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Reading materials –Chapter 5
Besanko and Braeutigam: Perloff: Chapter 5. pp. 111‐116. pp. 136‐141. The Theory of Demand pp. 152‐164. I will post these pages on Blackboard.
Individual Demand Individual Demand
Assume: Price Changes: •I = $20 Clothing •P = $2 (units per 10 C Using the figures developed in the previous chapters, we month) •PF = $2, $1, $.50 will be able to analyze the effect of a price change on the individual’s optimal choice. 6 A U And the analysis of these effects will give us the demand 5 1 C B Three separate indifference curves curve. U 4 3 are tangent to each budget line.
U2
Food (units per month) 41020 3 4
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Deriving an Individual’s Wine, Gallons Demand Curve per year 12.0 Individual Demand (a) Indifference Curves and Budget Constraints The price-consumption e3 Price-consumption curve curve traces out the 5.2 Clothing e2 10 4.3 (units per utility maximizing e I 3 month) 1 basket for the 2.8 I2
various prices for food. L1 (p = $12) I 1 L2 (p = $6) L3 (p = $4) 6 A b b b 0 26.7 44.5 58.9 Beer, Gallons per year (b) Demand Curve for Beer U1 C 5 p , $ per unit B b
4 U3
12.0 E1
U2
6.0 E2 E 4.0 3 Food (units D1, Demand for beer per month) 4120 026.7 44.5 58.9 Beer, Gallons per year 0 5 6
Finding a demand curve ‐ Example Finding a demand curve –Example Consider the Cobb‐Douglas utility function, u(x,y) = xy, with marginal utilities Consider now a utility function u xy 10x, where MU y 10, MU x, MU x y, MU y x, and we leave prices and x y income levels as general as possible, i.e., p , p , I x y And assume I=$100, and px=$1 but we leave unspecified py since we seek to find the demand curve for good y st 1 ) Budget constraint On your own (as a function of its price). 2nd) MRSx,y=price ratio Otherwise, we would be obtaining an optimal consumption bundle (a number for the optimal consumption of good y, rather than a function of py). 1st) Budget constraint nd 2 ) MRSx,y=price ratio Next slide
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Learning by doing 5.2: Learning by doing 5.2: Hence, the demand for good y is positive,
Uxyx10 MUy 10 and MUx Px X Py Y I xy 100 10 p y 0 only if 100-10pppy 0 100 10yy 10 Suppose that I 100 and pxy 1 p unknow then 2 p y
1) pxpyxpyxy1 100 y 100 B.L. 100 10 p y if p y <10 MU p y 10 1 y 2 p y 2) MRSxx x y 10 p plug into 1 y 0 otherwise MUyy p x p y Plugging (2) into (1) 100 10 py y10 pyy py 100 2 py y 10 p y 100 y 2py
We can now depict the demand cure for good y we just found Effects of Income Changes
Price (Vertical spike). of food An increase in income, from $10 to $20 to $30, with the prices fixed, shifts the consumer’s E G H demand curve to the right. $1.00
D3
D2
D1 Food (units 4 10 16 per month)
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An Inferior Good Individual Demand
Steaks 15 (units Income-Consumption per Curve Normal Good vs. Inferior Good month) Both hamburger and steak behave C as a normal good, Income Changes 10 between A and B... When the income‐consumption curve has a positive U 3 slope: …but hamburger The quantity demanded increases with income. becomes an inferior The income elasticity of demand is positive. B good when the income 5 consumption curve The good is a normal good. bends backward between B and C. U2 A U 1 Hamburgers 52010 30 (units per month)
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Individual Demand Effects of Income Changes: Engel Curves
Normal Good vs. Inferior Good Engel curves relate the quantity of goods consumed to income. Income Changes If the good is a normal good, the Engel curve is upward When the income‐consumption curve has a negative sloping. slope: If the good is an inferior good, the Engel curve is downward The quantity demanded decreases with income. sloping. The income elasticity of demand is negative. Let’s look at two examples: The good is an inferior good.
