Budget constraint, preferences, utility Varian, Intermediate Microeconomics, 8e, chapters 2, 3, and 4 1 / 43 In this lecture, you will learn • what budget set and budget line are • how their shape is influenced by taxes and food stamps • what preferences are and how they are derived • what the basic types of preferences are – why some indiference curves are straight and some curved, or circle-shaped • what we need a utility function for • how to find out whether to reconstruct a stadium 2 / 43 Budget constraint We assume that the consumer chooses a bundle (x1 , x2 ), where x1 and x2 are quantities of goods 1 and 2. Budget constraint is p1 x1 + p2 x2 ≤ m: • p1 and p2 are prices of goods 1 and 2 • m is income Budget set – bundles for which: p1 x1 + p2 x2 ≤ m. Budget line (BL) – bundles for which: p1 x1 + p2 x2 = m. 3 / 43 Budget set and budget line (graph) Budget line: p1 x1 + p2 x2 = m ⇐⇒ x2 = m/p2 − (p1 /p2 )x1 4 / 43 Composite good The theory works for more than two goods. How to plot it in a 2D graph? On the y axis we can plot the composite good = money value of all other consumed goods. 5 / 43 Change in income A rise in income from m to m0 =⇒ parallel shift out 6 / 43 Change in price A rise in price from p1 to p10 =⇒ pivot around the vertical intercept 7 / 43 Change in more variables Multiplying all prices and income by t does not change BL: tp1 x1 + tp2 x2 = tm ⇐⇒ p1 x1 + p2 x2 = m Multiplying all prices by t has the same effect as dividing income by t: tp1 x1 + tp2 x2 = m ⇐⇒ p1 x1 + p2 x2 = m t 8 / 43 Numeraire Any price or income can be normalized to 1 and adjust all variables so that the BL stays the same. Numeraire = an item with its value normalized to 1 Budget line p1 x1 + p2 x2 = m: • Good 1 is numeraire – the same BL: x1 + p2 m x2 = p1 p1 • Good 2 is numeraire – the same BL: p1 m x1 + x2 = p2 p2 • The income is numeraire – the same BL: p1 p2 x1 + x2 = 1 m m 9 / 43 Taxes and subsidies Three types of taxes: • Quantity tax – consumer pays the amount t for each unit. → Price of good 1 increases to p1 + t. • Value tax (ad valorem) – consumer pays a share τ of price. → Price of good 1 increases to p1 + τ p1 = (1 + τ )p1 . • Lump-sum tax – the value of the tax is independent from consumer’s choice. → Consumer income decreases by the size of the tax. Subsidy = a tax with a negative sign 10 / 43 Rationing If there is rationing imposed on good 1, no consumer is allowed to buy a higher quantity of good 1 than x̄1 . 11 / 43 Taxing consumption greater than x̄1 If consumer pays a tax only on the consumption of good 1 that is in excess of x̄1 ..., budget line is steeper to the right of x̄1 . 12 / 43 CASE: The food stamp program Before 1979 (left graph): • value subsidy – people pay a part of the value of the food stamp • rationing – maximum value of stamps (e.g. 153 $) After 1979 (right graph) – a specific number of food stamps for free 13 / 43 Preferences Consumers compare bundles according to their preferences. Preference relations – three symbols: • bundle X is strictly preferred to bundle Y : (x1 , x2 ) (y1 , y2 ) • bundle X is weakly preferred to bundle Y (bundle X is at least as good as bundle Y ): (x1 , x2 ) (y1 , y2 ) • consumer is indiferent between bundles X and Y : (x1 , x2 ) ∼ (y1 , y2 ) 14 / 43 Assumptions about preferences Assumptions that allow ordering of bundles according to preferences: • Completeness — any two bundles can be compared: (x1 , x2 ) (y1 , y2 ), or (x1 , x2 ) (y1 , y2 ), or both • Reflexivity — each bundle is at least as good itself: (x1 , x2 ) (x1 , x2 ) • Transitivity — if (x1 , x2 ) (y1 , y2 ) and (y1 , y2 ) (z1 , z2 ), then (x1 , x2 ) (z1 , z2 ) 15 / 43 Weakly preferred set and indifference curves 16 / 43 Two indifference curves cannot cross Two different IC such that X Y . Why cannot they cross? It follows from transitivity that if X ∼ Z and Z ∼ Y then X ∼ Y . 17 / 43 Examples of preferences – perfect substitutes Willingness to substitute one good for the other at a constant rate =⇒ constant slope of the indifference curve (not necessarily −1). 