ECON 500. ECON 500 Microeconomic Theory. Consumer Theory

Transcription

1 ECON 500 Microeconomic Theory Consumer Theory

2 Consumer Theory A theory of how consumers allocate incomes among different goods and services to maximize their well-being. Part I Part II Part III Preferences and Utility Constraints and Choice Comparative Statics

3 Part I. Preferences and Utility Axioms of Rational Choice I. Completeness II. Transitivity III. Continuity

4 Part I. Preferences and Utility Axioms of Rational Choice I. Completeness Consumers can compare and rank all possible baskets. Thus, for any two market baskets A and B, a consumer will prefer A to B, will prefer B to A, or will be indifferent between the two. By indifferent we mean that a person will be equally satisfied with either basket. Note that these preferences ignore costs. A consumer might prefer A to B but buy only B because it is cheaper.

5 Part I. Preferences and Utility Axioms of Rational Choice I. Completeness II. Transitivity If a consumer prefers basket A to basket B and basket B to basket C, then the consumer also prefers A to C. Transitivity ensures that the individual s choices are internally consistent.

6 Part I. Preferences and Utility Axioms of Rational Choice I. Completeness II. Transitivity III. Continuity If A is preferred to B, then situations suitably close to A must also be preferred to B. Enables the analysis of individuals responses to relatively small changes in income and prices

7 Part I. Preferences and Utility Utility Assuming completeness, transitivity, and continuity implies that people are able to rank all possible situations from the least desirable to the most. This ordinal relationship can be represented by what we call a utility function: If A is preferred to B, then the utility assigned to A exceeds the utility assigned to B: U(A) > U(B)

8 Part I. Preferences and Utility Utility Individuals preferences are assumed to be represented by a utility function of the form U(x 1, x 2,..., x n, Z ) Where x 1, x 2,, x n are the quantities of each of n goods that might be consumed in a period, and Z is everything else. Ceteris Paribus assumption allows us to drop Z. This function is unique only up to an order-preserving (monotone) transformation

9 Part I. Preferences and Utility Utility The ordinal nature of utility and the non-uniqueness of utility functions makes them: Useful in analyzing only the relative desirability of choices No so useful in analyzing absolute utility gains across bundles Not so useful in making interpersonal comparisons

10 Part I. Preferences and Utility Utility Arguments of utility functions U(W) = utility an individual receives from real wealth (W) U(c,h) = utility from consumption (c) and leisure (h) U(c 1,c 2 ) = utility from consumption in two different periods Two-good utility function U(x,y) More of any particular x i during some period is preferred to less

11 Part I. Preferences and Utility Alternative Market Bundles Bundle Units of Food Units of Clothing A B D E G H 10 40

12 Part I. Preferences and Utility ECON 500

13 Part I. Preferences and Utility Indifference Curve: Shows a set of consumption bundles that provide the consumer with the same level of utility. The consumer ranks these bundles equally, in other words the consumer is indifferent about consuming any bundle on the indifference curve

16 Part I. Preferences and Utility Marginal rate of substitution, MRS is measured as the negative of the slope of an indifference curve (U 1 ) at some point MRS reflects the individual s willingness to trade y for x and it changes as x and y change MRS dy dx U U 1

18 Part I. Preferences and Utility MRS=?

19 Part I. Preferences and Utility Indifference Curves and Transitivity

20 Part I. Preferences and Utility Indifference Curves and Transitivity The transitivity axiom implies that indifference curves cannot intersect. There is a single indifference curve that passes through every point in the goods space, in other words every bundle is associated with a unique utility level.

21 Part I. Preferences and Utility Indifference Curves and Transitivity

22 Part I. Preferences and Utility Convexity and Indifference Curves An alternative way of stating the principle of a diminishing marginal rate of substitution uses the mathematical notion of a convex set. A set of points is said to be convex if any two points within the set can be joined by a straight line that is contained completely within the set. The assumption of a diminishing MRS is equivalent to the assumption that all combinations of x and y that are preferred or indifferent to a particular combination x*, y* form a convex set.

23 Part I. Preferences and Utility Convexity of the indifference curve implies diminishing MRS

24 Part I. Preferences and Utility Convexity and Balance in Consumption If indifference curves are convex (if they obey the assumption of a diminishing MRS), then the line joining any two points that are indifferent will contain points preferred to either of the initial combinations. Intuitively, balanced bundles are preferred to unbalanced ones.