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Effects of Income Changes: Engel Curves Engel Curves Income ($ per 30 month) 30
Income Inferior ($ per Engel curves slope Engel curves slope month) upward for backward bending 20 normal goods. 20 for inferior goods.
Normal
10 10
Food (units Food (units 0 481216 per month) 0 481216 per month)
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Finding Engel Curves Finding Engel Curves Consider again the Cobb‐Douglas utility function Consider again the Cobb‐Douglas utility function
u ( x, y ) x y with MU x y and MU y x
p x x p y y I is the budget line. p x Then, plugging the above result, y x , into the budget line yields In order to find the Engel Curve for good x, we first need to find the p y demand curve of x : p x p x p y I p x p x I 2 p x I x y x y x p y Using the tangency condition, we obtain Solving for x, we obtain the demand curve for x MU p y p p x x x x y x I MU y p y x p y p y x 2 p x
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If, for instance p x $1, then the Engel Finding Engel Curves curve I 2p xx becomes I 2x : Since the Engel curve has the amount of the good x on the horizontal axis and income (I) on the vertical axis… I = 2x
I we first need to solve for I in the demand curve x , 2 p yielding an Engel curve of x
I 2 p x x
2 Slope of the Engel curve
Engel Curve in corner solutions • Of course, the Engel curve of good y implies y = 0 for all income • What if, instead, we face a utility function for perfect levels. substitutes, such as u(x,y) = x + y, which we know yields • That is, the equation of the Engel curve would be I = 0 for all y, as optimal consumption bundles at a corner? depicted in the figure. • Let’s practice: MU x 1 U (x, y) x y its MRS x, y is MRS x, y 1 MU y 1
Py Py MUy MUMU if p p, then 1 MRS xx yx P P MU x x x PPxy which implies that this consumer would be still willing to consume more units of x and less of y. Graph on next slide.
What about the Engel curve for good x?
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Income and Substitution Effects
A fall in the price of a good has two effects: Substitution & Income Substitution Effect Consumers will tend to buy more of the good that has become relatively cheaper, and less of the good that is now relatively more expensive.
Engel Curve, Positively sloped for all income levels.
Slope of 1: Every extra dollar is spent on good x.
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Income and Substitution Effects Income and Substitution Effects
A fall in the price of a good has two effects: Substitution Effect Substitution & Income The substitution effect is the change in an item’s Income Effect consumption associated with a change in its price, Consumers experience an increase in real purchasing power holding the utility level constant. when the price of one good falls. When the price of an item declines, the substitution effect always leads to an increase in the quantity of the item demanded.
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Substitution and Income Effects Income and Substitution Effects with NORMAL Goods Clothing
BL2 Income Effect The income effect is the change in an item’s BL1 consumption brought about by the increase in purchasing power, with the price of the item held constant. C When a person’s purchasing power increases, the BLd I 2 quantity demanded for the product may increase or A B decrease. I1
0 Substitution Income effect Food effect Total effect
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Income and Substitution Effects: INFERIOR Good Income and Substitution Clothing Effects: GIFFEN Good Since food is an BL2 inferior good, the income effect is C Clothing negative. However, BL1 the substitution effect A is larger than the income effect. C BLd I 2
A I2 B
B Substitution Effect I1 I 1 O Food Total effect Substitution effect Total Effect Food Income Effect Income effect
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Income and Substitution effects For a decrease in prices: Income and Substitution effects SE IE TE
Demand curve is Note that if the total effect of a price decrease is negatively sloped Normal ++ + positive (TE >0), the demand curve is negatively (as usual) good sloped. Potatoes (in Ireland Inferior +- + Indeed, a decrease in p implies an increase in the during the XIX quantity demanded. century) and good Tortillas (Mexico, This applies for both normal and inferior goods… 1990s). Giffen +- - But NOT for Giffen goods: They would have a positively sloped demand curve, i.e., a more expensive good would lead to an Demand curve is good positively sloped increase in the quantity demanded (they are a rare species of goods!)
