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O  =FtZMicroeconomics and Policy Analysis  PUAF 640/Professor Randi Hjalmarsson Fall 2008 Lecture 4  JClass Outline  Chapter 3.1, 3.2, 3A TReview  Deadweight loss of Christmas Lagrangian Utility Maximization Comparative Statics Price Changes Income Changes Demand Curves Again Individual Demand Market DemandLZ Z ,& "R]-Deadweight Loss of ChristmasPLast class: many individuals would be better off (higher utility) with cash transfer than food stamps. People like choices! Same type of story for Christmas presents& Bg,g-^._/ So, if I give you cash, you choose your optimal bundle (unconstrained), which is completely efficient. The amount of money I give you is the minimum amount necessary to move you from U0 to U1. Giving a gift is inefficient since you can reach U2 with a cash gift smaller than $x (see dashed budget constraint). Because you can achieve higher utility, U1, with a cash gift than you can with a gift (G) of equal value (U2), there is a deadweight loss. hZhQ2kA `0 The better the giver knows the recipient (and her preferences), the smaller the DW loss. Using survey data from his students, he finds empirical evidence of this. DW loss from grandparents and aunts/uncles much greater than that from friends and significant others. Value can actually be created instead of destroyed if the giver knows the recipients preferences better than the recipient does.6YY!Utility Maximization Review  When is utility maximized? Budget constraint is tangent to the indifference curve When MRSyx = MUx/MUy = Px/Py Suppose I give you an explicit utility function. How do we mathematically solve for consumer s equilibrium bundle?d$T1B$T1B>`u"Lagrangian Utility Maximization Maxx, y U(x, y) subject to I = Pxx + Pyy 3 steps to solve for x and y: Write down the  Lagrangian . Take partial derivatives with respect to each variable (x, y, and ) and set equal to zero. Solve system of equations. (3 equations, 3 unknowns))q_O ~0 EM#Step 1: What is the Lagrangian? Utility function to be maximized plus  times the budget constraint. Maxx, y U(x, y) subject to I = Pxx + Pyy Rewrite the budget constraint: I - Pxx  Pyy = 0 So, L = U(x, y) + (I - Pxx  Pyy ) f%& "        $  & %&Step 2: Take partial derivatives of the L with respect to x, y, and  and set equal to 0.:Z(&&&"&D| L = U(x, y) + (I - Pxx  Pyy ) Partial with respect to x: f?  "T!)Step 2: Take partial derivatives of the L with respect to x, y, and  and set equal to 0.:Z(&&&"&DzL = U(x, y) + (I - Pxx  Pyy ) Partial with respect to y: ^>  "T!+Step 2: Take partial derivatives of the L with respect to x, y, and  and set equal to 0.:Z(&&&"&D|L = U(x, y) + (I - Pxx  Pyy ) Partial with respect to  : t?   j."Step 3: Solve system of equations.3 unknowns: x, y,  Good way to begin: Solve (i) and (ii) for  and set equal to each other. Familiar? Equal marginal utility principle  Last $ spent on x and y give same marginal utility. Equilibrium condition! Budget constraint must hold!)@X4   ( 734-5 Brief Review of DerivativesDerivative of xa with respect to x. axa-1 Down decrease by 1. Derivative of cxa with respect to x. acxa-1 Derivative of c with respect to x. 0 $%##,>B0 #U(x,y)=x1/2y1/2 I=100, Px=5, Py=10l$& && & & && &&6 What is the optimum consumption bundle? Step 1  Write the Lagrangian. It is useful to keep it in general terms (do not yet substitute in income and prices). L = x1/2y1/2 + (I - Pxx  Pyy )IX!*X T= i1 -Step 2: Partial derivative with respect to x.".,&"&DL = x1/2y1/2 + (I - Pxx  Pyy ) ! B6 -Step 2: Partial derivative with respect to y.".,&"&BL = x1/2y1/2 + (I - Pxx  Pyy ) ! B7 ZStep 2: Partial derivative with respect to .".+&"&+FL = x1/2y1/2 + (I - Pxx  Pyy ) # B8"Step 3: Solve system of equations.3Rewrite (i) and (ii) and set them equal. Simplify.