26 Part I. Preferences and Utility Suppose the utility function s of the form U = An indifference curve for this function that set of combinations of x and y for which utility has the value 10 is: Rearranging: 10 = 100=x y y=100/x MRS = -dy/dx(along U 1 )=100/x 2 As x rises, MRS falls When x = 5, MRS = 4 When x = 20, MRS = 0.25

28 Part I. Preferences and Utility MRS and Marginal Utilities The total differential of a utility function U ( x, y ) rearranging: MRS of x for y is equal to the ratio of the marginal utility of x ( U/ x) to the marginal utility of y ( U/ y).

29 Part I. Preferences and Utility Cobb-Douglas Utility U(x,y) = x y Where and are positive constants The relative sizes of and indicate the relative importance of the goods We can normalize + = 1 by an invariant monotonic transformation U(x,y) = x y 1- Where = /( + ) and 1- = /( + )

30 Part I. Preferences and Utility Perfect substitutes U(x,y) = x + y Where and are positive constants The MRS will be constant along the linear indifference curves Perfect complements U(x,y) = min ( x, y) Where and are positive parameters L-shaped indifference curves

31 Part I. Preferences and Utility CES Utility (constant elasticity of substitution) U(x,y) = x / + y / when 1, 0 and U(x,y) = ln x + ln y when = 0 Perfect substitutes = 1 Cobb-Douglas = 0 Perfect complements = -

32 Part I. Preferences and Utility General Cobb-Douglas Form U(x,y) = x y MRS U x 1 x y y U y 1 x y x MRS depends on only the ratio of the amounts of the two goods, and not their total quantities. This type of utility functions are called homothetic. Slopes of the curves depend only on the ratio y=x, not on how far the curve is from the origin. Hence, we can study the behavior of an individual who has homothetic preferences by looking only at one indifference curve or at a few nearby curves without fearing that our results would change dramatically at very different levels of utility.

33 Part I. Preferences and Utility Many Goods Suppose utility is a function of n goods given by U(x 1, x 2,, x n ) Indifference surface in n dimensions is defined by U(x 1, x 2,, x n )=k MRS MRS is then given by dx U ( x, x,..., x ) 2 x1 1 2 n dx U ( x, x,..., x ) 1 U ( x, x,..., x ) k x n n

34 Part II. Constraints and Choice Basic Model of Consumer Choice: Utility Maximization This model assumes that individuals who are constrained by limited incomes will behave as if they are using their purchasing power in such a way as to achieve the highest utility possible. That is, individuals are assumed to behave as if they maximize utility subject to a budget constraint.

35 Part II. Constraints and Choice Basic Model of Consumer Choice: Utility Maximization Two Concerns: Do individuals make the lightning calculations required for utility maximization? Is the economic model of choice is extremely selfish?

36 Part II. Constraints and Choice To maximize utility, his or her preferences and a fixed amount of income to spend an individual will buy those quantities of goods: a) That exhaust his or her total income b) and for which the MRS is equal to the rate at which the goods can be traded one for the other in the marketplace MRS (of x for y) = the ratio of the price of x to the price of y (p x /p y )

37 Part II. Constraints and Choice Additional Assumptions I = Budget to allocate between good x and good y p x = price of good x p y = price of good y Budget constraint: p x x + p y y I Slope = -p x /p y If all of I is spent on good x(y), I/p x(y) units of good x(y) can be bought

38 Part II. Constraints and Choice ECON 500

40 Part II. Constraints and Choice Conditions (Necessary) for a maximum Point of tangency between the budget constraint and the indifference curve:

41 Part II. Constraints and Choice The tangency rule is necessary but not sufficient Unless we assume that MRS is diminishing If MRS is diminishing, then indifference curves are strictly convex and first order conditions are necessary AND sufficient If MRS is not diminishing, one must check second-order conditions to ensure that we are at a maximum

43 Part II. Constraints and Choice Interior vs. Corner Solutions If the rate at which x can be traded for y in the market (p x /p y ) is lower (higher) than the MRS of x for y for all possible bundles, individuals may end up maximizing utility by choosing to consume only x (y). At the optimal point the budget constraint is flatter (steeper) than the indifference curve

45 Part II. Constraints and Choice In the case of n-goods, consumer maximizes the utility subject to: To find the maximum s.t. the constraint we set up the Lagrangian: Taking the partial derivatives of Lagrangian w.r.t x 1,x 2,,x n, and λ, and setting them equal to zero yields the n+1 necessary conditions for an interior maximum.

