget Lemma For any number α > 0: B (αp, αm ) = B (p, m ) Problem Fill in: For any number α > 0: B (αp, m ) = B (p, ?) . Harald Wiese (University of Leipzig) Advanced Microeconomics 7 / 72 Budget Marginal opportunity cost for two goods Problem Verify that the budget line’s slope is given by p1 p2 (in case of p2 6= 0). De…nition If p1 0 and p2 > 0, MOC (x1 ) = p1 dx2 = dx1 p2 – the marginal opportunity cost of consuming one unit of good 1 in terms of good 2. Harald Wiese (University of Leipzig) Advanced Microeconomics 8 / 72 Budget Marginal opportunity cost x2 m p2 ∆ x1 ∆x 2 = MOC ∆x1 p = 1 ∆ x1 p2 m p1 Harald Wiese (University of Leipzig) Advanced Microeconomics x1 9 / 72 Endowment budget De…nition De…nition For p 2 R` and an endowment ω 2 R`+ : n B (p, ω ) := x 2 R`+ : p x – the endowment budget. Harald Wiese (University of Leipzig) Advanced Microeconomics p ω o 10 / 72 Endowment budget A two goods case budget line: p1 x1 + p2 x2 = p1 ω 1 + p2 ω 2 2 marginal opportunity cost: MOC = dx dx1 = p1 p2 x2 Problem p1ω1 + p 2ω2 p2 What happens to the budget line if dx2 p =− 1 dx1 p2 price p1 doubles; ω2 both prices double? ω1 Harald Wiese (University of Leipzig) p1ω1 + p 2ω2 p1 x1 Advanced Microeconomics 11 / 72 Application 1 Intertemporal consumption Notation: ω 1 and ω 2 – monetary income in t1 and t2 ; x1 and x2 – consumption in t1 and t2 ; household can borrow (x1 > ω 1 ), lend (x1 < ω 1 ) or consume what it earns (x1 = ω 1 ); r – rate of interest. Consumption in t2 : x2 = ω2 |{z} + second-period income = ω 2 + (1 + r ) ( ω 1 Harald Wiese (University of Leipzig) (ω 1 x1 ) | {z } amount borrowed (<0) or lended (>0) + r (ω 1 x1 ) | {z } interest payed (<0) or earned (>0) x1 ) Advanced Microeconomics 12 / 72 Application 1 Borrow versus lend borrow verwandt mit borgen und bergen („in Sicherheit bringen“) wie in Herberge („ein das Heer bergender Ort“) lend verwandt mit Lehen („zur Nutzung verliehener Besitz“) und leihen, verwandt mit lateinischstämmig Relikt („Überrest“) und Reliquie („Überbleibsel oder hochverehrte Gebeine von Heiligen“) und mit griechischstämmig Eklipse („Ausbleiben der Sonne oder des Mondes“ > „Sonnen- bzw. Mond…nsternis“) und auch mit griechischstämmig Ellipse (in der Geometrie ein Langkreis, bei dem die Höhe geringer ist als die Breite und insofern ein Mangel im Vergleich zum Kreis vorhanden ist – agr. elleipsis (έλλειψις) bedeutet „Ausbleiben“ > „Mangel“ Harald Wiese (University of Leipzig) Advanced Microeconomics 13 / 72 Application 1 Intertemporal consumption 2 ways to rewrite the budget equation: in future-value terms: (1 + r ) x1 + x2 = (1 + r ) ω 1 + ω 2 , in present-value terms: x1 + Harald Wiese (University of Leipzig) x2 ω2 = ω1 + . 1+r 1+r Advanced Microeconomics 14 / 72 Application 1 Intertemporal consumption budget line: (1 + r ) x1 + x2 = (1 + r ) ω 1 + ω 2 dx2 dx1 marginal opportunity cost: MOC = = 1+r x2 (1 + r ) ω1 + ω2 Problem dx2 = −(1 + r ) dx1 consumption point of a lender ω2 endowment point consumption point of a borrower ω1 Harald Wiese (University of Leipzig) ω1 + ω2 (1 + r ) Advanced Microeconomics What happens to the budget line if the interest rate decreases? x1 15 / 72 Application 2 Leisure versus consumption Notation: xR – recreational hours (0 household works 24 xR xR hours; 24 = ω R ) ! good 1; xC – real consumption ! good 2; w – the wage rate; ω C – the real non-labor income; p – the price index. Harald Wiese (University of Leipzig) Advanced Microeconomics 16 / 72 Application 2 Leisure versus consumption Household’s consumption in nominal terms: pxC = pω C + w (24 xR ) Household’s consumption in endowment-budget form: wxR + pxC = w 24 + pω