() 4<What next? Still haven t solved for x and y as function of givens (prices and income)VV&Found that y = (Px/Py)x Plug this into (iii), the budget constraint. I  Pxx  Pyy = 0 I  Pxx  Py (Px/Py)x = 0 I  Pxx  Pxx = 0 = I  2Pxx x = I/2Px Individual demand function for x. How x varies with changes in income and its own price. FZVZZ"Z7ZZ-    "7 3  w>Still need to solve for y:Plug solution for x back into budget constraint or equilibrium condition. y = (Px/Py)x and x = I/2Px x = I/2Px and y = I/2Py Demand functions for x and y.Q     PJ?/Last Step! Still need solve for optimal bundle.00&d Plug in income and prices. x = I/2Px and y = I/2Py Given: I=100, Px = 5, Py = 10. x = 10, y = 5.eP%,D P  @Practice Problems>Solve for an individual s demand functions for x and y if they have the following utility function: U(x, y) = xy 3.13 in Katz and Rosen Abe consumes guns (x) and butter (y). His utility function is U(x, y) = x  3/y. Px = 9, Py = 16, I = 900. Find utility maximizing quantities of x and y. Find demand functions for x and y. What happens to utility maximizing bundles if all prices and income increase by same %? dn nQd   Q  bdWa1AComparative Statics Process of comparing two equilibria  Statics implies that we are not analyzing the dynamics of how the consumer moves from the first to second equilibrium. Changes in equilibrium due to: Own price changes. Cross price changes. Income changes.<Z8ZCx8, BOwn Price Effects Impact of change in price of a good on quantity demanded of that good. If the price of x decreases, how will this effect the quantity demanded for x? What happens to BC when price changes? Rotates!8 < CSDecrease in price of x does not have to increase consumption of x. (more next week)TT&Increase in x and y x does not change.DCross Price EffectsImpact of change in price of one good on quantity demanded of another good. Substitutes Increase in Px increase in y. Heroin and methadone, coffee and tea, Toyotas and Hondas. Complements Increase in Px decrease in y. PB&J, cars and gasoline, coffee and cream If the 2 goods are unrelated, change in price of x has no impact on consumption of y..Z[ KVL   H    8  :,ffE#Cross Price Effects for Substitutes&Increase in price of x, increase in y.F#Cross Price Effects for Complements'Increase in price of x, decrease in y. G'Cross Price Effects for Unrelated Goods((&  Increase in price of x has no effect on y. Theory alone does not tell us how x and y are related. Must look at data. Could be different for different individuals or groups.HIncome Changes>Normal Goods Increase in I implies an increase in consumption&11IIncome ChangesIInferior Goods Increase in I implies a decrease in consumption (Amtrak) 699J&Normal Goods: Luxuries vs. NecessitiesLuxuries Goods for which fraction of total income spent on them increases as income increases Necessities Goods for which fraction of total income spent on them decreases as income increases Food, housing, utilities, etc.L U u U uK*Tracing Out Changes in x as Income Changes++&  Income  Consumption Curve Set of optimal bundles traced out as consumer income varies Engel Curve Traces out relationship between income and quantity consumed of one good. Note normal good.L<\<\LShapes of Engel Curves1Which one is a necessity and which is a luxury? MTracing Consumption with Price Changes  Derive Individual Demand CurveHH&  Consider price decreases. Price-Consumption Curve Set of commodity bundles traced out as price changes, all else equal. Individual demand curve N3F3FN Deriving Market Demand Curve&How do we combine individual demand curves to get the market demand curve? Market demand curve shows at each price, the quantity demanded by all participants in the market, holding all else equal. Market demand is found by horizontal summation of individual demand curves.tKzL6ECO!Market Demand for 2 Individuals &<Market Demand = D1 for P>=3 For P<3, Market Demand = D1 + D2@=$Q"/Market Demand Example (in class or discussion?)00&Individual s demand function: Qd = 24  6P 1000 individuals in a city (market). What is the market demand function? QM1d = 1000(24  6P) = 24000  6000P At a price of $1, a single individual demands 18 units. 