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Course: Microeconomics Text: Varian’s Intermediate Microeconomics 1 In Chapter 5, we talk about the optimal choice under the budget constraint. Here, we consider how the changes of exogenous variables affect decisions. Recall endogenous variables: quantities of goods x1, x2 exogenous variables: prices and income. 2 Consider the two-good case. Denote the ordinary demand functions as x1*(p1,p2,m), x2*(p1,p2,m). How does x1*(p1,p2,m) change as p1 changes, holding p2 and m constant? Suppose only p1 increases, from p1’ to p1’’ and then to p1’’’. 3 x2 Fixed p2 and m. p1x1 + p2x2 = m p1 = p1’ x1 4 x2 Fixed p2 and m. p1x1 + p2x2 = m p1 = p1’ p1= p1’’ x1 5 Fixed p2 and y. x2 p1x1 + p2x2 = m p1 = p1’ p1= p1’’’ p1= p1’’ x1 6 Own-Price Changes Fixed p2 and y. p1 = p1’ x1*(p1’) 7 Own-Price Changes p1 Fixed p2 and m. p1 = p1’ p1’ x1*(p1’) x1* x1*(p1’) 8 Own-Price Changes p1 Fixed p2 and m. p1’’ p1’ x1*(p1’) x1* x1*(p1’’) x1*(p1’) x1*(p1’’) 9 Own-Price Changes Fixed p2 and m. p1 p1’’’ p1’’ p1’ x1*(p1’’’) x1*(p1’) x1* x1*(p1’’) x1*(p1’’’) x1*(p1’) x1*(p1’’) 10 Own-Price Changes Fixed p2 and m. p1 p1’’’ Ordinary demand curve for commodity 1 p1’’ p1’ x1*(p1’’’) x1*(p1’) x1* x1*(p1’’) x1*(p1’’’) x1*(p1’) x1*(p1’’) 11 Own-Price Changes Fixed p2 and m. p1 price offer curve p1 p1’’’ Ordinary demand curve for commodity 1 p1’’ p1’ x1*(p1’’’) x1*(p1’) x1* x1*(p1’’) x1*(p1’’’) x1*(p1’) x1*(p1’’) 12 The curve containing all the utility-maximizing bundles traced out as p1 changes, with p2 and y constant, is the p1- price offer curve. The plot of the x1-coordinate of the p1- price offer curve against p1 is the ordinary demand curve for commodity 1. 13 A good is called an ordinary good if the quantity demanded of it always increases as its own price decreases and vice versa (negatively related), holding all other factors, such as prices, income and preference constant. Fixed p2 and y. x2 p1 p1 price offer curve Downward-sloping demand curve Good 1 is ordinary x1* x1 If the quantity demanded of a good decreases as its own-price decreases and vice versa (i.e. positively related) holding all other factors constant, then the good is called a Giffen Good. Note: we need to hold other factors constant. Thus, if the price change is also associated with change in income or preference, then even if there’s a positive relation between price and quantity, it is not characterized as Giffen good. Fixed p2 and y. x2 p1 p1 price offer curve Demand curve has a positively sloped part Good 1 is Giffen x1* x1 What does a p1 price-offer curve look like for a perfect-complements utility function? U( x1 , x 2 ) minx1 , x 2. m x ( p1 , p2 , m) x ( p1 , p2 , m) . p1 p2 * 1 * 2 m x ( p1 , p2 , m) x ( p1 , p2 , m) . p1 p2 * 1 * 2 With p2 and y fixed, higher p1 causes smaller x1* and x2*. As m p1 0,x x . p2 As p1 ,x x 0. * 1 * 2 * 1 * 2 x2 Fixed p2 and y. x1 Perfect Complement p1 Fixed p2 and y. x2 y/p2 p1 = p1’ p1’ x2* m p’1 p2 y x p1 ' p2 * 1 x1* m p1’ p2 x1 x1* Perfect Complement p1 Fixed p2 and y. x2 y/p2 p1 = p1’’ p1’’ p1’ x2* m * x1 p1" p2 m p1" p2 x1* m p1" p2 x1 x1* Perfect Complement Fixed p2 and y. x2 p1 = p1’’’ y/p2 p1 p1’’’ p1’’ p1’ x2* m p1 ' ' ' p2 m x p1 ' ' ' p2 * 1 ’’’ m x p1' ' ' p2 * 1 x1 x1* Perfect Complement Fixed p2 and y. p1 Ordinary demand curve for commodity 1 is m * x1 . p1 p2 p1’’’ x2 p1’’ y/p2 x2* p1’ m p1 p2 m p2 x1* m p1 p2 x1 x1* What does a p1 price-offer curve look like for a perfect-substitutes utility function? U( x1 , x 2 ) x1 x 2 . 