C Harald Wiese (University of Leipzig) Advanced Microeconomics 17 / 72 Application 2 Leisure versus consumption budget line: wxR + pxC = w 24 + pω C C marginal opportunity cost: MOC = dx dxR = w p xC Problem w 24 + pωC p dxC w =− dxR p What happens to the budget line if the wage rate increases? consumption point endowment point ωC recreational time Harald Wiese (University of Leipzig) labor 24 time Advanced Microeconomics xR 18 / 72 Application 3 Contingent consumption Notation: A – a household wealth; L = [x1 , x2 ; p, 1 p ] – lottery with insurance where p – the probability of a bad event; x1 = A D + K γK = A D + (1 γ ) K ! insured event x2 = A γK D – possible damage; L = [A D, A; p, 1 – lottery without insurance p] special cases K – insurance sum; γ – insurance rate γK – insurance premium. Harald Wiese (University of Leipzig) Advanced Microeconomics x1 = A D, x2 = A ! no insurance (K := 0). x1 = x2 = A γD ! full insurance (K := D). 19 / 72 Application 3 Contingent consumption budget line: x1 + 1 γ γ x2 = (A 1 γ γ A γ dx2 dx1 = 1 γ D) + marginal opportunity cost: MOC = x1 = A − D + K − γK x 2 = A − γK x2 (prob. 1 − p ) no insurance A A − γD dx2 γ =− dx1 1− γ Problem Interpret the part of the budget line full insurance right of the full-insurance point; 45° A − D A − γD Harald Wiese (University of Leipzig) x1 (prob. p ) Advanced Microeconomics left of the no-insurance point! 20 / 72 The household optimum 1 Budget 2 The household optimum 3 Comparative statics and vocabulary 4 Applying the Lagrange method (recipe) 5 Indirect utility function 6 Consumer’s rent and Marshallian demand Harald Wiese (University of Leipzig) Advanced Microeconomics 21 / 72 The household’s decision problem De…nition Best response function x R : x R (B ) : = x 2 B: there is no x 0 2 B with x 0 x or R x (B ) : = arg max U (x ) x 2B Any x from x R (B ) – a household optimum. Notation: also x R (p, m ) or x R (p, ω ) or just x (p ) Lemma For any number α > 0: x R (αp, αm ) = x R (p, m ) Harald Wiese (University of Leipzig) Advanced Microeconomics 22 / 72 The household’s decision problem Exercise 1 x2 x2 Problem A A indifference curve budget line B budget line indifference curve (a) x1 (b) x1 (d) x1 x2 x2 Assume monotonicity of preferences. Are the highlighted points A or B optima? budget line A budget line indifference curve B (c) Harald Wiese (University of Leipzig) x1 A indifference curve Advanced Microeconomics 23 / 72 The household’s decision problem Exercise 2 Problem Assume a household’s decision problem (B (p, ω ) , -). x R (B ) consists of the bundles x that ful…ll the two conditions: 1 The household can a¤ord x: p x 2 p ω There is no other bundle y that the household can a¤ord and that he prefers to x: y x )?? Substitute the question marks by an inequality. Harald Wiese (University of Leipzig) Advanced Microeconomics 24 / 72 MRS versus MOC dx2 dx1 Marginal willingness to pay: MRS = If the household consumes one additional unit of good 1, how many units of good 2 can he forgo so as to remain indi¤erent. movement on the indi¤erence curve Marginal opportunity cost: MOC = If the household consumes one additional unit of good 1, how many units of good 2 does he have to forgo so as to remain within his budget. Harald Wiese (University of Leipzig) Advanced Microeconomics dx2 dx1 movement on the budget line 25 / 72 MRS versus MOC dx2 dx1 | {z } MRS = dx2 dx1 | {z } > absolute value absolute value of the slope of of the slope of the indi¤erence curve the budget line = MOC ) increase x1 (if possible) x2 indifference curve budget line 1 unit of good 1 MOC MRS x1 Harald Wiese (University of Leipzig) Advanced