18 / 43 Examples of preferences – perfect complements Consumption in fixed proportions (not necessarily 1:1). 19 / 43 Examples of preferences – bads The consumer likes pepperoni but does not like anchovies, they are a bad for her. 20 / 43 Examples of preferences – neutrals The consumer likes pepperoni but is neutral about anchovies, they are a neutral for her. 21 / 43 Examples of preferences – satiation point Satiation point is the most preferred point (x̄1 , x̄2 ). When the consumer has too much of one of the goods, it becomes a bad. 22 / 43 Examples of preferences – discrete goods A discrete good is not divisible – consumption in integer amounts: • indiference curves“ – a set of discrete points ” • a weakly preferred set – a set of line segments 23 / 43 Well-behaved preferences Assumptions of well-behaved preferences: monotonicity and convexity Monotonicity – more is better (it excludes bads) =⇒ indifference curves have negative slope. 24 / 43 Well-behaved preferences (cont’d) Convexity – if (x1 , x2 ) ∼ (y1 , y2 ), then it holds for all 0 ≤ t ≤ 1 that (tx1 + (1 − t)y1 , tx2 + (1 − t)y2 ) (x1 , x2 ). Strict convexity – if (x1 , x2 ) ∼ (y1 , y2 ), then it holds for all 0 ≤ t ≤ 1 that (tx1 + (1 − t)y1 , tx2 + (1 − t)y2 ) (x1 , x2 ). 25 / 43 Marginal rate of substitution Marginal rate of substitution (MRS) = slope of the indifference curve: MRS = dx2 ∆x2 = ∆x1 dx1 Diminishing marginal rate of substitution – absolute value of MRS decreases as we increase x1 . 26 / 43 Interpretation of marginal rate of substitution Interpretation of MRS: • The amount of good 2 one is willing to pay for one unit of good 1. • If good 2 is measured in money: MRS = marginal willingness to pay = how many dollars you would just be willing to give up for an additional unit of good 1. 27 / 43 APPLICATION: Build a stadium for Minnesota Vikings? The club does not like the stadium – considers leaving Minnesota. Fenn a Crooker (SEJ, 2009) measure how much households are willing to pay for Vikings staying in Minnesota = MRS between composite good and Vikings in Minnesota. MRS of an average household: 531 $ Value of the stadium: 531 $ × 1,323 million households = 702 mil. $ Estimated costs are 1 billion $. The new stadium opens in 2016 – the state provided 500 million $. 28 / 43 Utility Two concepts of utility: Cardinal utility – attach a significance to the magnitude of utility: • difficult to assign the magnitude • not needed to describe choice behavior Ordinal utility – important is only the order of preference: • easy to set the utility – 1 rule: preferred bundle has a higher utility • we can derive a complete theory of demand We will use the ordinal utility. 29 / 43 Ordinal utility Utility function is a way of assigning a number to every possible consumption bundle such that more-preferred bundles get assigned larger numbers than less-preferred bundles. If (x1 , x2 ) (y1 , y2 ), then u(x1 , x2 ) > u(y1 , y2 ). Different ways to assign utilities that describe the same preferences: 30 / 43 Monotonic transformation Positive monotonic transformation f (u) = any increasing function of u. Describes the same preferences as the original utility function u. Examples of the function f (u): f (u) = 3u, f (u) = u + 3, f (u) = u 3 Example: Two bundles X and Y , preferences: X Y We assign utility so that u(X ) > u(Y ), e.g. u(X ) = 1, u(Y ) = −1 Do monotonic transformations f1 (u) = 3u a f2 (u) = u + 3 represent the same preferences as the original utility function u? Yes: • f1 (u) = 3u: f1 (u(X )) = 3 > −3 = f1 (u(Y )) • f2 (u) = u + 3: f2 (u(X )) = 4 > 2 = f2 (u(Y )) 31 / 43 Construction of indifference curves from utility function Utility function u(x1 , x2 ) = x1 x2 =⇒ indifference curves x2 = k x1 32 / 43 PROBLEM: The slope of indifference curves The slope of indifference curves for two utility functions: 1. What is the slope of IC x2 = 4/x1 v point (x1 , x2 ) = (2, 2)? Slope of indifference curves = MRS = −4 