Please note that all signs would change if we analyze an increase in prices. 33
Learning by doing 5. 4 Learning by doing 5. 4 (cont’d): Consider again the Cobb‐Douglas utility function
Uxy with marginal utilities MUyxy and MUx Basket C (at final prices px2 $4) :
Suppose that Ip $72 and yxx $1. In addition, assume that the initial p12 9 then decreases to p 4. 1) 4xpyy 72 MU p y 4 2) MRSxx y 4 x plug into 1 Basket A (at initial price px1 9): MUyy p x 1 1) 9xpy 72 y 4xx 4 72 8 x 72 x 9 plug into y MU x px y 9 2) MRS yx9 plug into 1 y 9(4) 36 MU y pxy 1 C (9,36) 9xx 9 72 18 x 72 x 4 plug into y y 9(4) 36 A (4,36) AC 49 of x , thus implying a total effect of TE=9‐4
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Learning by doing 5.4 (cont.) Learning by doing 5.4 (cont.) In order to disentangle the SE and IE from the TE = 5 we just Second, BLd must be tangent to IC1. Since the slope of BLd is the found, we need to first find the decomposition Basket B. same as BL2, then px 4 slope of BLd slope of IC1 p 1 We know that the decomposition bundle B satisfies two y 4 MU x properties: 1 MU y y First, B must give the same utility level as the initial basket A. 44yx x Hence, since basket A = (4, 36) yields a utility level of Combining the information we found from both properties, u = xy = 4 (36) = 144, then basket B must also satisfy xy = 144. xy 144 xx4 144 4 x2 144 x 6 yx 4 Hence, y 46 24 B 6, 24 Decomposition Basket B
Learning by doing 5.4 (cont.)
We can now summarize our results with a figure depicting the Summary of learning by doing 5.4: units consumed of good x (the good whose price changed): As price of good X decreases… ‐‐‐‐A‐‐‐‐B‐‐‐‐C The SE leads us to an increase in consumption of good X from 4 6 9 4 to 6. SE IE The IE will measure the consumption from decomposition TE { =5 basket B to basket C (From 6 to 9)
Note: When you examine the SE remember that at B you
SE X B X A 6 4 2 reach the same indifference curve as at A, but your BL has the same slope at bundle C. IE XC X B 9 6 3 0(normal good)
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Special Case: Special Case: SE and IE with quasilinear utility functions What happens with the SE and IE when we deal with a quasilinear utility functions, such as U (x, y) x y X is neither a normal good nor an inferior good.
SE=XB‐XA>0
IE=XC‐XB=0 (no income effect)
SE SE Hence, SE=E since IE =0 IE=0 Hence, SE=E since IE =0 IE=0
Learning by doing 5.6, where IE = 0 Let’s now consider a Quasi‐linear utility function Learning by doing 5.6, where IE = 0 1 UxyMU2xy and MU 1 x (At final prices)
Ip10, xx120.5, p 0.2, p y 1 Basket C: 1) 0.2xy 10 (At initial prices) 1 Basket A : MU p 0.2 1 1 2) MRS xxx x 25 1) 0.5xy 10 MUyy p 11x 5 1 0.2 25yy 10 5 MU p 0.5 1 1 C (25,5) 2) MRSxx x x 4 MUyy p 11x 2 0.5 4yy 10 8 A (4,8)
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Learning by doing 5.6 (cont.) Decomposition Basket B. It must satisfy two properties: Learn by doing 5.6 (cont.)
1) Same utility level as basket A=(4,8), so Uxy1 224812 Thus we know, 212xy We can now summarize our results about SE and IE: B coincides with C
2) BLd must be tangent to IC1 and parallel to BL2 1 MU p 0.2 1 1 xxx x 25 MUyy p 11x 5 slope of IC12 slope of BL (and BLd ) SE = xB –xA = 25 –4 = 21 225yy 12 2 IE = x –x = 25 –25 = 0 Hence, the decomposition bundle B is B 25, 2 C B Notice there is no IE because she consumes the same amount of good x at bundles B and C.