Microeconomics 26 / 72 MRS versus MOC MRS > MOC ) increase x1 (if possible) x2 budget line indifference curves household optimum Harald Wiese (University of Leipzig) Advanced Microeconomics x1 27 / 72 MRS versus MOC Alternatively: the household tries to maximize U x1 , pm2 p1 p 2 x1 . Consume 1 additional unit of good 1 utility increases by ∂U ∂x 1 reduction in x2 by MOC = utility decrease by ∂U ∂x 2 dx2 dx1 dx2 dx1 = p1 p2 and hence (chain rule) Thus, increase consumption of good 1 as long as ∂U ∂x1 |{z} marginal bene…t of increasing x1 or MRS Harald Wiese (University of Leipzig) ∂U dx2 ∂x2 dx1 | {z } > marginal cost of increasing x1 = ∂U ∂x1 ∂U ∂x2 > Advanced Microeconomics dx2 = MOC dx1 28 / 72 Household optimum Cobb-Douglas utility function U (x1 , x2 ) = x1a x21 a with 0 < a < 1 The two optimality conditions MRS = ∂U ∂x1 ∂U ∂x2 = a x2 1 a x1 ! p1 p 2 and = ! p1 x1 + p2 x2 = m yield the household optimum x1 (m, p ) = a m , p1 x2 (m, p ) = (1 Harald Wiese (University of Leipzig) a) Advanced Microeconomics m . p2 29 / 72 Household optimum Perfect substitutes U (x1 , x2 ) = ax1 + bx2 with a > 0 and b > 0 An increase of good 1 enhances utility if p1 a = MRS > MOC = b p2 holds. Therefore 8 m > ,0 , > > < n p1 x1 , pm2 x (m, p ) = > > > : 0, m p2 Harald Wiese (University of Leipzig) p1 p 2 x1 h 2 R2+ : x1 2 0, pm1 Advanced Microeconomics io a b a b a b > = < p1 p2 p1 p2 p1 p2 30 / 72 Household optimum Concave preferences U (x1 , x2 ) = x12 + x22 An increase of good 1 enhances utility if x1 2x1 = = x2 2x2 ∂U ∂x1 ∂U ∂x2 = MRS > MOC = p1 p2 holds. Therefore, corner solutions: 8 m > p1 < p2 > > n p1 , 0 , < o m m x (m, p ) = p1 = p2 p 1 , 0 , 0, p 2 > > > m : 0, p1 > p2 p2 Harald Wiese (University of Leipzig) Advanced Microeconomics 31 / 72 Household optimum Dixit-Stiglitz preferences for love of variety ` ∑ xj U (x1 , ..., x` ) = ε 1 ε j =1 ! ε ε1 with ε > 1 with ∂U (x1 ,x2 ,...,x ` ) ∂xg ∂U (x1 ,x2 ,...,x ` ) ∂xk ε = ε 1 ε ε 1 ε 1 = xg ε ε 1 ε xk ∑ ε 1 ` ε x j =1 j ε ε 1 1 ∑ ε 1 ` ε x j j =1 ε ε 1 1 1 1 1 = xg ε xk 1 ε = xg xk ε 1 1 ε 1 ε 1 ε 1 ε ε xg ε 1 ε xk 1 ε = xk xg 1 ε ! = pg pk and, in the case of two goods (` = 2), we obtain m m m x1 (m, p ) = , x2 (m, p ) = ε = 1 ε ε p1 + p2 p1 p2 + p11 ε p2ε p1 + p2 pp21 Harald Wiese (University of Leipzig) Advanced Microeconomics 32 / 72 Household optimum and monotonicity I Lemma Let x (p, m ) be a household optimum. Then local nonsatiation implies p x = m (Walras’law); and ... Proof. p x x2 m (why?). Assume: p x < m ) contradiction! m p2 x* y m p1 Harald Wiese (University of Leipzig) Advanced Microeconomics x1 33 / 72 Household optimum and monotonicity II Lemma Let x (p, m ) be a household optimum. Then ... strict monotonicity implies p >> 0; local nonsatiation and weak monotonicity imply p 0. Proof. Assume pg 0 ) household can be made better o¤ by consuming more of good g (strict monotonicity). Contradiction! Assume pg < 0 ) household can “buy” additional units of g without being worse o¤ (weak monotonicity). Household has additional funding for preferred bundles (nonsatiation). Contradiction! Harald Wiese (University of Leipzig) Advanced Microeconomics 34 / 72 Comparative statics and vocabulary 1 Budget 2 The household optimum 3 Comparative statics and vocabulary 4 Applying the Lagrange method (recipe) 5 Indirect utility function 6 Consumer’s rent and Marshallian demand Harald Wiese (University of Leipzig) Advanced Microeconomics 35 / 72 Vocabulary De…nition The (Marshallian) demand function for good g ! xg (pg ); The cross demand function for good g with respect to pk of good k 6= g ! xg (pk ) ; The Engel function for good g ! xg (m ) . In