Consumer Choice Theory of Consumer Behavior Herbert Stocker herbert.stocker@uibk.ac.at Institute of International Studies University of Ramkhamhaeng & Department of Economics University of Innsbruck Economics is about making the best of things. In other words, it is about choice subject to constraints. Layard / Walters (97): Microeconomic Theory Consumer Choice & Decisions Households and Firms Economists create good stories by being simple, explicit, and plausible about three things: the actors involved, their goals, and the choices available to them. (Fiona Scott Morton) Two actors with different goals: Households Buy and consume goods and services Own and sell factors of production Firms Produce and sell goods and services Hire and use factors of production
Circular Flow Diagram Households and Firms Wage L S Two principles of economics: Working Hours L D Optimization principle: people choose actions that are in their interest Households Price Q S Firms Equilibrium principle: people s actions must eventually be consistent with each other Q D Quantity Modeling of Decisionmaking Consumer Choice What we will do in this chapter: Find a general way to describe what consumers (households) want preferences Map these preferences in a utility function Describe the choices available (restrictions) budget constraint Use a technique to perform the optimization (e.g. Lagrange Multiplier) As a result we will get the demand curve of an individual household! Preferences (exogeneous!) Utility function Optimization Decisions (Demand functions) Budgetrestriction
Preferences The description of Preferences Consumers obtain benefits (utility) from the consumption of goods & services. Assume consumers have complete information about characteristics and availability of all goods & services. Consumers decide between different bundles of goods and services! Preferences Preferences Example for two different bundles: Bundle P kg of rice shirts 5 beer trip to Paris trips to Rome. ballpen, Bundle R 7 kg of rice shirts beer trips to Paris trip to Rome. 5 ballpen (bundles can be written as vectors) If a consumer chooses bundle P when bundle R is available it s natural to say that the consumer prefers bundle P to bundle R. We write P R, or Bundle P kg of rice shirts 5 beer trip to Paris trips to Rome. ballpen Bundle R 7 kg of rice shirts beer trips to Paris trip to Rome. 5 ballpen
Preferences All bundles of goods can be ranked based on their ability to provide utility: P R means the P-bundle is strictly preferred to the R-bundle. R P means the R-bundle is strictly preferred to the P-bundle. P R means that the P-bundle is regarded as indifferent to the R-bundle, P R means the P-bundle is at least as good as (preferred to or indifferent to) the R-bundle. Preferences Preferences are relationships between bundles! Preferences refer to the ranking of entire bundles of goods, not to individual goods. Individuals choose between bundles containing different quantities of goods. Theory works with more than two goods, but then we can t draw pictures. Therefore, we will restrict ourselves to two goods, bread and wine. Generally, we will assume that consumers always prefer more of any good to less; more is better! Preferences & Indifference Curves Wine P E R Bread More is better: Bundle R = (, ) is preferred to bundle E = (, ) is preferred to bundle P = (, ). More generally: The consumer prefers E to all combinations in the magenta box (e.g.p), while all those in the yellow box (e.g. R) are preferred to E. Preferences & Indifference Curves Wine B P A E D R C Bread Points such as A & D have more of one good but less of another compared to E; Need more information about consumer ranking! Consumer may decide they are indifferent between A, E and D. We can then connect those points with an indifference curve.
