Microeconomics I. Dr. S. Farshad Fatemi. Fall ( st Term) - Group 1 Chapter Two Consumer Choice

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1 Function ( st Term) - Group 1 Consumer Choice Dr. Graduate School of Management and Economics Sharif University of Technology Fall / 23

2 Function In this chapter, we start our study of consumer demand in the context of a market economy. Consumer is the most fundamental decision unit of microeconomic theory. Market economy is the setting in which the goods and services are available for purchase at the known prices. (consumers are price takers and their individual decisions have no effect on these prices) 2 / 23

3 Commodities Function Commodity: a good or service available for purchase in the market. Commodity Vector: a list of amounts of the different commodities. The number of commodities is considered to be finite (l = 1, 2,..., L). x = x 1. x L R L Time and location can be built into the definition of a commodity. In principle, the elements of x can obtain negative values as well. Negative consumption of a commodity can be interpreted as the net outflow or sale for a consumer. 3 / 23

4 Consumption Set Function Consumption Set: is a subset of commodity space (X R L ) which includes only the consumption bundles which an individual can conceivably consume considering the physical constraints imposed by the environment. To keep things straightforward, we continue with the simplest sort of consumption sets: X = R L + Here, R L + is defined as non-negative then it includes zeros as well. Then R L + can be written as: R L + = {x R L : x l 0 for l = 1, 2,..., L} R L + is a convex set. if x, x R L + then x = αx + (1 α)x R L + for α [0, 1] 4 / 23

5 Budget Set Function Budget Set: Consumer s limit on how much she can spend. Two assumptions: Principle of completeness (universality) of markets: All commodities are traded in the market at prices which are publicly quoted. p = p 1. p L R L Note: prices can be negative. But we always assume p 0. Price-taking assumption: the prices are not affected by consumer s choice. 5 / 23

6 Budget Constraint Function If w is the consumer s wealth level, then the consumer s budget constraint can be interpreted as: p.x = L p l x l w l=1 6 / 23

7 The Walrasian Budget Set Function Combining this economic-affordability constraint with the requirement that x R L + : Definition (MWG 2.D.1): The Walrasian budget set or Competitive budget set is the set of all feasible consumption bundles for the consumer who faces market prices p and has wealth w: B p,w = {x R L + : p.x w} The upper boundary of the budget set is {x R L + : p.x = w} which is called the budget hyperplane (or budget line for L = 2). The Walrasian budget set is a convex set. Note: B p,w is convex because R L + is convex. With a more general X : B p,w is convex as long as X is. 7 / 23

8 Function Function The Walrasian (/market/ordinary) demand correspondence assigns a set of chosen consumption bundles for each price-wealth pair: x(p, w). If x(p, w) is single-valued then it is called demand function. Definition (MWG 2.E.1): The Walrasian demand correspondence x(p, w) is homogeneous of degree zero if x(αp, αw) = x(p, w) for p, w & α > 0 8 / 23

9 Function Function Definition (MWG 2.E.2): The Walrasian demand correspondence x(p, w) satisfies Walras law if p 0 & w > 0, we have p.x = w for x x(p, w) From now on, we assume x(p, w) is always single-valued, continuous, and differentiable. 9 / 23

10 Function How demand is affected by changes in the consumer s wealth and prices. Engel function: For fixed prices p, the function of wealth x( p, w) is called the consumer s Engel function. Wealth (income) expansion path: The image of Engel function in the commodity space. 10 / 23

11 Wealth Effect Function Wealth (income) effect: The change in quantity demanded as a result of a change in wealth (income): Matrix interpretation: x l (p, w) w D w x(p, w) = (p, w) x 1(p,w) w x 2(p,w) ẉ. x L (p,w) w RL 11 / 23

12 Wealth Effect Function Normal good: A commodity l is normal at (p, w) if x l (p, w) 0 w Inferior good: A commodity l is inferior at (p, w) if x l (p, w) w < 0 12 / 23

13 Price Effect Function Price effect: The change in quantity demanded as a result of a change in prices: Definition of the function Defenition of the Offer Curve. The price effect of the price of good k on the demand for good l: x l (p, w) p k (p, w) l = k: own price effect; l k: cross price effect. 13 / 23

14 Giffen Good Function Giffen good: A commodity l is Giffen at (p, w) if: x l (p, w) p l > 0 The definition of a Giffen good involves an income (wealth) effect, as well. A Giffen good is always an inferior good (Why?). Matrix interpretation: x 1(p,w) p 1 D p x(p, w) =. x L (p,w) p 1 x 1(p,w) p L.... x L (p,w) p L 14 / 23