46 Part II. Constraints and Choice First order conditions:

47 Part II. Constraints and Choice Rearranging the first order conditions yields that for any two goods x i and x j where the left hand side is the ratio of the marginal utilities, which have derived to be the Marginal Rate of Substitution between x i and x j

48 Part II. Constraints and Choice Interpreting the Lagrange Multiplier or At the utility-maximizing point, each good purchased should yield the same marginal utility per dollar spent on that good.

49 Part II. Constraints and Choice Cobb-Douglas Example (α + β = 1) Lagrangian FOCs

50 Part II. Constraints and Choice From the first two FOCs Rearranging Substituting in the budget constraint Solving for optimal consumption bundle

51 Part II. Constraints and Choice Indirect Utility Function We can solve for the optimum consumption bundles from the FOCs as a function of prices and income We can substitute the optimum bundles back into the utility function to find maximum utility attainable given prices and income

52 Part II. Constraints and Choice Lump Sum Principle Lump sum income taxes or subsidies are superior to good-specific taxes or subsidies in terms of their impact on consumer well-being. The intuition is that the lump sum transfers leave the individual free to decide how to allocate the final income across various good. Good-specific taxes or subsidies not only impact purchasing power but also distort choices by artificially distorting prices.

54 Part II. Constraints and Choice Duality Utility maximization subject to a budget constraint is equivalent to expenditure minimization subject to a utility target. This duality allows us to employ the expenditure minimization approach in certain cases where it is more useful due to the direct observability of expenditures. The dual problem is to choose x 1,, x n to minimize subject to

56 Part III. Comparative Statics Demand functions for the two goods case are In this formulation prices and income are exogenous to this process, i.e. the individual has no control at this stage of the analysis. Changes in the parameters will shift the budget constraint and cause this person to make different choices. Our focus is analyzing the partial derivatives x/ I and x/ p x as well as some useful properties of these demand functions.

57 Part III. Comparative Statics Homogeneity Functions that obey this property are said to be homogeneous of degree zero. If we were to double all prices and income simultaneously (or multiply them all by any positive constant), then the optimal quantities demanded would remain unchanged. Changing all prices and income changes only the units by which we count, not the real quantity of goods demanded.

58 Part III. Comparative Statics Changes in Income As a person s purchasing power rises, it is natural to expect that the quantity of each good purchased will also increase. Changes in income shift the budget lines in a parallel fashion, reflecting that the relative prices of x and y remain unchanged. Because the ratio p x /p y stays constant, the utility-maximizing conditions also require that the MRS stay constant as the individual moves to higher levels of satisfaction. However, the point of tangency might or might not be on an upward sloping expansion path with increasing consumption levels. If xi/ I=0 over some range of income variation then the good is a normal (or noninferior ) good in that range. A good xi for which xi/ I < 0 over some range of income changes is an inferior good in that range.

59 Part III. Comparative Statics ECON 500

61 Part III. Comparative Statics Changes in Prices Changing a price involves changing not only the intercepts of the budget constraint but also its slope. Therefore, when a price changes, two analytically different effects come into play. One of these is a substitution effect: even if the individual were to stay on the same indifference curve, consumption patterns would be changed so as to equate the MRS to the new price ratio. A second effect, the income effect, arises because a price change necessarily changes an individual s real income. The individual cannot stay on the initial indifference curve and must move to a new one.

64 Part III. Comparative Statics In the case of inferior goods, income and substitution effects work in opposite directions, and the combined result of a price change is indeterminate. A fall in price, will always cause an individual to tend to consume more of a good because of the substitution effect. But if the good is inferior, the increase in purchasing power caused by the price decline may cause less of the good to be bought. Unlike the situation for normal goods, it is not possible here to predict even the direction of the effect of a change in p x on the quantity of x consumed and if the income effect of a price change is strong enough, the change in price and the resulting change in the quantity demanded could actually move in the same direction, which is called Giffen s Paradox

65 Part III. Comparative Statics Individual s (Uncompensated) Demand Curve The demand curve looks at the relationship between x and p x while holding p y, I, preferences and all other determinants constant.

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67 Part III. Comparative Statics Compensated Demand Curve A compensated demand curve shows the relationship between the price of a good and the quantity purchased on the assumption that other prices and utility are held constant. It therefore illustrates only substitution effects and isolates the analysis from income effects. The choice between compensated and uncompensated demand curves depends on the application at hand. Mathematically, the curve is a two-dimensional representation of the compensated demand function:

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69 Part III. Comparative Statics Suppose the utility function is of the form Then the Marshallian demand functions are The indirect utility function is then Rearranging and substituting into the demand function reveals

70 Part III. Comparative Statics Relationship between Compensated and Uncompensated Demand

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