0,if p1 p2 x ( p1 , p2 , m) m / p1,if p1 p2 * 1 and 0,if p1 p2 x ( p1 , p2 , m) m / p2 ,if p1 p2 . * 2 p1 Perfect Substitutes Fixed p2 and y. x2 p1 = p1’ < p2 p1’ m x p1 ' * 1 x*2 0 m x1 x p1 * 1 x1* Perfect Substitutes p1 Fixed p2 and y. x2 p1 = p1’’ = p2 p1’ x1* x1 Perfect Substitutes p1 Fixed p2 and y. x2 p1 = p1’’ = p2 p1’ x1* x1 p1 Perfect Substitutes Fixed p2 and y. x2 p1 = p1’’ = p2 p2 = p1’’ m x p2 * 2 * x 2 0 x1* x*1 0 p1’ m 0 x p2 * 1 m p2 x1 x1* Perfect Substitutes Fixed p2 and y. p1 p1’’’ x2 p2 = p1’’ m x p2 * 2 x*1 0 p1’ x*1 0 x1 x1* Perfect Substitutes Fixed p2 and y. p1 Ordinary demand curve for commodity 1 p1’’’ x2 x1* p2 = p1’’ m p2 p1 price offer curve m p1 p1’ m 0 x p2 * 1 x1 x1* Usually we ask “Given the price for commodity 1 what is the quantity demanded of commodity 1?” But we could also ask the inverse question “At what price for commodity 1 would a given quantity of commodity 1 be demanded?” Taking quantity demanded as given and then asking what must be price describes the inverse demand function of a commodity. p1 Given p1’, what quantity is demanded of commodity 1? Answer: x1’ units. p1’ x1’ x1* p1 Given p1’, what quantity is demanded of commodity 1? Answer: x1’ units. p1’ x1’ The inverse question is: Given x1’ units are demanded, what is the price of commodity 1? Answer: p1’ x1* If an increase in p2, holding p1 constant, increases demand for commodity 1 then commodity 1 is a gross substitute for commodity 2. reduces demand for commodity 1 then commodity 1 is a gross complement for commodity 2. Substitutes: if the price is higher for one good, you turn to buy other goods to give you satisfaction. E.g. drinks: bubble tea vs fruit juice; entertainment: movie vs magazine. Complements: things tend to be used together so when one of the prices increases, the demand for the other will decrease. E.g.: camera and memory cards 38 A perfect-complements example: m x p1 p2 * 1 so x m 0. 2 p2 p1 p2 * 1 Therefore commodity 2 is a gross complement for commodity 1. p1 p1’’’ Increase the price of good 2 from p2’ to p2’’ and p1’’ p1’ m p2 ' x1* p1 Increase the price of good 2 from p2’ to p2’’ and the demand curve for good 1 shifts inwards -- good 1 is a complement for good 2. p1’’’ p1’’ p1’ m p2 ' ' x1* For perfect substitutes, how will the quantity demanded x1 change when p2 increases? Note: it only changes the point where x1 starts to be positive. 42 p1 p1’’’ p1’’ p1’ Generally, an increase the price of good 2 from p2’ to p2’’ and the demand curve for good 1 shifts outwards -- good 1 is a substitute for good 2. x1* How does the value of x1*(p1,p2,m) change as m changes, holding both p1 and p2 constant? Income Changes Fixed p1 and p2. m’ < m’’ < m’’’ Income Changes Fixed p1 and p2. m’ < m’’ < m’’’ x2’’’ x2’’ x2’ x1’ x1’’’ x1’’ Income Changes Fixed p1 and p2. m’ < m’’ < m’’’ Income offer curve x2’’’ x2’’ x2’ x1’ x1’’’ x1’’ A plot of income against quantity demanded is called an Engel curve. Income Changes Fixed p1 and p2. m’ < m’’ < m’’’ Income offer curve Engel curve; good 2 y m’’’ m’’ m’ y x2’’’ x2’ x2* x2’’ x2’’’ m’’’ x2’’ ( a b) p m (a b2) px2* 2 * m b x2 b x2’ x1’ x1’’’ x1’’ Engel curve; good 1 m’’ m’ x1’ x1’’’ x1’’ x1* p1 p1’’’ When income increases, the curve shifts outward for each give price, if the good is a normal good. p1’’ p1’ x1* A good for which quantity demanded rises with income is called normal. Therefore a normal good’s Engel curve is positively sloped. Generally, most goods are normal goods. A good for which quantity demanded falls as income increases is called income inferior. Therefore an income inferior good’s Engel curve is negatively sloped. E.g.: low-quality goods. Income Changes; Goods y 1 & 2 Normal Engel curve; good 2 y’’’ y’’ y’ Income offer curve y x2’’’ x2’ x2* x2’’ x2’’’ y’’’ x2’’ Engel curve; good 1 y’’ x2’ y’ x1’ x1’’’ x1’’ x1’ x1’’’ x1’’ x1* Income Changes; Good 2 Is Normal, Good 1 Becomes Income Inferior x2 x1 Income Changes; Good 2 Is Normal, Good 1 Becomes Income Inferior x2 x1 Income Changes; Good 2 Is Normal, Good 1 Becomes Income Inferior x2 Income offer curve