Preferences & Indifference Curves Any bundle lying northeast of an indifference curve is preferred to any market basket that lies on the indifference curve. Points on the curve are preferred to points southwest of the curve. Indifference curves slope downward to the right; If they sloped upward, they would violate the assumption that more is preferred to less! Some points that had more of both goods would be indifferent to a basket with less of both goods. Indifference curves and -map To describe preferences for all combinations of goods/services, we use a set of indifference curves - an indifference map. Each point represents a bundle of different quantities bread and wine. Each indifference curve in the map connects the bundles among which the consumer is indifferent. Wine Preferred bundles Bread Indifference curves Assumptions about preferences Indifference curves graph the set of bundles that are indifferent to some bundle. Indifference curves are like contour lines on a map. Note that indifference curves describing two distinct levels of preference cannot cross (because they are like contour lines on a map; for a proof use transitivity) Assumptions about preferences: complete: any two bundles can be compared, reflexive: any bundle is at least as good as itself, transitive: if Q R and R S, then Q S; Whenever these assumptions are fulfilled the preferences can be represented in a utility function. Often, two additional assumptions are useful... 5
Well-behaved preferences Monotonicity: more of either good is better; implies indifference curves have negative slope. Convexity: averages are preferred to extremes; slope gets flatter as you move further to right (example of non-convex preferences?) Preferred bundles Preferences Convex Preferences averages are preferred to extremes d.h. goods are consumed together, e.g. bread and wine. This is the normal case! Special Preferences Special Case: Satiation (bliss point) Concave Preferences Goods are normally not consumed together (e.g. beer and wine). time horizon! corner solutions! Not Monotonic! Above the bliss point utility decreases, nobody will consume there! Bliss-Point
Special Preferences Perfect Substitutes: e.g. U(, ) = + have a constant rate of trade-off between the two goods; e.g. red pencils and blue pencils. Perfect Complements: U(, ) = min{, } always consumed together, e.g. right shoes and left shoes; coffee and cream. Marginal Rate of Substitution (MRS) The Marginal Rate of Substitution (MRS) measures how the consumer is willing to trade off consumption of good X for consumption of good Y. The MRS is the slope along an indifference curve, keeping utility constant MRS = (for du = ) Sign: natural sign is negative, since indifference curves will generally have negative slope. Marginal Rate of Substitution (MRS) Preferences: Example Diskrete: Infinitesimal: Fibre Health-conscious consumer: Fibre A sweet tooth consumer Slope: Slope: d d Sugar Sugar 7
Marginal Rate of Substitution (MRS) MRS measures marginal willingness to pay (what the consumer is willing to give up for one additional unit); However, its irrespective of what the consumer is able to pay, therefore no demand yet! Utility If axioms are fulfilled (i.e. preferences are complete, reflexive and transitive) preferences can be expressed more elegantly with a utility function. Utility Two ways of viewing utility: Old way: measures how satisfied you are not operational, many other problems New way: summarizes preferences, i.e. the ranking of bundles. Utility functions are just a shorter and more elegant way to summarizes preferences. only the ordering of bundles counts, so this is a theory of ordinal utility gives a complete theory of demand; operational Utility Function A utility function assigns a number to each bundle of goods so that more preferred bundles get higher numbers, that is, U(, ) > U(R, R ) if and only if (, ) (R, R )
Utility Function Cobb-Douglas Utility Function Utility functions are not unique: if U(, is a utility function that represents some preferences, and f (U) is any increasing function, then f (U(, ) represents the same preferences, because U(, ) > U(R, R ) only if f [U(, )] > f [U(R, R )], so if U(, ) is a utility function then any positive monotonic transformation of it is also a utility function that represents the same preferences. A very simple and well behaved utility function: Cobb-Douglas Function: U = U(, ) = Q a x Qb y (a and b are positive parameters determining the kind of preferences) Example: U 7.5 5.5 U = Qx. Qy.7 Q Q Cobb-Douglas Utility Function Indifference Curves (red) can also be drawn with utility functions connect points with equal utility: Cobb-Douglas Utility Function Indifference Curves (red) are like contour lines: U Q Q U Q Q 9