15 Elasticities of Function Price elasticity of demand: ε lk (p, w) = x l (p,w) x l (p,w) p k p k = p k x l (p, w). x l(p, w) p k Elasticity is independent of the units unlike the derivative itself. Luxury and Necessity goods (defined based on income elasticity). 15 / 23

16 Elasticities of Function Proposition (MWG 2.E.1): If the Walrasian demand function x(p, w) is homogeneous of degree zero, then L x l (p, w) p k + x l(p, w) w = 0 for l = 1,..., L and (p, w) p k w k=1 or or D p x(p, w)p + D w x(p, w)w = 0 (p, w) L ε lk (p, w) + ε w (p, w) = 0 for l = 1,..., L and (p, w) k=1 An equal percentage change in all prices and wealth leads to no change in demand. 16 / 23

17 Cournot and Engel Aggregation Function Proposition (MWG 2.E.2 & 2.E.3): If the Walrasian demand function x(p, w) satisfies the Walras law, then (Cournot aggregation) L l=1 x l (p, w) p k p l + x k (p, w) = 0 for k = 1,..., L and (p, w) and (Engel aggregation) L l=1 x l (p, w) p l = 1 (p, w) w Cournot aggregation: The total expenditure cannot change in response to a change in prices. Engel aggregation: The total expenditure must change by an amount equal to any wealth change. 17 / 23

18 Function Assume x(p, w) is single-valued, homogeneous of degree zero, and satisfies Walras law. Definition (WARP) (MWG 2.F.1): The Walrasian demand function x(p, w) satisfies the weak axiom of revealed preferences if this property holds for any two price-wealth situation (p, w) and (p, w ) : p.x(p, w ) w and x(p, w ) x(p, w) p.x(p, w) > w 18 / 23

19 Slutsky Wealth Compensation Function Slutsky Wealth Compensation: If the consumer initially faces price-wealth pair of (p, w) and chooses x(p, w), then the price vector changes to p ; the Slutsky wealth compensation adjusts the consumer s wealth to make the initially chosen bundle as affordable as before (x(p, w) is still on the budget hyperplane): w = p.x(p, w) or w = w + (p p).x(p, w) 19 / 23

20 Function Proposition (MWG 2.F.1): Suppose that the Walrasian demand function x(p, w) is homogeneous of degree zero and satisfies Walras law; then it satisfies WARP if and only if the following property holds: For any compensated price change from initial situation (p, w) to a new one (p, p.x(p, w)) we have: (p p).[x(p, w ) x(p, w)] 0 if x(p, w ) x(p, w) Students need to go through the proof in MWG The proposition simply states that demand and price move in opposite directions. 20 / 23

21 Function dx = D p x(p, w)dp + D w x(p, w)dw by Slutsky wealth compensation dw = x(p, w).dp dx = [D p x(p, w) + D w x(p, w)x(p, w) T ]dp from the proposition we have dp.dx 0, so dp.[d p x(p, w) + D w x(p, w)x(p, w) T ]dp 0 the middle part is an L L matrix: s 11 (p, w) s 1L (p, w) S(p, w) =..... s L1 (p, w) s LL (p, w) where s lk (p, w) = x l(p, w) + x l(p, w) x k (p, w) p k w Slutsky (substitution) matrix substitution effect 21 / 23

22 Function Proposition (MWG 2.F.2): If a differentiable Walrasian demand function x(p, w) is homogeneous of degree zero and satisfies Walras law and the WARP, then at any (p, w), the Slutsky matrix satisfies v.s(p, w)v 0 v R L A matrix satisfying this property is called negative semi-definite. Then s ll (p, w) 0 which means The own substitution effect is always nonpositive. Recall the definition of a Giffen good ( x l (p,w) p l > 0) then since s ll (p, w) = x l(p, w) + x l(p, w) x l (p, w) 0 p l w for a good to be Giffen it should be an inferior good ( x l (p,w) w < 0). 22 / 23

23 Function Proposition (MWG 2.F.3): Suppose that the Walrasian demand function x(p, w) is differentiable, homogeneous of degree zero, and satisfies Walras law, then for any (p, w): S(p, w)p = 0 Note: Two theories of demand (i. based on rational preference maximisation; ii. Based on assumption of homogeneity of degree zero, Walras law, and WARP) are not equivalent. Why? We will come back to this in details in the next chapter, while discussing how demand is generated from preferences. For now, they are different, because the Walrasian budget sets do not include every possible budget set (in particular all budgets formed by 2 or 3 commodity bundles). 23 / 23