case of an endowment budget the household is called a net supplier of good g if xg (p, ω ) < ω g and a net demander if xg (p, ω ) > ω g . Harald Wiese (University of Leipzig) Advanced Microeconomics 36 / 72 Vocabulary De…nition A good g is: ordinary if ∂xg ∂pg 0 (non-ordinary otherwise) ! slope of demand curve; normal if ∂xg 0 ∂m (inferior otherwise) ! slope of Engel curve; Problem Consider the demand function x = a mp , a > 0. Is the good an ordinary and/or a normal good? Harald Wiese (University of Leipzig) Advanced Microeconomics 37 / 72 Vocabulary De…nition A good g is: a substitute of good k if ∂xg ∂pk 0; ∂xg ∂pk 0. a complement of good k if Problem Consider the demand functions x1 = a pm1 and x2 = (1 a) pm2 , 0 < a < 1, and …nd out whether good 1 is a substitute or a complement of good 2! Harald Wiese (University of Leipzig) Advanced Microeconomics 38 / 72 Price-consumption curve and demand curve Deriving the demand curve graphically x2 dx 2 pD =− 1 dx1 p2 B p1B x1B x1C priceconsumption curve D C Problem p1C p1D x1D Assuming that good 1 and good 2 are complements, sketch a price-consumption curve and the associated demand curve for good 1. x1 p1 p1B demand curve p1C p1D x1B x1C x1D Harald Wiese (University of Leipzig) x1 Advanced Microeconomics 39 / 72 Price-consumption curve and demand curve Deriving the demand curve analytically 1 2 Assume the utility function U (x1 , x2 ) = x13 x23 with household optimum x1 = 2m 1m , x = . 3 p1 2 3 p2 The demand curve for good 1 is x1 = f (p1 ) = 1 m 3 p1 . x2 = h (x1 ) = 23 pm2 is already the price-consumption curve–x2 is a constant (boring) function x1 . Note: It is important that h is not a function of p1 . Problem Determine, analytically, the price-consumption curve for the the case of perfect complements, U (x1 , x2 ) = min (x1 , 2x2 )! Can you also …nd the demand function for good 2? Assume p1 > 0 and p2 > 0! Harald Wiese (University of Leipzig) Advanced Microeconomics 40 / 72 Vocabulary Saturation quantity and prohibitive price De…nition Let xg (pg ) be the quantity demanded for any pg 0. ) xgsat pg p gproh := xg (0) demand curve – the saturation quantity; p1proh := min fpg 0 : xg (pg ) = 0g xsat g xg – the prohibitive price. Harald Wiese (University of Leipzig) Advanced Microeconomics 41 / 72 Income-consumption curve and Engel curve Deriving the Engel curve graphically x2 mD income-consumption curve mC D m B C B x1B x1C x1D Problem Assuming that good 1 and good 2 are complements, sketch an income-consumption curve and the associated Engel curve for good 1! x1 m Engel curve mD mC mB x1B x1C x1D Harald Wiese (University of Leipzig) x1 Advanced Microeconomics 42 / 72 Income-consumption curve and Engel curve Deriving the Engel curve analytically Assume the household optimum x1 = The Engel curve for good 1 is 1 m 3 p1 , x1 = q (m ) = x2 = 2 m 3 p2 . 1m . 3 p1 Income-consumption curve: Solve good 1’s demand for m and obtain m = 3p1 x1 . Subsituting in x2 3p x yields x2 = 32 pm2 = 23 p12 1 = 2 pp21 x1 and hence the income-consumption curve p x2 = g ( x1 ) = 2 1 x1 . p2 Note: It is important that g is not a function of m. Problem Determine, analytically, the income-consumption curve and the Engel-curve function for U (x1 , x2 ) = min (x1 , 2x2 )! Harald Wiese (University of Leipzig) Advanced Microeconomics 43 / 72 De…ning substitutes and complements Exercise Problem Determine ∂x1 (p,m ) ∂p 2 and ∂x2 (p,m ) ∂p 1 for the quasi-linear utility function: U (x1 , x2 ) = ln x1 + x2 Assume positive prices and m p2 ( x1 > 0 ) ! > 1, in order to avoid a corner solution! Conclusion: good g can be the substitute of good k while k is not a (strict) substitute of g ! Harald Wiese (University of Leipzig) Advanced Microeconomics 44 / 72 Vocabulary Price elasticity of demand De…nition Let xg (pg ) be the demand at price pg (other prices are held constant). ) dxg xg dp g pg εxg ,pg := = dxg pg dpg xg |{z} mathematically doubtful – the price elasticity of demand. Problem Calculate the price elasticities of demand for the demand functions: xg (pg ) = 100 Harald Wiese (University of Leipzig) pg and xk (pk ) = Advanced Microeconomics 1 . pk 45 / 72 Vocabulary Price elasticity of demand - application Drug users have inelastic demand: jεx ,p j < 1 or εx ,p > 1 Then a price increase increases expenditure: d (px (p )) dp dx dp p dx 1+ x dp = x +p = x = x (1 + εx ,p ) = x (1 jεx ,p j) > 0. Political implication: Making drugs expensive by taxing them or by criminalizing selling or buying increases expenditure and hence drug-related crime (stealing money in order to …nance the addiction). Harald Wiese (University of Leipzig) Advanced Microeconomics 46 / 72 Vocabulary Income elasticity of demand De…nition Let xg (m ) be the demand at income m. The income elasticity (of demand) is denoted by εxg ,m and given by εxg ,m := dxg xg dm m = dxg m . dm xg De…nition We call a good g a luxury good (such as caviar) if εxg ,m a necessity good (such as oat groats) if 0 Harald Wiese (University of Leipzig) Advanced Microeconomics 1 holds; εxg ,m 1 holds. 47 / 72 Vocabulary Income elasticity of demand - exercise inferior goods normal goods 0 necessity goods 1 luxury goods ε x ,m 1 Problem Calculate the income elasticity of demand for the Cobb-Douglas utility 1 2 function U (x1 , x2 ) = x13 x23 ! How do you classify (demand for) good 1? Harald Wiese (University of Leipzig) Advanced Microeconomics 48 / 72 Vocabulary Average income elasticity of demand - lemma Lemma Assume local nonsatiation and the household optimum x . Then the average income elasticity is 1: ` ∑ sg εx ,m = 1 g g =1 where the weights are the relative expenditures, sg := Harald Wiese (University of Leipzig) Advanced Microeconomics pg xg m . 49 / 72 Vocabulary Average income elasticity of demand - proof According to Walras’law, the household chooses x on the budget line, p x (m ) = m. (How about ` = 1?) Find the derivative of the budget equation m = respect to m to obtain ` dxg 1 = ∑ pg dm g =1 ` ∑g =1 pg xg (m) with and, by multiplying the summands with ` 1= ∑ pg g =1 Harald Wiese (University of Leipzig) xg m m xg = 1, ` p x dx ` dxg xg m g g g m = ∑ = ∑ sg εxg ,m dm m xg g =1 m dm xg g =1 Advanced Microeconomics 50 / 72 Vocabulary Aggregate demand De…nition Let x i (p ) be the demand functions of individuals i = 1, ..., n. ) n x (p ) := ∑i =1 x i (p ) – aggregate demand. p p x1 (p ) Harald Wiese (University of Leipzig) p x2 (p ) Advanced Microeconomics x(p ) 51 / 72 Vocabulary Aggregate demand Problem xg1 (pg ) = max (0, 100 pg ) , xg2 (pg ) = max (0, 50 2pg ) and xg3 3pg ) . (pg ) = max (0, 60 Find the aggregate demand function! Hint: Find the prohibitive prices …rst! Harald Wiese (University of Leipzig) Advanced Microeconomics 52 / 72 Vocabulary Inverse demand function De…nition h i Let x1 : 0, p1proh ! [0, x1sat ] be an injective demand function. ) p1 = x1 1 h i : 0, x1sat ! 