Special Preferences Special Preferences Perfect Substitutes: e.g. U(, ) = + e.g. red pencils and blue pencils; have a constant rate of trade-off between the two goods. Perfect Complements: U(, ) = min{, } always consumed together, e.g. right shoes and left shoes; coffee and cream). U Q Q U Q Q Marginal Utility Extra utility from some extra consumption of one of the goods, holding the other good fixed this is a derivative, but a special kind of derivative, a partial derivative ( ). This just means that you look at the derivative of U(, ) keeping fixed, treating it like a constant. U du d dqy = Marginal Utility Examples: U = + MU x U = U = QxQ a y a MU x U = aqx a Qy a U = QxQ a y a MU y U = ( a)q Q xq a y a y
Marginal Utility & MRS Marginal Utility & MRS Note that marginal utility depends on which utility function you choose to represent preferences: if you multiply utility times, you multiply marginal utility times, but thus it is not an operational concept. However, MU is closely related to the Marginal Rate of Substitution (MRS), which is an operational concept. With calculus one can show that the MRS is the ratio of marginal utilities: MRS d d = MU x MU y U U The MRS is an indicator for the willingness to pay. A budget constraint will show the ability to pay. When we combine the MRS with the ability to pay, i.e. the budget constraint, we can derive demand. Budget Constraint The Budget Constraint What we can afford The Budget Constraint M = P x + shows for given prices P x and all combinations of and a household with given income can afford. Assume he spends all money. Rewriting: = M P x Slope: d d = P x
Budget Constraint M = P x + M M = P x + = M P x = P x Budget Constraint The price ratio P x / shows how many units of the second good can be obtained on the market for one unit of the first good. Example: when Q B is the quantity of bread, and Q W the quantity of wine then P B /P W gives the price of one unit bread in units of wine. Example: P B P W = Euro kg Bread Euro lt Wine = Euro Euro lt Wine kg Bread =.5 lt Wine kg Bread Budget Constraint Changes in the Budget Line 5 β α 5 7 9 d d = P x = tanβ = d d M = P x + = M P x = P x = tanα =, 5 one unit of costs.5 units of (= tanα)! or, one unit of costs units of (= tanβ). What happens when all prices and the income multiply? (e.g. inflation) Multiply all prices and income with a constant t: tm = tp x + t but this is the same as the initial budget constraint M = P x + therefore a perfectly balanced inflation doesn t change consumption possibilities!
Changes in the Budget Line Changes in the Budget Line What happens when all prices double, but the income remains constant? Multiply all prices with a constant t: this is the same as M = tp x + t M t = P x + therefore it makes no difference whether all prices double or income is halved, multiplying all prices by a constant t is just like dividing income by t. What happens when a specific tax is levied on? A specific tax (quantity tax) T raises the price of to P x + T, d.h. the budget line becomes steeper. What happens when a ad-valorem subsidy s is paid on? the budget line becomes M = ( s)p x + i.e. becomes cheaper, the budget line flatter! Changes in the Budget Line Changes in the Budget Line What happens when the consumer gets one unit of for free? What happens when the consumer gets the second two units of for half the price of the first two units? 5
Desicions (in a neoclassical perspective) Combining preferences and budget constraint... Optimal Choice Preferences (exogeneous!) Utility function Budgetrestriction Optimization Decisions (Demand functions) Desicions: neoclassical point of view Preferences M = P x + Consumer Choice Cobb-Douglas utility function and linear budget constraint: U = U(, ) max : U(, ) s.t. M = P x + U L = U(, ) + λ[m P x ] Q x = (P x,, M), Q y = (P x,, M) Q Q
Optimization Lagrange Method Joseph Louis Lagrange (7 - ): Problem: max U(, ), s.t.: M = P x + Two Possibilities: Substitution method (rather awkward) Lagrange method (simple and elegant) an Italian-French mathematician and astronomer who made important contributions to all fields of analysis and number theory was arguably the greatest mathematician of the th century. Developed a simple method for constrained optimization. Lagrange Method. Step: Problem max U(, ), s.t.: M = P x +. Step: Lagrange function (goal function plus Lagrange multiplier λ times the restriction in implicit form) L = U(, ) + λ [M P x ] }{{} = Lagrange Method. Step: Set partial derivatives of the Lagrange function with respect to the endogenenous (decision-) variables and as well as the Lagrange multiplier λ equal to zero. L L = U λp x! = = U λ! = L λ = M P x! = 5
