Chapter 1 Consumer Theory Part II

Transcription

1 Chapter 1 Consumer Theory Part II Economics 5113 Microeconomic Theory Kam Yu Winter 2018

2 Outline 1 Introduction to Duality Theory Indirect Utility and Expenditure Functions Ordinary and Compensated Demand Functions 2 Properties of Consumer Demand Relative Prices and Real Income Income and Substitution Effects Implications 3 Elasticities Definitions Average Elasticities Kam Yu (Lakehead) Chapter 1 Consumer Theory Part II Winter / 25

3 Introduction to Duality Theory Indirect Utility and Expenditure Functions Relations between V and E Let V (p, y) and E(p, u) be the indirect utility function and the expenditure function derived from a continuous and strictly increasing utility function. Then for all p 0, y 0, and u 0, 1 E(p, V (p, y)) = y, 2 V (p, E(p, u)) = u. Notes: 1 Reading assignment: See page 42 in JR for a proof. 2 For a given price vector p, the indirect utility function v : R + R + maps income y into utility u, v(y) = u. Since v is strictly increasing in y, the inverse function exists and is the expenditure function. That is v 1 (u) = e(u) = y. Kam Yu (Lakehead) Chapter 1 Consumer Theory Part II Winter / 25

4 Introduction to Duality Theory Indirect Utility and Expenditure Functions Example (Also see Example 1.4 in JR) Consider the Cobb-Douglas utility function U(x 1, x 2 ) = x α 1 x 1 α 2, and 0 α 1. The ordinary demand functions are (see exercise 20, chapter 7 in Yu) x 1 = d 1 (p, y) = αy p 1, The indirect utility function is x 2 = d 2 (p, y) = (1 α)y p 2. ( ) α α ( ) 1 α 1 α V (p, y) = y. (1) p 1 To find the expenditure function, let u = V (p, y). Solving for y using equation (1) gives ( p1 ) ( ) α 1 α p2 E(p, u) = y = u. (2) α 1 α p 2 Kam Yu (Lakehead) Chapter 1 Consumer Theory Part II Winter / 25

5 Introduction to Duality Theory Ordinary and Compensated Demand Functions Relations between d and h Let d(p, y) and h(p, u) be the ordinary and compensated demand functions derived from a continuous and strictly increasing utility function. Then for all p 0, y 0, and u 0, Notes: h(p, V (p, y)) = d(p, y), (3) d(p, E(p, u)) = h(p, u). (4) 1 Reading assignment: See page in JR for a proof. 2 For a given price vector p, the compensated demand function for good i is a composite function, that is, h i = d i e, or h i (u) = d i (e(u)) = d i (y). 3 Similarly, d i (y) = h i (v(y)) = h i (u). Kam Yu (Lakehead) Chapter 1 Consumer Theory Part II Winter / 25

6 Introduction to Duality Theory Ordinary and Compensated Demand Functions Example Continued Apply Shephard s lemma to the expenditure function in equation (2), we get the compensated demand functions ( ) α p 1 α ( ) 2 1 α p α 1 h 1 (p, u) = u, h 2 (p, u) = u. 1 α p 1 α p 2 Using equations (1) and (3), the ordinary demand functions are ( ) α p 1 α ( ) 2 α α ( ) 1 α 1 α d 1 (p, y) = y = αy, 1 α p 1 p 1 p 2 p 1 ( ) 1 α p α ( ) 1 α α ( ) 1 α 1 α (1 α)y d 2 (p, y) = y = α p 2 p 1 p 2 p 2 as before. Exercise: Use the ordinary demand functions and the expenditure function in the example to demonstrate equation (4). Kam Yu (Lakehead) Chapter 1 Consumer Theory Part II Winter / 25

7 Properties of Consumer Demand Relative Prices and Real Income Homogeneity Revisited Recall that the budget constraint is unaffected by multiplying a positive scalar to prices and income. That is for any α > 0, αp T x αy is the same as p T x y. As a result the ordinary demand function d and the indirectly utility function V are homogeneous of degree zero in (p, y). Let α = 1/p i, where p i is the price of any good i (called the numéraire, such as gold or silver). Then the price vector becomes ( ) ( 1 p1 p =,..., p i 1, 1,..., p ) n. p i p i p i p i The ratio p j /p i is called the price of good j relative to good i, and is expressed in quantity of good i needed to buy one unit of good j. Similarly, y/p i is the real income in terms of good i. Kam Yu (Lakehead) Chapter 1 Consumer Theory Part II Winter / 25