A more compact representation of the SE and IE using the compensated demand curve From Perloff’s textbook (posted on Blackboard) Let us use the Compensated Demand Curve to analyze SE and IE: In our exercise from the expenditure minimization problem we Using h(p1, p2,U ) we obtain the minimal expenditure that the obtain the compensated (or Hicksian) demand. (See the last consumer needs to incur in order to reach indifference curve U at part of the Ch. 4 lectures) prices p and p 1 2, X 1 X2
E ppU12,, phppU 1112 ,, phppU 2212 ,,
Min p1 x1 p 2 x 2 Hence, E _ h(P ,P , ū) h1 ( p1, p 2 ,U ) 1 2, p subject to u ( x ) u 1
This is just an amount of good 1 bought by this consumer,
i.e., equivalent to q1.
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A more compact representation of the SE and IE using the compensated demand curve This equation is often regarded on the “Slutsky equation” Plugging the minimal expenditure into the consumer’s income level I and describes the SE and IE as follows: we obtain h D D q1 p1 p1 E hppU12,, Dpp 12 ,, EppU 12 ,, minimal expenditure to reach U SE TE IE
And differentiating with respect to p1, That is, SE = TE + IE h Note that captures the SE since the compensated h D D E p1 ,U p p E p demand operates on a “target” indifference curve for 1 1 1 different price levels. D D Second, q1 reflects the IE since indicates how an h D D E E q1 increase in expenditure increases the demand for good 1. p1 p1 E
p1 E Multiplying all terms by and the last by , we obtain Hence, we can summarize the above big expression more q1 E h p D p D p E compactly as follows 1 1 q 1 * p q p q E 1 q E QI, 1 1 1 1 * And rearranging, * QI, D E p1q1 E q E TE SE 1 IE q ,I where: 1) ε is the standard definition of price elasticity of demand 2) ε* is the definition of price‐elasticity applied to the compensated demand curve
3) εq,I is the income‐elasticity of demand, and 4) θ is the budget share of that good.
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For an inferior good, ε < 0, implying Examples Q,I * Q,I Since SE IE for inferior (not 1) Garlic, θ≅ 0. Hence IE≅ 0, and TE=SE Giffen) goods. 2) Housing, θ=0.4 (40% of monthly expenditures), εQ,I=1.38, ε=‐0.6 Hence, ε<0 which implies that a consumer’s demand is negatively sloped (as usual). * Then we can find the value of ε by using the above For a Giffen good, ε < 0 (and extremely negative), expression ε*=ε+θε = -0.6+(0.4*1.38) = -0.04 Q,I Q,I implying that * Intuitively, if the consumer is not compensated for a 1% price Q,I increase in housing, his demand drops by 0.6%. Because SE IE for Giffen If, however, he is compensated (so he reaches his initial indifference goods. curve at the new prices) his demand only decreases by 0.04%. Hence, ε> 0 which implies that demand is positively sloped! (only true for Giffen goods).
Can all goods be inferior? NO!! Another way to show this result, by using elasticities: The figure represents an increase in income, shifting the consumer’s budget line from BL to BL . If his optimal consumption bundle 1 2 Consider a list of optimal consumption bundles x1(p, I), x2 (p, I), …, xn(p, I). under BL1 is e, where does his optimal bundle under BL2 lie? Hence, this consumer’s budget is exhausted at this optimal consumption bundle, that is
Budget Line P1 X 1 P2 X 2 ...Pn X n I for n goods
In order to examine what’s the effect of an income change let’s differentiate with respect to I,
dx dx dx P 1 P 2 ...P n 1 1 dI 2 dI n dI Here?
Some good must be normal (not all inferior): If the consumer was buying less of both x and y, he would pick a bundle inside BL1, thus not exhausting all his income.