Lagrange Method. Step: Solve the equation system for the endogenenous variables, and λ Q x = (P x,, M), Q y = (P x,, M) λ = λ(p x,, M) These solutions are the demand functions for an individual household and describe the optimal decisions of an household under given restrictions. Additionally, the first order conditions allow some more insights in the problem of optimal consumer choice... Optimal Choice L = U(, ) + λ [M P x ] L L λ = = U λp x! = = U λ! = L λ = M P x! = U P x = U or MU x P x = MU y Optimal Choice Optimal Choice Since on an indifference curve utility is constant by definition it follows hence Therefore: du = = MU x d + MU y d MRS = d d = MU x MU y Utility (U) Utility function x MRS = dqy = MUx = dqx MUy Px Py A Indifference curves P x = U U MU x MU y = d d MRS Good Y ( ) x Good X ( ) Budgetconstraint
Optimal Choice Optimal Choice Condition for optimality: Income- Consumption-Curve Slope: d du= d Slope: Px dm= MRS = Price ratio Implications of MRS condition: Why do we care that MRS = price ratio? If everyone faces the same prices, then everyone has the same local trade-off between the two goods. This is independent of income and tastes. Since everyone locally values the trade-off the same, we can make policy judgments. Is it worth sacrificing one good to get more of the other? Prices serve as a guide to relative marginal valuations! Demand and Changes in Income Income-consumption curve: normal good Engel Curve: good normal Demand and Changes in Income Clothes Income- Consumption Curve Income Engel Curve Food Food 7
Demand and Changes in Income Inferior good: Engel Curve: inferior good Beefsteak Income- Consumption Curve Income Engel Curve normal inferior Demand and Changes in Price Hamburger Hamburger Cobb-Douglas Preferences Special Cases P x max, s.t.: U(, ) = M = P x + Budget constraint for M =, = : = P x + = P x Solu- P x =,,,,,,,,,5, tion: Qx = M = P x P x The usual methods for maximization (e.g. Lagrange method) is not applicable when preferences are concave or indifference curves are not differentiable in the relevant point (e.g. kinky, linear,... ) Examples: Perfect Substitutes ( corner solution) Perfect Complements An analytical solution is in these cases more difficult (Kuhn-Tucker conditions!
Perfect Substitutes Perfect Complements P x max, s.t. U(, ) = + M = P x + [Graph: M = und = ] MRS = d d =, P x = /,, /, /5 [ ] wennp x > Qx =, M P x if P x =, M P x if P x. P x max, U(, ) = min{, } s.t. M = P x + [Graph: M =, = ] Lagrange not applicable!!! Insert efficiency-condition = in budget constraint: M = (P x + ) Qx M = P x + Preferences and Demand The kind of assumed preferences determines the properties of the demand functions! For example, Cobb-Douglas preferences imply a linear income-consumption curve. a horizontal price-consumption curve. the price elasticity of demand is always the income elasticity of demand is always + cross price elasticities are always zero expenditure shares are always constant. Effects of Price Changes Slutsky- and Hicks Decomposition 9
Consumer Choice Price Changes The theory of consumer choice addresses the following questions: What happens with labor supply when wages increase? Do people save more when interest rates go up? Do the poor prefer to receive cash or in-kind transfers? Do all demand curves slope downward? A fall in the price of a good has two effects: First, relative prices change second, the purchasing power changes Slutsky-decomposition: what happens with demand, when relative prices change, but the purchasing power is held constant Hicks-decomposition: what happens with demand, when relative prices change, but the utility is held constant Slutsky-decomposition Hicks-Decomposition max U = s.t. M = P x + (for M = und = ) SE EE Optimal decision when P x = : =, 5, = Optimal decision when P x = : =, = Slutsky Substitution Effect (=SE): new price ratio, but constant purchasing power! Income effect (=EE): constant price ratio, but purchasing power increases! max U = s.t. M = P x + (mit M = und = ) SE EE Optimal decision when P x = : =, 5, = Optimal decision when P x = : =, = Hicks Substitution Effect (=SE): new price ratio, but constant utility! Income Effect (=EE): constant price ratio, but higher income!
Substitution- and Income Effects Giffen-Good When preferences are convex the substitution effect can never be positive! The income effect can either be positive or negative. If the income effect is negative inferior goods. If the income effect is negative and larger as the substitution effect Giffen-good. SE EE GE Although becomes cheaper less of is demanded! Market Demand Market Demand Market Demand (D): is the horizontal sum of individual demands. D = Q x (P x,, M )+Q x (P x,, M )+ Q N x (P x,, M N ) (the subscript denotes the good, the superscript the consumerder i; N is the total number of consumers.)
Market Demand Attention: Quantities can never be negative, only zero! P The market demand function has kinks! Q d = P Q d =.5P Q d =.5.5P Thanx! Any questions?.5.5 5 5.5 Q forp.5.5p for P D =.5 P for P 5.5 P for P