0, p1proh x1 7! p1 (x1 ) where p1 (x1 ) is the unique price resulting in x1 . – the inverse demand function. Harald Wiese (University of Leipzig) Advanced Microeconomics 53 / 72 Vocabulary Inverse demand function pg p gproh p g (xˆg ) inverse demand curve demand curve pˆg xˆg Harald Wiese (University of Leipzig) x g (pˆg ) xsat g Advanced Microeconomics xg 54 / 72 The Lagrange method 1 Budget 2 The household optimum 3 Comparative statics and vocabulary 4 Applying the Lagrange method (recipe) 5 Indirect utility function 6 Consumer’s rent and Marshallian demand Harald Wiese (University of Leipzig) Advanced Microeconomics 55 / 72 Applying the Lagrange method (recipe) " L (x, λ) = U (x ) + λ m ` ∑ pg xg g =1 # with: U – strictly quasi-concave and strictly monotonic utility function; prices p >> 0; λ > 0 – Lagrange multiplier - translates a budget surplus m ∑g` =1 pg xg > 0 into utility Increasing consumption has a positive e¤ect via U, but a negative e¤ect via decreasing budget and λ Harald Wiese (University of Leipzig) Advanced Microeconomics 56 / 72 Applying the Lagrange method (recipe) Di¤erentiate L with respect to xg : ∂U (x1 , x2 , ..., x` ) ∂L (x1 , x2 , ..., λ) = ∂xg ∂xg ! λpg = 0 or ∂U (x1 , x2 , ..., x` ) ! = λpg ∂xg and hence, for two goods g and k ∂U (x1 ,x2 ,...,x ` ) ∂xg ∂U (x1 ,x2 ,...,x ` ) ∂xk Harald Wiese (University of Leipzig) ! = pg ! or MRS = MOC pk Advanced Microeconomics 57 / 72 Applying the Lagrange method (recipe) Problem Set the derivative of L with respect to λ equal to 0. What do you …nd? λ – the shadow price of the restriction: λ= dU . dm But: U does not have m as an argument so that Harald Wiese (University of Leipzig) Advanced Microeconomics dU dm is not quite correct. 58 / 72 Indirect utility function 1 Budget 2 The household optimum 3 Comparative statics and vocabulary 4 Applying the Lagrange method (recipe) 5 Indirect utility function 6 Consumer’s rent and Marshallian demand Harald Wiese (University of Leipzig) Advanced Microeconomics 59 / 72 Indirect utility function De…nition De…nition Consider a household with utility function U. ) V : R` R+ ! R, (p, m) 7! V (p, m) := U (x (p, m)) – indirect utility function. Harald Wiese (University of Leipzig) Advanced Microeconomics 60 / 72 Utility function versus indirect utility function function arguments utility function indirect utility function x 2 R`+ optimal bundles m and p 2 R` x (p, m ) x (p, m ) Problem Determine the indirect utility function for the Cobb-Douglas utility function U (x1 , x2 ) = x1a x21 a (0 < a < 1)! Harald Wiese (University of Leipzig) Advanced Microeconomics 61 / 72 Revisiting the Lagrange multiplier Aim: λ = dU dm !λ= dV dm Di¤erentiating the budget function with respect to m yields ∂x ∑g` =1 pg ∂mg = dm dm = 1 (as we know from the the proof about the average income elasticity) Di¤erentiating the indirect utility function V (p, m ) = U (x (p, m )) with respect to m leads to ` ` ` ∂xg ∂xg ∂U ∂xg ∂V = ∑ = ∑ (λpg ) = λ ∑ pg =λ ∂m ∂x ∂m ∂m ∂m g g =1 g =1 g =1 Harald Wiese (University of Leipzig) Advanced Microeconomics 62 / 72 Revisiting the Lagrange multiplier The optimization condition can be rewritten as: ∂U (x1 , x2 , ..., x` ) ! = λpg = ∂xg | {z } marginal utility ∂V pg {z } |∂m marginal cost Interpreting the marginal cost of consuming one additional unit of good g : You consume one additional unit of good g , your expenditure increases by pg so that the income left for other goods decreases by pg and hence utility decreases by Harald Wiese (University of Leipzig) ∂V ∂m pg . Advanced Microeconomics 63 / 72 Consumer’s rent and Marshallian demand 1 Budget 2 The household optimum 3 Comparative statics and vocabulary 4 Applying the Lagrange method (recipe) 5 Indirect utility function 6 Consumer’s rent and Marshallian demand Harald Wiese (University of Leipzig) Advanced Microeconomics 64 / 72 Marginal willingness to pay Assume: x2 – “all the other goods” (money); p2 = 1. ) Marginal willingness to pay for one extra unit of x1 is MRS = p1 = p1 . p2 Inverse demand function measures (cum grano salis) the