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For a setting with only two goods (as in the above figure): or or 1 1 x1 ,I 2 x2 ,I
1,1 0 We can now multiply and divide by xi *I Can all goods be inferior? Let’s see what would happen if they were all X I dx inferior: P 1 1 ... 1 1 0 and 0 X 1 I dI 1 x1 ,I x2 ,I Rearranging,
P X dx I 1 1 1 ... 1 If that was the case, we wouldn’t be able to obtain a sum that adds up to 1: I dI X 1 { { x , I 1 1 1, xI 2 xI , 1 12
Example Consider you work for a firm selling a good in a developing country. For simplicity, assume that most households only spent money on food (f) and other goods (o), which is the goods you sell. In addition, consider you Consider that you have information about the average know the income elasticity of demand for food, f ,I [0,1] but you don’t
know the income elasticity of other goods, o,I . budget share spent on food: as the country developed, it decreased from = 0.6 to = 0.37. We can then However, from the above equation; we know that evaluate the upper bound for these values of , as 1[ f ,I ] (1) 1. Hence, 0, I is follows: f ,I 0,I 0,I 1 Don’t know: 0,I
IF 0.6 In 1983 1 1 Upper bound for since (0,1) 2 .5 1 o, I Upper bound In 1983 IF 0 1 0.37 In 2005 1 1 0.6 f ,I 0,I 1 1 1 .59 In 2005 1 0 .37 1 1 1 lower bound for IF f ,I 0,I 1 o, I .
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Measuring Welfare Changes Consumer Surplus
The consumer surplus 20 of purchasing 6 concert 19 tickets is the sum of the Price surplus derived from ($ per 18 each one individually. ticket) 17 6 5 16 4 Consumer Surplus 15 6 + 5 + 4 + 3 + 2 + 1 = 21 14 Market Price 13
While still imprecise, the range for O,I became more accurate as declined. Business strategy: a 1% in GDP in the developing country will lead to a 0123456Rock Concert Tickets maximum increase in your sales of 1.59%. 62
Learning by doing (cont.) Q 1) Point A is the crossing of the demand function p=10‐ 4 Learning by doing (Consumer Surplus): and market price p=3. Then, Q=# of gallons of milk purchased at P dollars QQ 310 7Q 28point A ($3, 28 units) Q=40‐4p —> But we want p to be in the y‐axis, then we need to solve for p 44 (obtaining the indirect demand function), as follows: Q 2) CS is the shaded blue triangle G in the previous slide 4 p 40 Q p 10 4 10‐3 1 { CS (28)(10 3) 98 2
A { height
28 {
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‐So far we measure CS using the area of the triangle below the Demand curve, because demand curve was linear (straight line) What if the demand function is not a straight line?
Consumer Surplus with a Cobb‐Douglas Utility Function Consider the following Cobb‐Douglas utility function Uxy 0.6 0.4 What if it is not? And assume that, while income remains constant at I = 300, the price of good x increases from
ppIxx15toto 20 300 Example of a non‐linear 1) Find the consumer’s demand curve for any P 180 x, demand curve:x As a practice, use the tangency condition to find that the market px demand for good x is 0.6I 0.6(300) 180 x pppxxx
Measuring CS of a Cobb‐Douglas Utility What if the demand function not a straight line? Function (which yields a Non‐Linear Demand)
Hence, the decrease in CS when px increases from Not a straight line. $15 to $20 is
20 180 20 CS d p 180 ln p 180 ln 20 ln15 51.79 xx15 15 px
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What if we incorrectly assume linear demand (i.e., so CS=rectangle+triangle): Query #1 1) Area of A=(20‐15)9=45 (Rectangle) 2) Area of B=(20‐15)*(1/2)*6=15 (Triangle) 3) Total loss in CS= 45+15= 60>51.79 (overestimation of the Suppose the consumer’s demand curve for a good can be loss in CS) expressed as P=50-4Qd. Suppose that the market is initially in equilibrium at a price of $10. What is the consumer surplus at the original equilibrium price? a) 100 b) 150 c) 200 X=180/p x d) 250
Query #1 ‐ Answer Query #1 - Answer Graphically Answer: C ($200)
The consumer surplus is the difference between the maximum P=50‐4Q amount a consumer is willing to pay for a good (y‐intercept) and the amount they must actually pay (equilibrium price) below the demand curve. CS=1/2 *(50‐10)*10= (1/2) *40*10=$200 From the demand curve, we see that when Qd = 0, P=$50. When P=0, Qd=12.5. Those are our y‐and x‐intercepts, respectively. From there, we can subtract the maximum price consumers are willing to pay, $50, form the equilibrium price, $10, which gives us a difference of $40. When we plug P*=$10, the equilibrium given to us, into the demand curce, we find that Q*=10 Consumer surplus is simply the area of the triangle CS=(.5)(40)(10)=$200
