Simple Model Economy. Business Economics Theory of Consumer Behavior Thomas & Maurice, Chapter 5. Circular Flow Model. Modeling Household Decisions

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1 Business Economics Theory of Consumer Behavior Thomas & Maurice, Chapter 5 Herbert Stocker herbert.stocker@uibk.ac.at Institute of International Studies University of Ramkhamhaeng & Department of Economics University of Innsbruck Simple Model Economy Circular flow model 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 Model Modeling Household Decisions Households Wage L S L D Working Hours Price Q S Q D Quantity Firms Plan of the 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: Unconstrained vs. constrained maximization (see Th&M: Ch., Appendix) Result: Demand curve of an individual household!

2 Consumer Choice Preferences (exogeneous!) Utility function Optimization Budget restriction The description of Preferences Decisions (Demand function) Preferences Preferences Assumptions Consumers obtain benefits (utility) from the consumption of goods & services Consumers have complete information about characteristics and availability of all goods & services Consumers decide between different bundles of goods and services! Example for two different bundles: Bundle P Bundle R kg of rice 7 kg of rice shirts shirts 5 beer beer trip to Paris trips to Rome, trips to Paris trip to Rome.... ballpen 5 ballpen Bundles can be written as vectors!

3 Preferences If a consumer chooses bundle P when bundle R is available, we can say that the consumer prefers bundle P to bundle R. We write or Bundle P kg of rice shirts 5 beer trip to Paris trips to Rome.. ballpen P R, 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: P-bundle is strictly preferred to the R-bundle R P: R-bundle is strictly preferred to the P-bundle P R: P-bundle is regarded as indifferent to the R-bundle P R: P-bundle is at least as good as (preferred to or indifferent to) the R-bundle Preferences Preferences are relationships between bundles! Individuals choose between bundles containing different quantities of goods In the following we rely on only two goods (Theory also works with more than two goods) Assumption: 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 = (,), which is preferred over 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

4 Preferences & Indifference Curves Wine B P A R E C D Bread What about bundles B & C? Bundles A & D have more of one good but less of another, compared to E More information about consumer ranking is needed! Consumer might be indifferent between A, E and D We can then connect those points with an indifference curve Preferences & Indifference Curves Any bundle lying above (northeast of) an indifference curve (e.g. C) is preferred to any bundle lying on the indifference curve Points on the curve are preferred to points below (southwest of) the curve (e.g. B) Indifference curves slope downward to the right (negatively sloped) Otherwise, they would violate the assumption that more is preferred to less: Bundles with more of both goods would be indifferent to baskets 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 Indifference map Each point represents a bundle of different quantities of bread and wine Each indifference curve connects thebundlesamongwhich the consumer is indifferent Wine Preferred bundles Bread Notice: The higher the indifference curve, the higher an individual s utility! Question: Is it possible that indifference curves cross? Assumptions about preferences 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 Preferences can be represented in a utility function In addition, it is often useful to assume convex preferences

5 Preferences Special Preferences Convex preferences Averages are preferred to extremes i.e., goods are consumed together (e.g., bread and wine) This is the normal case! Concave Preferences Goods are normally not consumed together (e.g., beer and wine) time horizon! corner solutions! Special Case: Satiation (bliss point) Special Preferences Not Monotonic! Bliss-Point Perfect Substitutes e.g., U(,) = + Perfect Complements U(,) = min{,} Above the bliss point utility decreases, nobody will consume there! Constant rate of trade-off between two goods (e.g., red pencils and blue pencils) Always consumed together in fixed proportions (e.g., right shoes and left shoes; coffee and cream)

6 U Q Q Utility Function The shape of indifference curves Utility 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 Cobb-Douglas Utility Function 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 Averysimpleand wellbehaved utility function: Cobb-Douglas Function U = U(,) = Q a xq b y (a and b are positive parameters determining the kind of preferences) Example: U =..7

7 U Q Q U 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 Special Preferences Special Preferences Perfect Substitutes: e.g., U(,) = + Constant rate of trade-off between the two goods Perfect Complements: U(, ) = min{, } Always consumed together

8 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 d= Marginal Utility Examples: U = + U = Q a y a U = Q a y a MUx U = MUx U = a a a MUy U = ( a)q a y a Marginal Rate of Substitution (MRS) Marginal Rate of Substitution (MRS): Measures how a consumer is willing to trade off consumption of good X for consumption of Y Slope along an indifference curve, keeping utility constant MRSxy d d = MUx MUy U U Sign: Generally, indifference curves have a negative slope (for du = ) Marginal Rate of Substitution (MRS) Discrete Infinitesimal Slope: Slope: d d MRS diminishes along an indifference curve!

