Week 4 Consumer Theory

Transcription

1 Week 4 Consumer Theory Serçin ahin Yldz Technical University 16 October 2012

2 The Consumer's Problem We view the consumer as having a consumption set, X = R n +. His preferences are described by the preference relation dened on R n +. the consumer's feasible alternatives consist a feasible set, B R n +. And consumer is motivated to choose the most preferred feasible alternative: x B such that x x, x B.

3 The Consumer's Problem Assumption: The consumer's preference relation is complete, transitive, continuous, strictly monotonic, and strictly convex on R n +. Therefore, it can be represented by a real-valued utility function, u, that is continuous, strictly increasing, and strictly quasiconcave on R n +.

4 The Consumer's Problem Assumption: The consumer's preference relation is complete, transitive, continuous, strictly monotonic, and strictly convex on R n +. Therefore, it can be represented by a real-valued utility function, u, that is continuous, strictly increasing, and strictly quasiconcave on R n +.

5 The Consumer's Problem An individual consumer is operating within a market economy. A price p i > 0 prevails for each commodity i, i = 0, 1,..., n. The vector of market prices p 0, as xed from the consumer's point of view. The consumer is endowed with a xed money income y 0.

6 The Consumer's Problem Then the feasible set, B, which is called the Budget set is: B = { x x R n +, p.x y }

7 The Consumer's Problem The consumer's utility-maximisation problem is written as max u(x) s.t. p.x y x R n + The solution vector depends on the parameters to the consumer's problem. Because it will be unique for given values of p and y, we can properly view the solution to the consumer's maximisation problem as a function from the set of prices and income to the set of quantities, X = R n +, which is known as ordinary, or Marshallian demand functions. When income and all prices other than the good's own price are held xed, the graph of the relationship between quantitiy demanded of x i and its own price p i is the standard demand curve for good i

8 The Consumer's Problem

9 The Consumer's Problem If we strengthen the requirements on u(x) to include dierentiability, we can use calculus methods. If we form the Lagrangian: Kuhn-Tucker conditions: max u(x) s.t. p.x y x R n + L(x, λ) = u(x) λ [p.x y]

10 The Consumer's Problem By strict monotonicity, these conditions reduce to For any goods j and k, we can combine the conditions to conclude that

11 The Consumer's Problem

12 Indirect Utility and Expenditure The Indirect Utility Function The relationship among prices, income and the maximised value of utility can be summarised by a real-valued function, v : R n + R + R, which is called the indirect utility function. v(p, y) = max u(x) s.t. p.x y x R n + It is the maximum value corresponding to the consumer's utility maximisation problem. v(p, y) = u(x(p, y))

13 Indirect Utility and Expenditure The Indirect Utility Function

14 Indirect Utility and Expenditure The Indirect Utility Function Properties of the Indirect Utility Function If u(x) is continuous and strictly increasing on R n +, then v(p, y) is

15 Indirect Utility and Expenditure The Expenditure Function To construct the expenditure function, we ask: What is the minimum level of money expenditure the consumer must make facing a given set of prices to achieve a given level of utility? e(p, u) min p.x s.t. u(x) u x R n + for all p 0 and all attainable utility levels u. If x h (p, u) solves this problem, the lowest expenditure necessary to achieve utility u at prices p will be exactly equal to the cost of the bundle x h (p, u), or e(p, u) = p.x h (p, u)

16 Indirect Utility and Expenditure The Expenditure Function

17 Indirect Utility and Expenditure The Expenditure Function The solution, x h (p, u), to the expenditure-minimisation problem is precisely the consumer's vector of Hicksian demands (or compensated demands ).

18 Indirect Utility and Expenditure The Expenditure Function Properties of the Expenditure Function If u( ) is continuous and strictly increasing, then e(p, u) is

19 Indirect Utility and Expenditure Relations Between The Two Let v(p, y) and e(p, u) be the indirect utility function and expenditure function for some consumer whose utility function is continuous and strictly increasing. Then for all p 0, y 0, and u U: 1 e(p, v(p, y)) = y 2 v(p, e(p, u)) = u Mathematically, both the indirect utility function and the expenditure function are simply the appropriately chosen inverses of each other. 1 e(p, u) = v 1 (p : u) 2 v(p, y) = e 1 (p : y)