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Measuring Welfare Changes Measuring Welfare Changes After a price decrease, the consumer is better off. But how can we measure by how much? Equivalent Variation: A measure of how much additional Compensating Variation: A measure of how money a consumer would need before a price reduction to much money a consumer would be willing to give be as well off as after the price decrease. up after a reduction in the price of a good to be just If prices are going to decrease, then consumers are going to be better as well off as before the price decrease. off because they can purchase more. If prices increase, then I need more income to maintain ▼p How much income should we give them today, to make them ∆p the same level of utility. just as well‐off as they will be tomorrow after the price decrease? { How much money would I need? CV EV If prices decrease, then I don’t need all my original income If prices are going to increase, then consumers are going to be worse to remain at my original utility. off. ▼p ∆p { How much money would I be willing to give up in How much money would we need to take from the consumer order to remain at the original utility level? CV today, before the price increase, in order for him to be just as worse‐off as he is going to be tomorrow? EV
Compensating Variation of a price decrease
0K— consumer’s income 0L— income needed to In order to provide a graphical representation of CV and purchase basket B at the EV, we normalize the price of good y, so that p = $1. new prices of x y 0K‐ 0L=KL is the CV This allows us to identify the vertical intercept of any Definition: The amount of budget line as the individual income. money that the consumer will be willing to give up, CV Why? Recall that the vertical intercept of BL is , so if after the price change, in order to maintain the original utility that he had py = $1, becomes I. before the price change.
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Equivalent Variation of a price Query #2 decrease 0J— income needed to buy basket E at the old prices of x. Compensating variation is: 0J ‐ 0K=JK is the EV EV Definition: The amount a) the change in income necessary to hold the consumer of money that we need at the final level of utility as price changes to give to the consumer, before the price change, b) always the area under the demand curve and above in order to make him the price paid just as well‐off as he will be after the price change. c) the change in income necessary to restore the Generally, CV ≠ EV, consumer to the initial level of utility except with quasilinear d) the difference in the consumer’s income between the utilities (where we will purchase of the original basket and the new basket at find that CV = EV). the old prices
Query #2 ‐ Answer Example –CV and EV with quasi linear utility CV and EV with no Income effect: Answer C Consider the following quasi linear utility function Compensation variation is a measure of how much Uxy2 Ip10, 0.5, p0.2, p1 money a consumer would be willing to give up after a xx12 y Assume that I = 10 and py = 1, but the price of good x reduction in the price of a good, just to be as well off decreases from p = 0.5 to p = 0.2. before the price decrease. x1 x2 Compensating Variation (CV) Answer A describes the Equivalent variation. CV= Cost of initial basket A ($10) ‐ Cost of the The income effect causes the magnitude of the decomposition basket B (0.2(25)+2=$7) compensation and equivalent variations to be different. CV= 10‐7=$3 Recall that basket B=(25,2)
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Example –CV and EV with quasi linear utility In search of basket E…
Equivalent Variation (EV) We know that basket E must reach the same utility EV= Cost of buying basket E at initial prices –Cost of level as basket C. buying initial basket A ($10) Since basket C is C = (25, 5), its associated utility is u = But, where is basket E? 2 25 + 5 = (2*5) + 5 = 15. Hence, we need that basket E satisfies 2 + y = 15.