8 Properties of Consumer Demand Income and Substitution Effects Decomposition of the Effect of a Price Change Suppose that in period 0 a consumer with income y buy two goods with prices p 0. Let x 0 be optimal bundle. In period 1 the price of good 1 goes down from p 0 1 to p1 1. The optimal bundle now is x 1. The total effect of the price change on consumption is x 1 x 0. We can decompose the total effect into two components, one represents a relative price change and the other real income change. The first component, called substitution effect, is obtained by keeping the utility, or standard of living, constant at the period 0 level, u 0, with the price change. Effectively it is the difference between the optimal bundle given by the compensated demand function, x s = h(p 1, u 0 ), and the original optimal bundle x 0. The residual change, x 1 x s, called income effect, reflects the real income change due to the price change. Kam Yu (Lakehead) Chapter 1 Consumer Theory Part II Winter / 25

9 52 Properties of Consumer Demand Income and Substitution EffectsCHAPTER 1 x 2 y/p 2 0 u 0 u 1 (a) x 2 0 x 2 1 x 2 s SE TE IE p 1 0 /p2 0 p 1 1 /p 2 0 p 1 1 /p2 0 p 1 x 1 0 SE x 1 s IE x 1 1 x 1 TE p 1 0 (b) p 1 p 1 1 SE IE x 1 (p 1, p 2 0, y) TE x 1 h (p 1, p 2 0, u 0 ) x 1 0 x 1 s x 1 1 x 1 x 1 Figure The Hicksian decomposition of a price change. Kam Yu (Lakehead) Chapter 1 Consumer Theory Part II Winter / 25

10 Properties of Consumer Demand Income and Substitution Effects The Slutsky Equation Effects of a price change p j on x i is nicely summarized by the equation d i (p, y) = h i(p, u) d j (p, y) d i(p, y). p j p j y The term on the left is the total effect. The first term on the right is the substitution effect, and the second income effect. Intuition for the income effect: Consider a small change in price of good j equal to p j. This effectively change the real income by y = ( p j )x j. Express this in p j = y/x j. Then the effect on good i is x i p j = x j x i y. Kam Yu (Lakehead) Chapter 1 Consumer Theory Part II Winter / 25

11 Properties of Consumer Demand Income and Substitution Effects Proof of the Slutsky Equation Recall the duality relation h i (p, u) = d i (p, E(p, u)). Differentiate both sides of the equation with respect to p j : h i (p, u) p j = d i(p, y) + d i(p, y) p j y By Shephard s lemma, E(p, u)/ p j = h j (p, u). E(p, u) p j. (5) At (p, y), the ordinary demand and compensated demand are the same, that is, h j (p, u) = d j (p, E(p, u)) = d j (p, y). Substitute this into the last term in equation (5) and rearrange, we get the Slutsky equation. Kam Yu (Lakehead) Chapter 1 Consumer Theory Part II Winter / 25

12 Properties of Consumer Demand Implications More Properties of Consumer Demand Consider the case that i = j in the Slutsky equation: d i (p, y) = h i(p, u) d i (p, y) d i(p, y). p i p i y Because the expenditure function E is concave in p, the compensated demand function of good i is decreasing in its own price, that is, h i (p, u)/ p i 0. The income effect in the Slutsky equation can be positive or negative. We define good i to be 1 a normal good if d i (p, y)/ y > 0, 2 an inferior good d i (p, y)/ y < 0. Law of Demand: If good i is normal, then the ordinary demand function is decreasing in its own price, that is, d i (p, y)/ p i < 0. Question: What is the contrapositive of the law of demand? Kam Yu (Lakehead) Chapter 1 Consumer Theory Part II Winter / 25

13 Properties of Consumer Demand Implications Symmetric Substitution The right-hand side of equation (5) is often called the i-jth Slutsky term, s ij (p, y) = d i(p, y) d i (p, y) + x j. p j y The term on the left-hand side of equation (5), by Shephard s lemma, is h i (p, u) = 2 E(p, u). p j p j p i It follows that the n n Slutsky matrix formed by the Slutsky terms is equal to the Hessian of the expenditure function with respect to the price vector, S(p, y) = 2 pe(p, u). Therefore S is symmetric and negative semi-definite. S is also observable. Kam Yu (Lakehead) Chapter 1 Consumer Theory Part II Winter / 25

14 Elasticities Definitions Income Elasticity of Demand Prices, quantities, and income have units attached to them, like $/kg (food), $/Tax Return (H&R Block), etc. Sometimes we want the consumers response in price and income changes expressed in a dimensionless number. Define income elasticity of demand for good i as % change in quantity demanded for good i η i = % change in income = d i(p, y)/d i (p, y) y/y = d i(p, y) y y d i (p, y). Kam Yu (Lakehead) Chapter 1 Consumer Theory Part II Winter / 25