20 Indirect Utility and Expenditure Relations Between The Two Duality Between Marshallian and Hicksian Demand Functions Under the assumption that consumer's preference relation is complete, transitive, continuous, strictly monotonic and strictly convex on R n +, we have the following relations between the Hicksian and Marshallian demand functions for p 0, y 0, u U, and i = 1,..., n: x i (p, y) = xi h (p, v(p, y)) xi h(p, u) = x i(p, e(p, u))

21 Indirect Utility and Expenditure Relations Between The Two

22 Properties of Consumer Demand Relative Prices and Real Income 'Money is a veil.' The relative price of some good is, the number of units of some other good that must be forgone to acquire 1 unit of the good in question. For any arbitrary good i and j, this is given by the price ratio p i p j because The real income is the maximum number of units of some commodity the consumer could acquire if he spent his entire money income.

23 Properties of Consumer Demand Relative Prices and Real Income This is sometimes expressed by saying that the consumer's demand behaviour displays an absence of money illusion.

24 Properties of Consumer Demand Relative Prices and Real Income Under the assumption that consumer's preference relation is complete, transitive, continuous, strictly monotonic and strictly convex on R n +, the consumer deand function x i (p, y), i = 1,..., n, is homogeneous of degree zero in all prices and income, and it satises budget balancedness p.x(p, y) = y, (p, y) Homogeneity allows us to completely eliminate the yardstick of money from any analysis of demand behaviour. This generally done by arbitrarily designating one of the n goods to serve as numeraire in place of money.

25 Properties of Consumer Demand Income and Substitution Eects

26 Properties of Consumer Demand Income and Substitution Eects The Hicksian decomposition of the eect of a price change suggests that the total eect (TE) of a price change can be decomposed into two separate conceptual categories: The substitution eect (SE) is that (hypothetical) change in consumption that would occur if relative prices were to change to their new levels but the maximum utility the consumer can achieve were kept the same as before the price change. The eect on quantity demanded of the change in purchasing power resulting from the change in prices is called the income eect (IE).

27 Properties of Consumer Demand Income and Substitution Eects

28 Properties of Consumer Demand Income and Substitution Eects The Slutsky Equation: Let x(p, y) be the consumer's Marshallian demand system. Let u be the level of utility the consumer achieves at prices p and income y. Then Negative Own-Substitution Terms: Let xi h (p, u) be the Hicksian demand for good i. Then

29 Properties of Consumer Demand Income and Substitution Eects A good is called normal if consumption of it increases as income increases, holding prices constant. A good is called inferior if consumption of it declines as income increases, holding prices constant. The Law of Demand: A decrease in the own price of a normal good will cause quantity demanded to increase. If an own price decrease causes a decrease in quantity demanded, the good must be inferior.

30 Properties of Consumer Demand Income and Substitution Eects called the substitution matrix, contain all the Hicksian substitution terms. Then the matrix σ(p, u) is negative semidenite. Symmetric Substitution Terms: Let x h (p, u) be the consumer's system of Hicksian demands and suppose that e( ) is twice continuously dierentiable. Then Negative Semidenite Substitution Matrix: Let x h (p, u) be the consumer's system of Hicksian demands, and let

31 Properties of Consumer Demand Income and Substitution Eects Symmetric and Negative Semidenite Slutsky Matrix: Let x(p, y) be the consumer's Marshallian demand system. Dene the ij-th Slutsky term as and form the entire n n Slutsky matrix of price and income responses as follows: Then s(p, y) is symmetric and negative semidenite.

32 Properties of Consumer Demand Some Elasticity Relations The income elasticity of demand for good i measures the percentage change in the quantity of i demanded per 1 per cent change in income, and denoted by the symbol η i. The price elasticity of demand for good i measures the percentage change in the quantity of i demanded per 1 per cent change in the price p j, and denoted by the symbol ɛ ij. If j = i, ɛ ii is called the own-price elasticity of demand for good i. If j i, ɛ ij is called the cross-price elasticity of demand for good i with respect to p j.

33 Properties of Consumer Demand Some Elasticity Relations The income share or proportion of the consumer's income, spent on purchases of good i is denoted by the symbol s i. Aggregation in Consumer Demand: Let x(p, y) be the consumer's Marshallian demand system. Then the following relations must hold among income shares (s i ), price (ɛ i ) and income elasticities (η i )of demand: 1 Engel aggregation: 2 Cournot aggregation: n s i η i = 1 i=1 n s i ɛ ij = s j, j = 1,..., n i=1

34 Properties of Consumer Demand Some Elasticity Relations