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Relaxing the assumptions would not alter basic results. Assume I is exogenous. Income is a given in the model. The individual does not decide how much to work. Assume no saving or borrowing and that all goods are consumed when purchased. No stockpiling. Abstracting from intertemporal decisions. Just 1 time period. ]nQn^n>n]Q\   >4 !%@Budget Constraints  AssumptionsAssume only two goods. Common to label them x1 and x2 or x and y. Primarily for convenience. Much easier to draw two dimensional graphs! Assume the consumer is a price taker. Each consumer faces a budget constraint. They can t spend more than their income. BnGn(n)) G$    )  &Budget Constraint0Amount spent = Amount earned (income). Budget constraint for two commodities: Pxx + Pyy = I More generally, for k commodities: P1x1 + P2x2 +& + Pkxk = I  N#N#>N8'(Graph Budget Constraint (BC) for 2 goods))&Pxx + Pyy = I What will BC look like? May help to rewrite BC such that y is a function of x. Pyy = I - Pxx y = (I/Py) - (Px/Py)x What are x and y intercepts? What is the slope? Always label intercepts and axes!]%RU  RlT S ( <Budget Constraint Graph cont d$Graph shows the feasible set of bundles that can be consumed. Individual can afford to buy any bundle that lies on or below the budget constraint (shaded area). Budget constraint itself represents the different bundles of goods that would exhaust the individual s income, I.<>ZcZrZ>cr Z) Important Notes about the BCThe corner points (x and y intercepts) represent bundles in which the consumer spends all of her income on one good. What does the slope of the budget constraint tell us? (Recall m = -Px/Py) Negative of the slope indicates the rate at which the market permits you to substitute one good for another. Let Px=$2 and Py =$3, then  m = 2/3. For every unit of x you give up, you get 2/3 unit of y. Or give up 2 units of y to get 3 units of x. Px/Py is the price of x in terms of y. The price of one unit of x is 2/3 units of y.m%   M   g  OPs{P* ASteve has income of $100 and faces prices of: Px=$4 and Py =$10. @B/& & & &&,.FWhat is Steve s budget constraint? Pxx + Pyy = I; 4x + 10y = 100 Graph BC. First, solve for y: 10y = 100 - 4x y = 10  (4/10)x What is the slope? -4/10 = -2/5 What does the slope tell us? One has to give up 2 units of y to get 5 units of x. Rate at which market trades good x for good y. #PPP!PPPPeP#  !  e,# P+ $Budget Constraint and Income Changes%%&Pxx + Pyy = I What happens to BC if income changes? Parallel shift of the budget constraint. Increase in income shifts BC out  can afford more What happens to the slope? Doesn t change! Ratio of prices hasn t changed!z4\0,'50$ , Back to Steve& Initially: I = $100, Px=$4 and Py =$10 What if I increases to $200? New budget constraint: I2 = Pxx + Pyy 200 = 4x + 10y y = 20  (4/10)x Slope is the same = -2/5 What happens to the intercepts? Y-intercept = 20, x-intercept = 50 They doubled. Income doubled, so the max you can buy of one good (if spend all money on that good) also doubles.[09   9   #9P?-#Budget Constraint and Price Changes&What happens to BC if price of good x increases? Think about the corners. If spend all money on x, can you buy more or less x? Less x. If you spend all money on y, can you buy more or less y? Same y. So, BC rotates in when price increases! (See BC2) When price decreases? Rotates out. (See BC3)JZ5ZZ9ZZHZZJ59 !  Z.*Rotate around x axis if price of y changes++&/Preferences and UtilityNow know which bundles are available to individual, but still need to figure out how he decides on a bundle. The bundle that is chosen is determined by an individual s preferences. How do we represent an individual s preferences? Build preferences according to three assumptions or axioms (rules generally accepted as true).EZE0Axiom 1 - CompletenessCompleteness: When confronted with two bundles, the consumer can always tell which one is preferred, or if she is indifferent. While any set of preferences can be complete, they might not be rational. Thus, axioms 2 and 3 rule out some inconsistencies.24J41Axioms 2 and 3Axiom 2  Transitivity If x is preferred to y, and y is preferred to z, then x is preferred to z. Likewise, if indifferent between x and y, and indifferent between y and z, then must be indifferent between x and z. Axiom 3  Non-satiation More is always better (gives more satisfaction)! This assumption is often for convenience, as there is nothing irrational about getting satiated.T2Utility Function&@Axioms allow definition of a utility function on all goods  defines an individual s preferences. Utility function  Measures how well off an individual is given that the individual consumes various amounts of the consumption goods. Utility = U(x1, x2) or U(x, y) Utility = U(x1, x2, x3,& , xk) Utility function is a way of ranking bundles. Note that magnitude of difference between U1 and U2 don t tell us anything.