15 Elasticities Definitions Properties of Income Elasticity Income elasticity can be positive or negative: inferior goods: η < 0 necessities: 0 η 1 luxuries: η > 1 Kam Yu (Lakehead) Chapter 1 Consumer Theory Part II Winter / 25

16 Elasticities Definitions Luxury, Normal, and Inferior Goods Observations during the 2008 recession: Kam Yu (Lakehead) Chapter 1 Consumer Theory Part II Winter / 25

17 Elasticities Definitions Price Elasticity of Demand Price elasticity: % change in quantity demanded for good i ɛ ij = % change in price of good j = d i(p, y) p j p j d i (p, y). If i = j, then ɛ i is called the own-price elasticity of demand for good i. If i j, then ɛ ij cross-price elasticity of demand for good i with respect to good j. Technically, ɛ i is always negative because of the law of demand. Often we only care about the absolute value, ɛ i, that is, the value without the negative sign. Kam Yu (Lakehead) Chapter 1 Consumer Theory Part II Winter / 25

18 Elasticities Definitions Own-Price Elasticity Definitions: inelastic: 0 ɛ i < 1 unitary elastic: ɛ i = 1 elastic: ɛ i > 1 Some examples: Good or Service Elasticity ɛ i Cigarettes 0.42 Salmon 2.47 Gasoline 0.50 Chicken 1.67 Peak hour bus services 0.23 Kam Yu (Lakehead) Chapter 1 Consumer Theory Part II Winter / 25

19 Elasticities Definitions Substitutes and Complements Two goods are substitutes if the price of one good goes up result in an increase in demand for the other good. Examples: Zippers and buttons Butter and margarine Natural gas and fuel oil Pork and chicken Two goods are complements if the price of one good goes up result in an decrease in demand for the other good. Examples: Movie tickets and restaurant meals Lettuce and salad dressing Blue-ray players and HDTV Kam Yu (Lakehead) Chapter 1 Consumer Theory Part II Winter / 25

20 Elasticities Definitions Examples of Cross-Price Elasticity Kam Yu (Lakehead) Chapter 1 Consumer Theory Part II Winter / 25

21 Elasticities Average Elasticities Engel Aggregation Define the expenditure share of good i to be s i = p i x i n j=1 p jx j = p ix i y. It is obvious that s i 0 and n i=1 s i = 1. Engel aggregation: n s i η i = 1. i=1 The intuition behind this result is that the weighted average change in consumption with respect to income change is equal to the change in income itself. Kam Yu (Lakehead) Chapter 1 Consumer Theory Part II Winter / 25

22 Elasticities Average Elasticities Proof of Engel Aggregation Recall that straight monotonicity of the utility function implies that the budget constraint is binding. This translates into the so-called budget balancedness condition: For all p 0 and y 0, n p i d i (p, y) = y. (6) i=1 Differentiate both sides with respect to y, we get n i=1 p i d i (p, y) y = 1. Multiply and divide each term on the left by yd i (p, y) gives the result. Kam Yu (Lakehead) Chapter 1 Consumer Theory Part II Winter / 25

23 Elasticities Average Elasticities Cournot Aggregation The Cournot aggregation states that for j = 1,..., n, n s i ɛ ij = s j. i=1 It means that the weighted average of the effect of change in p j on all goods depends on good j s income share. Kam Yu (Lakehead) Chapter 1 Consumer Theory Part II Winter / 25

24 Elasticities Average Elasticities Proof of Cournot Aggregation : Differentiate both sides of the budget balancedness in (6) with respect to p j, we have x j + p j d j (p, y) p j + The last equation implies that n i=1 Multiple each side by p j /y, we obtain n i=1 p i y n i j p i d i (p, y) p j = 0. p i d i (p, y) p j = x j. d i (p, y) p j p j = s j. Multiplying and dividing each term on the left by x i gives the result. Kam Yu (Lakehead) Chapter 1 Consumer Theory Part II Winter / 25

25 Elasticities Average Elasticities From Statistics Canada s Survey of Household Spending Canadian households spent an average of $62,183 on goods and services in 2016, up 2.8% from Spending on shelter accounted for 29.0% of this total, followed by transportation (19.2%) and food (14.1%). Provincially, the highest average spending on goods and services was reported by households in Alberta ($74,044), followed by Ontario ($66,220) and Saskatchewan ($65,411). Households in New Brunswick ($50,175) reported the lowest average spending. On average, couples with children spent $88,273 on goods and services, compared with $34,674 for one-person households. Kam Yu (Lakehead) Chapter 1 Consumer Theory Part II Winter / 25