dx2 = 2 = −1 dx1 x1 √ 2. What is the slope of IC x2 = 10 − 6 x1 v point (4, 5)? Slope of indifference curves = MRS = dx2 −3 −3 =√ = dx1 x1 2 33 / 43 Examples of utility functions – perfect substitutes The consumer is willing to exchange • coke and pepsi at a ratio 1:1 important is the total number: e.g. u(K , P) = K + P • 2 buns for 1 baguette baguette has a double weight: e.g. u(R, H) = R + 2H 34 / 43 Examples of utility functions – perfect complements The consumer demands • left and right shoes at a fixed ratio 1:1 lower quantity matters: e.g. u(L, P) = min{L, P} • rum and coke at a fixed ratio 1:5 goal: same numbers in the bracket – we need only 1/5 of coke: e.g. u(R, K ) = min{5R, K } 35 / 43 Examples of utility functions – quasilinear preferences Indifference curves are vertically parallel (a practical property) Utility function u(x1 , x2 ) = v (x1 ) + x2 , e.g. u(x1 , x2 ) = √ x1 + x2 36 / 43 Examples of utility functions – Cobb-Douglas preferences • A simple utility function representing well-behaved preferences. • Utility function of the form u(x1 , x2 ) = x1c x2d . 1 • More convenient to use the transformation f (u) = u c+d and write x1a x21−a , where a = c/(c + d). 37 / 43 Marginal utility Marginal utility (MU) is the change in utility from an increase in consumption of one good, while the quantities of other goods are constant. Partial derivatives of u(x1 , x2 ) with respect to x1 or x2 . Přı́klady: • u(x1 , x2 ) = x1 + x2 → MU1 = ∂u/∂x1 = 1 • u(x1 , x2 ) = x1a x21−a → MU2 = ∂u/∂x2 = (1 − a)x1a x2−a The value of MU changes with a monotonic transformation of the utility function. If we multiply utility times 2, MU increases times 2. 38 / 43 Relationship between MU and MRS We want to measure MRS = slope of IC u(x1 , x2 ) = k, where k is a constant. We are interested in (∆x1 , ∆x2 ), for which the utility is constant: MU1 ∆x1 + MU2 ∆x2 = 0 MRS = MU1 ∆x2 =− ∆x1 MU2 We can calculate MRS from the utility function. E.g. for u = MRS = − √ x1 x2 : 0,5x1−0,5 x20,5 ∂u/∂x1 x2 =− =− 0,5 −0,5 ∂u/∂x2 x1 0,5x1 x2 The value of MRS does not change with monotonic transformation. MU1 1 If we multiply utility function times 2, MRS= − 2MU 2MU2 = − MU2 . 39 / 43 APPLICATION: Utility from commuting People decide whether to take bus or car. Each type of transport represents a bundle with different characteristics, e.g.: • x1 is walking time • x2 is time taking a bus or car • x3 is the total cost of commuting • ... Assume that the utility function has a linear form U(x1 , ..., xn ) = β1 x1 + ... + βn xn . Then we use statistical techniques to estimate the parameters βi that best describe choices. 40 / 43 APPLICATION: Utility from commuting (cont’d) Domenich and McFadden (1975) estimated the following utility function: U(TW , TT , C ) = −0,147TW − 0,0411TT − 2,24C • TW = total walking time in minutes • TT = total driving time in minutes • C = total cost in dollars The parameters can be used for different purposes. For instance, we can: • calculate the marginal rate of substitution between two characteristics • forecast consumer response to proposed changes • estimate whether a change is worthwhile in a benefit-cost sense 41 / 43 What should you know? • Budget set = consumption bundles available at given prices and income • Budget line are bundles for which the entire income is spent. • If the preference relation is complete, reflexive and transitive, consumer can order bundles according to preferences. • Monotonicity and convexity are reasonable assumptions – easier to find the optimum bundle. 42 / 43 What should you know? (cont’d) • Utility function assigns numbers to different bundles so that the bundles are ordered according to preferences. • The numbers have no meaning in itself. Monotonic transformation of u represents the same preferences.. • MRS measures the slope of IC. • The slope of IC measures the willingness to pay for good 1 (in units of good 2) • The slope of BL measures the opportunity cost of good 1(in units of good 2) 43 / 43