x1 Income Changes; Good 2 Is Normal, Good 1 Becomes Income Inferior x2 y Engel curve for good 1 x1 x1* Another example of computing the equations of Engel curves; the perfectly-complementary case. U( x1 , x 2 ) minx1 , x 2. The ordinary demand equations are m x x . p1 p2 * 1 * 2 m x x . p1 p2 * 1 * 2 Rearranged to isolate y, these are: m ( p1 p2 ) x * 1 m ( p1 p2 ) x * 2 Engel curve for good 1 Engel curve for good 2 Income Changes Fixed p1 and p2. x2 x1 Income Changes Fixed p1 and p2. x2 y’ < y’’ < y’’’ x2’’’ x2’’ x2’ x1’ x1’’’ x1’’ x1 Income Changes Fixed p1 and p2. x2 Engel curve; good 2 y y’’’ y’’ y’ < y’’ < y’’’ y’ y x2’’’ x2’ x2* x2’’ x2’’’ y’’’ x2’’ Engel curve; good 1 y’’ x2’ y’ x1’ x1’’’ x1’’ x1 x1’’’ x1’ x1’’ x1* Income Changes Fixed p1 and p2. * y (p1 p2 )x 2 Engel curve; good 2 y y’’’ y’’ y’ y x2’’’ x2’ x2* x2’’ y’’’ * y (p1 p2 )x1 Engel curve; good 1 y’’ y’ x1’’’ x1’ x1’’ x1* Another example of computing the equations of Engel curves; the perfectly-substitution case. U( x1 , x 2 ) x1 x 2 . The ordinary demand equations are 0, if p1 p2 x ( p1 , p2 , m) m / p1, if p1 p2 * 1 0, if p1 p2 x ( p1 , p2 , m) m / p2 , if p1 p2 . * 2 Suppose p1 < p2 y y * x 2 0. * y p1x1 x1* Engel curve for good 1 0 Engel curve for good 2 It is the opposite when p1 > p2. x2* Quasi-linear preferences U( x1 , x 2 ) f ( x1 ) x 2 . For example, U( x1 , x 2 ) x1 x 2 . Income Changes; Quasilinear Utility x 2 ~ x1 x1 Income Changes; Quasilinear y Utility x 2 Engel curve for good 2 x2* Engel curve for good 1 y ~ x1 x1 ~ x1 x1* Income Changes; Good 2 Is Normal, Good 1 Becomes yIncome Inferior x2 Engel curve for good 2 y x2* Engel curve for good 1 x1 x1* If income and prices all change by the same proportion k, how does the consumer demand change? (e.g. same rate of inflation for prices and income) Recall that prices and income only affects the budget constraint: p1 x1 + p2 x2 = m. Now it becomes: kp1 x1 + kp2 x2 = km. Which clearly gets back to the original one. Thus, xi (kp1 , kp2 , km) = xi (p1 , p2 , m) We sometimes call this no money illusion. 70 How are the demand curves and Engel curves look like for a Cobb-Douglas utility? Recall the Cobb-Douglas utility function: a b U( x1 , x 2 ) x1 x 2 . On solving using the MRS=p1/ p2 and the budget constraint, you will have: 71 Then the ordinary demand functions for commodities 1 and 2 are a m x ( p1 , p2 , m) a b p1 * 1 b m x ( p1 , p2 , m) . a b p2 * 2 Own-Price effect: inversely related to its own price. Cross-Price effect: no cross price effect Income Effect: Engle curve is a straight line passing through the origin. 73 Own-Price Changes Fixed p2 and y. p1 Ordinary demand curve for commodity 1 is am x (a b) p1 * 1 x2* bm ( a b) p2 x1* x1*(p1’’’) x1*(p1’) x1*(p1’’) am bm * x ; x2 . (a b) p1 ( a b ) p2 * 1 Rearranged to isolate m, these are: (a b) p1 * m x1 a ( a b) p2 * m x2 b Engel curve for good 1 Engel curve for good 2 m m (a b) p1 * m x1 a Engel curve for good 1 x1* ( a b ) p2 * m x2 b x2* Engel curve for good 2 Demand Elasticity measures the percentage change of demand as a result of one percent change in exogenous variable. Own price elasticity of demand: dxi / xi pi dxi dpi / pi xi dpi i i 77 Cross Price Elasticity of demand: p j dxi dxi / xi i dp j / p j xi dp j j Income Elasticity of demand: dxi / xi m dxi dm / m xi dm m i 78 ii 0 Ordinary Good: Giffen Good: 0 i i Gross Substitutes: 0 Gross Complements: 0 j i j i Normal Good: Inferior Good: im 0 m i 0 79 The consumer demand function for a good generally depends on prices of all goods and income. Ordinary: demand decreases with own price Giffen: demand increases with own price Substitute: demand increases with other price Complement: demand decreases with other price Normal good: demand increases with income Inferior good: demand decreases with income 80