In search of basket E… In summary, the CV and EV coincide, i.e., CV=EV= $3, when the IE is absent (which occurs when we have a We also know that at E, the slope of U2 = slope of BL1 (old price ratio) quasilinear utility function). 1
MU 13 x x 0.5 1 1 MU x 4 y 11x 2 $3=EV Plugging this result, x = 4, into the condition we found { above 2 + y = 15, yields $3=CV 2 4yyE 15 11 (4,11) { Therefore Basket E is E = (4, 11). Hence, the cost of buying E is (0.5)4+11=13 As a consequence EV = Cost of basket E –Cost of basket A = 13 –10 = $3
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What if we use CS to measure change in welfare that arises from the decrease in the price of good x? Hence, since the demand If we did… function is a curve, the increase Uxy2 in CS is given by the integral: 0.2 0.2 Ip10, xx120.5, p 0.2, p y 1 1111 CS2 d px 3 pp0.2 0.5 0.5 xx0.5 1 Gain in CS The demand for x is x . p2 x which is exactly the same as the CV and EV since we have a quasilinear utility function.
Note: Using CS only yields the same welfare change as CV and EV if income effects are absent, i.e, IE = 0. Otherwise CV≠EV≠CS.
What if IE > 0? Consider the following Cobb‐Douglas utility function What if IE > 0? U(x,y)= xy From L.D. 5.4 we found A= (4,36) a) CV I = $72 C= (9, 36) CV= Cost of initial bundle A –Cost of decomp. basket B at the final prices = B= (6, 24) Py= $1 ’ Px= $9Px = $4 = 72 –(46)+(124)= 72‐48= $24
ABC Cost of bundle A Cost of bundle B(6,24) at the final prices $9*4+$1*36=$72 46 9
SE = 2IE = 3
Since IE ≠ 0 we can then expect that CV ≠ EC ≠ CS.
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B) EV On the other hand, we know that at E, EV= Cost of basket E of final prices –Cost of initial basket A y 9 slope of U2= slope of BL1 y 9x We know that the cost of buying basket A is $72, but… x 1 Combining y = 9x with the condition we found above, xy = 324, yields x (9x) = 324, or x = 6 units. Where is basket E? 1) On one hand, we know that E reaches the same This entails that y = 9x = 9*6 = 54 units. utility level U2 as basket C. Therefore, basket E is E = (6, 54). Since basket C is C = (9, 36), its utility level is U2 = 9*36 = 324. Hence, the cost of buying basket E at initial prices (recall initial Hence, basket E must satisfy x*y = 324. prices were Px =$9 and PY =$1) is (96)+(154)= $108
Then, EV= 108‐72= 36
c) Comparing CV and EV in this exercise, CV= 24 ≠ EV= 36
What if we measure the change Hence, the gain in CS resulting from the decrease in the price of good x in consumer welfare using CS? from $9 to $4 is given by integral: 9 36 CS dp 36ln(P ) 9 36 ln9 ln 4 $29.20 First, we find the demand curve for good x when utility p x x 4 function is 4 x 0.81
U(x,y)=xy $9 I=$72
Py=$1 Gain in CS I 72 36 x 2px 2px px
(non‐linear in px)
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Application: Automobile export restrictions for Japanese cars in 1984 Summarizing: Prices when up about 20% for JPN cars sold in U.S. CS 29 These measures of welfare change are different CV=$14 billion: additional income needed after the CV 24 prices change (due to the export restrictions) is $14 (Since IE>0), but… is this usual? EV 36 billion.
Not so much since and QI, are low (making income effects low for many goods). C Qp,, QP QI , Since in 1984 there were 11 million new cars bought, the CV per car buyer is 14,000 TE SE IE $1,272 11
Application: Should I really pay my workers more? Income and Substitution effects in the labor market Y, Goods I 2 per day BL2
I1 If the income effect is sufficiently small, total effect of an increase in the wage rate would be still positive. (more working hours applied) C However, if IE is really large, TE would be negative BLd B (workers would supply less working hours). A BL1
024N * N1 N 2 N, Leisure hours per day 24H * H1 H 2 0 H, Work hours per day Substitution effect Total effect (-) Income effect
95
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Backward‐bending Labor Supply Curve Analysis of labor supply
Consider a worker with utility function uwH()(24) H1
Where H:# hours worked per day w: Wage per hour
The first term represents utility function from the goods that the worker can purchase with their total salary, while the second term represents the utility from leisure, i.e., remaining hours in a day that he/ she doesn’t work.