>.Lb  +  , |3)Extra Assumptions about Utility Functions**&BAs long as x and y are  goods , utility increases as x and/or y increases. x is hamburgers and y is  all other goods Assumption is that holding y fixed, an increase in hamburgers results in an increase in utility level. In other words, marginal utility of x is positive Marginal utility is the change in utility associated with increasing consumption of x, holding y constant. MUx > 0 For a utility function U(x,y): tKnnK"k%,z5)Extra Assumptions about Utility Functions**&Diminishing Marginal Utility As consumption of a good increases, utility increases but at a slower and slower rate; i.e. at a decreasing rate. First hamburger may be great, second hamburger satisfying, third hamburger okay,& Utility functions are continuous and smooth. zn-n- 7"What does utility curve look like?$#   Why does the curve slope up? Assumption 1  positive marginal utility Why is the curve bowed? Assumption 2  diminishing marginal utility MU at A > MU at B, i.e. as x increases At A, MU is high  small change in x yields big change in U. At B, MU is low.`)SN)SN 8Indifference curvesOften more useful than utility function Indifference curves plot out all possible combinations of the various goods that yield an identical utility level. Indifference curve for utility level, U*, is set of all x and y such that: U*=U(x,y)h(sK )tK 9&What do indifference curves look like?:-Can we have the following indifference curve?..&Z No! Why? Non-satiation assumption At B, consume higher levels of x and y, therefore should have higher utility than at A (not the same) Exceptions? Suppose x is pollution and y is  all other goods If person doesn t like pollution, then could have such a curve. But, means we are relaxing the assumption that all goods are  goods. T ZZ ZZ  ;-Can we have the following indifference curve?..&Z No. Why? Transitivity property According to graph, you are: Indifferent between A and C Indifferent between B and C So, should also be indifferent between A and B But, B is preferred to A (since get more x and more y) Transitivity assumption is not satisfied!j ZZZ8ZZ 8<!Properties of Indifference Curves""&RIndifference curves always slope down. If you consume more of x, then you need less of y to stay at the same utility level (non-satiation). Indifference curves cannot cross. By transitivity. Indifference curves are bowed in towards the origin. Because of diminishing marginal utility. Understand by looking at indifference curve slope.'nZeZZ"nZZZ5nZ]Z'e" 5]=#Marginal Rate of Substitution (MRS)$$&{Negative of the slope of the indifference curve So, curves are bowed in implies that: MRS decreases as x increases and y decreases. What does MRS measure? Willingness to trade x for y while remaining just as well off (indifferent) MRSyx = 5 MRS of y for x Individual needs 5 units of y in exchange for 1 more unit of x. MRSyx is # of units of y willing to give up for 1 more x. VZ.ZZZZN.O 4,U6>MRS cont d At A: MRS is  (slope of dashed line) y is high relative to x. MRS is high since individual would be willing to give up a lot of y just to get 1 more unit of x. Because of diminishing marginal utility. When have just a little x, the next unit of x is worth a lot. But, when have a lot of y, next unit of y is worth just a little.:?MRS cont dZ At B: x is high relative to y. MRS is low Individual is only willing to give up a little y to get 1 more unit of x. Because of diminishing marginal utility. When have a lot of x, the next unit of x is not worth much. But, when have little y, next unit of y is worth a lot. Suppose slope at A = -2. Individual is willing to give up 2 more units of y to get 1 more x and stay at same U. Z%ZZZXZZ% D     @MRS Relationship to MUWhere does this come from? Recall that MUy is the change in utility associated with consumption of additional unit of y, holding x constant. $)d'dB >MRS Relationship to MU (cont d) Take away enough y so that you move from f to g How does utility change? Change is y*MUy Give you enough x to bring you to go from g to h. How does utility change? Change is x*MUx What happens to total utility? U=0 (f and h on same curve) U = 0 = y*MUy +x*MUx0*2*0(2(SV D!>MRS Relationship to MU (cont d)U = 0 = y*MUy +x*MUx Rearrange terms: The negative of the slope of the indifference curve equals ratio of marginal utilities.6mfnE"$Special types of indifference curves Perfect Substitutes How would you characterize MRS? MRSyx is constant. Amount of y individual needs to be given in exchange for one unit of x is the same at points A and B. Note that it not necessarily one for one.T3#4F#$Special types of indifference curvesZ Perfect Complements x and y must be consumed in fixed proportion. Proportion is defined by slope of line through origin. Individual doesn t get extra utility from more x until he gets more y. B makes individual no better off than A.(ZG$`Describe each consumer s preferences for x and y11&=Give up a lot of y for one more x. y is less important than x># ?Give up just a little y for more x. x is less important than y.