9 Derivation of MRS Consider U = U(,) Totally differentiating this utility function yields du = U d + U d = Re-arranging this expression gives What we can afford The Budget Constraint d U = U MRSxy d Budget Constraint Budget Constraint The Budget Constraint M = Px +Py shows for given prices Px and Py all combinations of and a household with given income can afford No lending and no borrowing Rewriting gives = M Px Py Py d Slope: = Px d Py M Py M = Px +Py = M Px Py Py = Px Py Changes in income and in prices changes the shape of the budget line!

10 Budget Constraint The price ratio Px/Py shows how many units of the second good can be obtained on the market for one unit of the first good. Example: when QB is the quantity of bread, and QW the quantity of wine then PB/PW gives the price of one unit bread in units of wine. Example: Euro PB kg Bread = = Euro lt Wine.5 lt Wine = PW Euro Euro kg Bread kg Bread lt Wine Budget Constraint 5 β α d = Py d Px = tanβ = M = Px +Py = M Px Py Py d = Px d Py = tanα =,5 one unit of costs.5 units of (= tanα)! or, one unit of costs units of (= tanβ). Changes in the Budget Line Changes in the Budget Line What happens when all prices and the income multiply? (e.g. inflation) Multiply all prices and income with a constant t: tm = tpx +tpy but this is the same as the initial budget constraint M = Px +Py therefore a perfectly balanced inflation doesn t change consumption possibilities! What happens when all prices double, but the income remains constant? Multiply all prices with a constant t: this is the same as M = tpx +tpy M = Px +Py t 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.

11 Changes in the Budget Line Changes in the Budget Line What happens when a specific tax is levied on? A specific tax (quantity tax) T raises the price of to Px +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)px +Py i.e. becomes cheaper, the budget line flatter! What happens when the consumer gets one unit of for free? Changes in the Budget Line What happens when the consumer gets the second two units of for half the price of the first two units? 5 Combining preferences and budget constraint Optimal Choice

12 Q Q U Decisions (in a neoclassical perspective) Decisions: neoclassical point of view Preferences (exogeneous!) Utility function Optimization Decisions (Demand function) Budget restriction Preferences U = U(,) max : U(,) s.t. M = Px +Py M = Px +Py L = U(,)+λ[M Px Py] Q x = (Px,Py,M), Q y = (Px,Py,M) Consumer Choice Cobb-Douglas utility function and linear budget constraint Optimization Problem max, U(, ) s.t.: M = Px +Py Two possibilities for optimization Substitution method (rather awkward) Lagrange method (simple and elegant)

13 Lagrange Method Joseph Louis Lagrange (7 - ): 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 = Px +Py Step : Lagrange function (goal function plus Lagrange multiplier λ times the restriction in implicit form) L = U(,)+λ[M Px Py] }{{} = Lagrange Method Step : Set partial derivatives of the Lagrange function with respect to the endogenous (decision) variables and as well as the Lagrange multiplier λ equal to zero: L L = U = U! λpx =! λpy = L λ = M Px Py! = First order conditions (FOC) Lagrange Method Step : Solve the equation system for the endogenous variables, and λ Q x = (Px,Py,M), Q y = (Px,Py,M) λ = λ(px,py,m) The solutions to Q x and Q y are the demand functions for an individual household Describe the optimal decisions of an household under given restrictions

