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The Theory of Demand  Pp 9/8/2016 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 1 9/8/2016 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 7 2 9/8/2016 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 I100 and pxy1 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 y10 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. $1.00 D3 D2 D1 Food (units 4 10 16 per month) 12 3 9/8/2016 An Inferior Good Individual Demand Steaks 15 (units Income-Consumption per Curve Normal Good vs. Inferior Good month) Both hamburger and steak behave C as a normal good, Income Changes 10 between A and B... When the income‐consumption curve has a positive U 3 slope: …but hamburger The quantity demanded increases with income. becomes an inferior The income elasticity of demand is positive. B good when the income 5 consumption curve The good is a normal good. bends backward between B and C. U2 A U 1 Hamburgers 52010 30 (units per month) 13 14 Individual Demand Effects of Income Changes: Engel Curves Normal Good vs. Inferior Good Engel curves relate the quantity of goods consumed to income. Income Changes If the good is a normal good, the Engel curve is upward When the income‐consumption curve has a negative sloping. slope: If the good is an inferior good, the Engel curve is downward The quantity demanded decreases with income. sloping. The income elasticity of demand is negative. Let’s look at two examples: The good is an inferior good. 15 16 4 9/8/2016 Effects of Income Changes: Engel Curves Engel Curves Income ($ 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) 17 18 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 5 9/8/2016 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? 6 9/8/2016 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. 26 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. 27 28 7 9/8/2016 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 29 30 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 31 32 8 9/8/2016 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.
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