@$I%Utility MaximizationCan now answer question of how individual chooses bundle to buy from all affordable bundles. What bundle will he choose? He will choose from the affordable bundles (feasible set) the one that gives the highest level of utility. Maximizes utility subject to his budget constraint. B^^J&Where is the optimal bundle? DA is optimal. Why? It is equilibrium  individual has no incentive to change bundle (can t achieve higher utility) More x would increase utility, but can t afford it. Likewise for y. At B, taking away x and giving more y would yield higher utility (and is affordable). B is not equilibrium.LVVL'Equilibrium conditionIndifference curve and budget constraint are tangent. Have equal slopes. Individual and market value goods at the same rate. Chooses x and y such that the MRS equals the rate at which the goods can be traded for each other in the market. B6ZZZ6 N( Rewrite equilibrium condition as^Marginal Utility Principle - Bundle that maximizes total utility is such that marginal utility of the last $ spent on each commodity is the same. Why? Consider what would happen if take away last $ spent on x and spend it on y. Because of diminishing MU, would lose more utility from lost x than would gain from additional y. Not utility maximizing.4Z{Z{V0JPractice Problem  Discussion SectionJack and Jill have decided to allocate $1000 of their individual incomes to purchasing magazines on home theater equipment and magazines on running/exercising. Jack prefers home theater magazines more than exercise magazines, while Jill prefers the exercise option more. In the same graph, draw sets of indifference curves for Jack and Jill which depict these different preferences. Discuss why the two sets of curves are different from each other using the concept of the marginal rate of substitution. If Jack and Jill face the same prices for these magazines, will their marginal rates of substitution of home theater magazines for exercise magazines be the same or different at their optimal goods basket? Assume no corner solutions.FZZO)@Policy Application  Food StampsHow does food stamp program affect consumer behavior? U.S. food stamp program Type of in-kind transfer 19 million participants in 1985 and cost > $11 billion Almost 26 million participants in 2005 and cost > $28 billion Let s consider a very simplified food stamp program. Family will pay $80 to get $150 worth of food stamps.XOZZkZO 6P+(Food Stamps (cont d)hLet I = 250. What is budget constraint without food stamps? PfoodF + Paog(AOG) = I Don t know prices but we know amount spent on each good (like letting prices = $1) 250 = F + AOGp=S= S,=g Q*;What does BC look like for family eligible for food stamps?<<& SRecall: pay $80 for $150 of food stamps. If spend all $250 on AOG, then get no food stamps (y-intercept is the same). Need $80 to purchase food stamps. Can consume AOG = $170 If purchase no additional food, then can consume $150 worth of food (point A). If spend remaining $170 also on food (no AOG), then can spend $320 on food in total).<fVfVR,AWhat would BC look like if family gets $70 cash transfer instead?BB&S-*Under which policy is consumer better off?++& Family does better with cash than food stamps. Achieves U2 versus U1. How would you characterize this household? Have weak preferences for food. Willing to give up a lot of food to get a little more AOG. Does this have to be the case?/K;/  K;T.Z Individuals with preferences such as these are indifferent between the 2 programs. 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ff3LmNLm___PPT10t.B+|"D' = @B D' = @BA?%,( < +O%,( < +D' =%(D}' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* %(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*>%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*>%(DY' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(+P+0+ 0 ++0+ 0 ++0+ 0 ++0+0 ++0+0 ++0+0 +R  0 p P(   r   S m  `}  m    S (m  `<$D 0 m H   0޽h ? ff3LmNLm:2___PPT10. +YD' = @B D' = @BA?%,( < +O%,( < +D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* u%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* u%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* ,%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* ,Q%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* Q%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* %(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* %(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*  %(+2  0  R(  x  c $m  `}  m   c $m   <$D 0 m r  S m p ` m B  s *D0 0 ,$@  0B  s *D 0  ,$@ 0  0m @,$ 0 7x0 2  0ȉmpp W,$  0 7y0 2   0m ,$  0 B25$0 2   0m @ g ,$  0 B10$0 2B   c $D 0  ,$D  0B  @ s *DP `  ,$@ 0   0m w ,$ 0 L Slope = -2/5$ 0 2 H  0޽h ? ff3LmNLm))___PPT10).>x+ؾDK(' = @B D(' = @BA?%,( < +O%,( < +D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*#%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*#B%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*B`%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*`p%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*p%(D' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* %(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =%(Du' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =%(Dh' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*#%(++0+0 ++0+0 ++0+ 0 ++0+ 0 ++0+ 0 +R5  0 X P  (  x  c $q  `}  q   c $q @  <$D 0 q r  S Xq  ` q B  s *D` ` ,$@ 0B  s *D `  ,$@ 0  0XqP 7,$ 0 7y0 2B  c $D`  ,$@ 0   0m p ,$  0 WI1/Px60 2   0\q` G,$ 0 WI1/Py60 2   0#q ,$ 0 CBC1$0 2   0(q  ,$  0 7x0 2B  c $D`  ,$@ 0  0-q,$ 0 KBC2,0 2 B  s *D P ,$@ 0  00qp W,$ 0 gI2/PyF0 2    08q  ,$ 0 gI2/PxF0 2  H  0޽h ? ff3LmNLm((___PPT10z(. XR+怲D&' = @B DI&' = @BA?%,( < +O%,( < +D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(DI ' =%(D ' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*4%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*4]%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*]%(D' =%(D3' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.