Rearranging, 24 H (1)H 24 H H H H * 24
1‐ 1 Taking F.O.C. in utility function u = (wH) (24‐H) with b) Consider , What is H*? respect to the number of hours worked, H; 3 1 u 1 1 H* 24 8 hours (wH) w(24H) (wH) (1)(24H) 0 3 H
(wH)1(24H)1 (wH) (1)(24H)
wwH()(24)11 H 1 (24H) 1 H
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Tax Revenue and Labor Supply
The above analysis about SE and IE in labor markets can be used to analyze tax policy. Since labor supply is insensitive to variations in the TaxRevenue wh ( ) wage rate, w, then the substitution and income effects where hw ( ) is the number of hours worked when wage net of taxes is (1- ) must completely offset each other, i.e., |SE| = |IE|, implying a null total effect, TE = 0. Some people argue that increasing tax rates would reduce incentives to work. We all agree with that, but the question is by how much. • If the reduction in working hours is sufficiently large, total tax revenue will actually decrease. • If, in contrast, such a reduction is only minor, total tax revenue will increase. This relationship is graphically represented with the Laffer curve, depicting tax rates on the horizontal axis and total tax collection on the vertical axis.
The Laffer curve Tax Revenue and Labor Supply T T 0 0 Let us analyze under which conditions we are in the increasing or decreasing portion of the Laffer curve. T T TaxRevenue wh ( ) where hw ( ) is the number of hours worked when wage net of taxes is (1- ) T dh wh (w) w 2 dw positive effect on T from higher rate negative effect on T from fewer hours worked
T 0 For (a decrease in tax rates to cause an increase in revenue), we need dh1 dh w wh() w2 ddh ()
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Tax Revenue and Labor Supply 1 dh w Hence, for total tax collection ,T, to raise from a small fall in dh() the tax rate, we need
Multiplying both terms by (1 ) , we obtain 1 supply, 1 dh w 1 dh() Example 1: 25%if your income is about average $35,000 a year 1 dh 10.25 3 Unlikely: Bush admin. or nowadays. dh() supply, 0.25 More compactly: elasticity of Example 2: τ=80% supply of labor A 1% increase in net wages, w, 1 produces an ε percentage 1 0.8 supply,w 0.25 Likely: Japan, and Sweden 90s, US supply, increase in working hours. Supply,w 0.8 during Kennedy
τ τ* Maximum Additional Tax Revenue United States 28 63(!) 30 EU‐14 41 62 8 Ireland 27 68 30 United Kingdom 28 59 17 Market Demand Portugal 31 59 14 We now seek to aggregate the individual demands of Spain 36 62 13 all consumers in a given industry to obtain the market Germany 41 64 10 (or aggregate) demand. Netherlands 44 67 9 Greece 41 60 7 We will need to horizontally sum their individual France 46 63 5 demands. That is, for a given price, we add up the units Italy 47 62 4 demanded by each consumer. Belgium 49 61 3 Let us look at one example Finland 49 62 3 Austria 50 61 2 Sweden 56 63 1 Denmark 57 55 1 Only recommended decrease Source: Trabandt and Uhlig (2009)
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Market Demand Overlap of D and Market
h We horizontally sum the Demand since only H consumes individual demand of casual Network Externalities ‐ Example Consumes consumers and health‐conscious
consumers, obtaining the Neither Bandwagon effect: A positive network externality that aggregate demand. refers to the increase in aggregate demand for a good as 15 3pp if 5
Qph () more consumers buy the good. (e.g., online games) h
0 if p 5 consumes
62pp if 3 Only Qpc () 0 if p 3 m Snob effect: A negative network externality that refers D The market demand will be so to the decrease in aggregate demand for a good as more 21 5pp if 3 (Both)
h consumers buy the good. (e.g., gym memberships)
Qp( ) 15 3 p if 3 p 5 (Only h consumes) consume,
M >D 0 if p 5 (Neither consumes) Both
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