14 Lagrange Method: Example U = U(,) = With M = Px +Py, we have L = +λ[m Px Py] The FOC read as L! = λpx = () L! = λpy = () L λ = M Px Py! = () Lagrange Method: Example From () and (), we have = Px Py or = Px Py Inserting in () gives ( ) Px M Px Py = Py Solving for we obtain demand for X Q x = M Px Similarly, demand for Y is = M Py Optimal Choice The FOC allow some more insights in the problem of optimal consumer choice L = U(,)+λ[M Px Py] L L = U = U! λpx =! λpy = L λ = M Px Py! = U U λ = = Px Py or MUx = MUy Px Py Example Suppose, M =. All income is spend on units of X with PX =, and on units of Y with PY = : = + Further, assume that MUx = and MUy = MUx Px = = 5 < = = MUy Py It is optimal for the consumer to spend more money on Y: Spending one EUR on Y increases utility by units Reducing consumption of X by one unit induces a utility loss of only 5 units Note: Budget remains constant at Consumer maximizes utility if income is allocated in a way that the marginal utility per money unit spent on each good is identical

15 Optimal Choice Optimal Choice Since on an indifference curve utility is constant by definition it follows x Indifference curves Hence, Therefore: du = = MUxd +MUyd MRS = d = MUx d MUy Utility (U) Utility function MRS = d = MUx = d MUy Px Py A U Px = U Py MUx MUy = d MRSxy d Good Y () x Good X () Budgetconstraint Optimal Choice Optimal Choice Condition for optimality: Income- Consumption-Curve d Slope: d du= Slope: Px Py 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!

16 MRS condition: Recap The MRS is an indicator for the willingness to pay A budget constraint shows the ability to pay Demand and Changes in Income When we combine the MRS with the ability to pay, i.e., the budget constraint, we can derive demand Demand and Changes in Income Demand and Changes in Income Income-consumption curve: Normal good Clothes Income- Consumption Curve Engel Curve: Normal good Income Engel Curve Inferior good Beefsteak Income- Consumption Curve Engel Curve: Inferior good Engel Curve Income normal inferior Food Food Hamburger Hamburger

17 Cobb-Douglas Preferences Demand and Changes in Price Demand Curves Px max U(,) =, s.t.: M = Px +Py Budget constraint for M =, Py = : = Px + = Px Px =,,,,,,,,,5, Solution: = M = Px Px This is the usual demand curve D = Q(Px,M) Special Cases Perfect Substitutes The usual methods for maximization (e.g., Lagrange method) are 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 (i.e., Kuhn-Tucker conditions) Px max U(,) = +, s.t. M = Px +Py [Graph: M = und Py = ] MRS = d Px =, = /,, /, /5 d Py [ ] wennpx > Py =, M if Px = Py, Px M if Px Py. Px

18 Perfect Complements Preferences and Demand Px max U(,) = min{,}, s.t. M = Px +Py [Graph: M =, Py = ] Lagrange not applicable!!! Insert efficiency-condition = in budget constraint: M = (Px +Py) M = Px +Py 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 Consumer Choices: Examples Effects of Price Changes Slutsky- and Hicks Decomposition The theory of consumer choice, inter alia, 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?

19 Price Changes Slutsky-decomposition 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? max U = s.t. M = Px +Py (for M = and Py = ) SE IE Optimal decisionwhen Px = : =.5, = Optimal decisionwhen Px = : =, = Slutsky Substitution Effect (=SE): new price ratio, but constant purchasing power! Income effect (=IE): constant price ratio, but purchasing power increases! Hicks-Decomposition Substitution- and Income Effects max U = s.t. M = Px +Py (for M = and Py = ) SE IE Optimal decisionwhen Px = : =.5, = Optimal decisionwhen Px = : =, = Hicks Substitution Effect (=SE): new price ratio, but constant utility! Income Effect (=IE): constant price ratio, but higher income! 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

20 Giffen-Good Although becomes cheaper less of is demanded! Market Demand SE IE TE Market Demand Market Demand Market Demand, D, is the horizontal sum of individual demands Dx = Q x (Px,Py,M )+Q x (Px,Py,M ) + Q N x (Px,Py,MN ) Note: The subscript denotes good x and y, the superscript consumer i i =,,...,N Attention: Quantities can never be negative, only zero! P The market demand function has kinks! Q D = Q d = P Q d =.5P Q d =.5.5P forp.5.5p for P.5 P for P 5.5 P for P