marginal willingness to pay for one extra unit of a good. Harald Wiese (University of Leipzig) Advanced Microeconomics 65 / 72 Marginal willingness to pay pg p gproh p hg consumer`s rent inverse demand curve = willingness to pay pˆg aggregate willingness to pay xˆg xsat g xg Problem p (q ) = 20 4q, p = 4 aggregate willingness to pay? consumer’s rent? Harald Wiese (University of Leipzig) Advanced Microeconomics 66 / 72 The Marshallian willingness to pay and the Marshallian consumer’s rent De…nition Let pg be an inverse demand function. The Marshallian willingness to pay R x̂ for the quantity x̂g : 0 g pg (xg ) dxg for a price decrease from pgh to p̂g < pgh : for a price decrease from pgproh to p̂g < CR Marshall (p̂g ) = = Z xg (p̂g ) 0 Z xg (p̂g ) 0 Harald Wiese (University of Leipzig) R pgh p̂ g proh pg pg (xg ) dxg (pg (xg ) Advanced Microeconomics xg (pg ) dpg (consumer’s rent) p̂g xg (p̂g ) p̂g ) dxg = Z pgproh p̂ g xg (pg ) dpg 67 / 72 The diamond-water paradox Why are diamonds more expensive than water although water is more „useful“? pD pW water Harald Wiese (University of Leipzig) diamants Advanced Microeconomics 68 / 72 The Marshallian willingness to pay An imperfect concept Marshallian willingness to pay for the quantity x̂g — > “sum” the prices for all the units from 0 up to x̂g But: According to Marshallian demand, the consumers pay one price for all the units. R x̂ In contrast, the integral 0 g pg (xg ) dxg presupposes that the consumer pays (in general) di¤erent prices for the …rst, the second, and so on, units. Now, if higher prices are to be paid for the …rst units, the household has less income available to spend on the additional units: demand is lowered consumer’s willingness to pay or consumer’s rent is lower Marshallian willingness to pay exaggerates “real” consumers’rent — > next chapter Harald Wiese (University of Leipzig) Advanced Microeconomics 69 / 72 Further exercises Problem 1 Discuss the units in which to measure price, quantity, expenditure. If you are right, expenditure should be measured in the same units as the product of price and quantity. Problem 2 Sketch budget lines: Time T = 18 and money m = 50 for football F (good 1) or basket ball B (good 2) with prices pF = 5, pB = 10 in monetary terms, tF = 3, tB = 2 and temporary terms Two goods, bread (good 1) and other goods (good 2). Transfer in kind with and without probhibition to sell: m = 300, pB = 2, pother = 1 Transfer in kind: B = 50 Harald Wiese (University of Leipzig) Advanced Microeconomics 70 / 72 Further exercises Problem 3 Assume two goods 1 and 2. Abba faces a price p for good 1 in terms of good 2. Think of good 2 as the numeraire good with price 1. Abba’s utility p functions U is given by U (x1 , x2 ) = x1 + x2 . His endowment is 1 ω = (25, 0) . Find Abba’s optimal bundle. Hint: Distinguish p 10 and 1 ! p < 10 Problem 4 ∂x (p,m ) Show by way of example that 1∂p1 < 0 and happen. Hint: use perfect complements. Harald Wiese (University of Leipzig) Advanced Microeconomics ∂x1 (p,ω ) ∂p 1 > 0 may well 71 / 72 Further exercises Problem 5 Derive the indirect utility functions of the following utility functions: (a) U (x1 , x2 ) = x1 x2 , (b) U (x1 , x2 ) = min fa x1 , b x2 g where a, b > 0 holds, (c) U (x1 , x2 ) = a x1 + b x2 where a, b > 0 holds. Problem 6 Consider the preferences given by the utility function U (x1 , x2 ) = x1 + 2x2 . Find x (p, m ) . Sketch the demand function for good 1. Sketch the Engel curve for good 1 for the case of p1 < 21 p2 while observing the usual convention that the x1 -axis is the abscissa! Harald Wiese (University of Leipzig) Advanced Microeconomics 72 / 72