1000 individuals demand 18000 units. Sum of the quantities demanded by all consumers at every price level. Quantity (not prices!) is multiplied by 1000.ztT~E.JR#<Market Demand Example (cont d)d500 more individuals move into the city. Each has the following individual demand function: Qd = 32  4P What is the total market demand for the city? For the second group of individuals, QM2d = 500(32  4P) = 16000  2000P Need to combine QM1d and QM2d to get total market demand. Look at graphically. N\Z ZUZ$Z;ZZ\  %  \S$<Market Demand Example (cont d)QM1d = 24000  6000P QM2d = 16000  2000P Total Demand: If P>4, QTd = QM2d = 16000  2000P If P<=4, QTd = QM2d + QM1d QTd = 40000  8000P*:V >C#  T%&Policy Application: Charitable Giving :People all over the world contribute to charity. In U.S. of households contribute an average of $978 annually (2% of personal income) (a few years ago). Is charitable giving consistent with utility maximization? Yes, if donating money to others gives satisfaction, then charitable giving can be seen as a good.*ZcZcU&?(1) Illustrate how individual decides how much charity to give.@@&Individual has I = $25,000 (after tax) Chooses between: x  donations to charity y  composite of other goods consumed by self. What does BC look like? Pxx + Pyy = I Assume Py = $1. What is Px (charity)? For each $ donated, one gives up $1 of own consumption. So, Px = $1. Think of price of charity as the opportunity cost of charity.8H&8H     =   D  b HEV'Graph utility maximizing bundle W(!(2) Tax Deductions and Charity ""&What is the effect (on behavior) of allowing individuals to deduct charitable contributions from their taxable incomes? Key: tax deduction changes the effective price of donating! So, BC rotates. Why? Suppose 25% tax on income. Without tax deduction, $1 spent on charity is $1 less spent on self. With deduction, you donate $1 to charity and don t pay the $0.25 tax on that $. So, your taxes decrease by $0.25 . By donating that $1, you can spend $0.75 less on yourself. Opportunity cost of charity is 0.75 = PxyL7y L 5 X)FTax Deductions and Charity (cont d) Y*.Effect depends on shape of indifference curves//& Z+TPractice Question  for discussion section++&>Tom receives a $40 gift card from his father that he can redeem in a movie theater. Suppose the theater provides only movies and popcorn. Watching one movie costs $8 and a bucket of popcorn costs $4. Represent Tom s spending capacity at the theater as a budget constraint (in a linear equation form). Then graph this budget constraint, with movies (M) on the horizontal axis and popcorns (P) on the vertical axis.2nn[, Suppose Tom maximizes his utility by watching 3 movies. How many buckets of popcorn should he buy to maximize the utility? Show your calculations. Using the diagram from part (a), graph how his indifference curves might look. Imagine that Tom s brother uses up $20 from the value of the card  so that he is left with only $20 value on the gift card. Show how this will alter your diagram from part (c), assuming both movie and popcorn are normal goods. How will your diagram differ if popcorn was an inferior good? Show this graphically too, and if required on a separate diagram. (In<G/\   0` 3333ff3` 3333f33ff3` "3333̙ff3` Kf3̙` &e̙3g3f` f333̙po7` ___f3̙;/f9` ff3Lm` ff3LmNLm>?" dd@*?nAd@q<nAqFLK#M n?" dd@   @@``PR    M`2p>> & (      Hli? ?" `}  X Click to edit Master title style!!  @   HXl? ?" `  RClick to edit Master text styles Second level Third level Fourth level Fifth level!    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