Consumer Theory Lecture 2

Introduction Consumer Theory In this chapter, we formalize the concept of consumer choice. Lecture 2 We see how an individual, with certain preferences, makes decisions when faced with a constraint. Reading: Perlo¤ Chapter 3 It is the foundation of the demand curve we saw yesterday. August 2015 1 / 70 Outline 2 / 70 Preferences Preferences - Assumptions we make about how individuals rank di¤erent options. Utility - A convenient way to summarize preferences. Individuals prefer certain sets of goods to others Budget Constraint - What can people a¤ord? In order to formalize this abstract concept in a useful way, we need to make a few assumptions about these preferences Constrained Consumer Choice - Given income, prices and preferences, what is the best choice I can make? Behavioral Economics - Do individuals behave in reality how we have assumed? 3 / 70 4 / 70 Preferences Preferences But rst lets go over some notation. Consumers rank bundles in terms of desirability. a bundle (or basket) is simply of a combination of goods and services. a % b. I like bundle a at least as much as b. I weakly prefer a to b bundle a is two haircuts and a can of Pepsi. a  b. I de nitely like a more than b. I strictly prefer a to b bundle b is four cans of Pepsi and no haircuts. a  b. I like the bundles equally. I am indi¤erent between a and b. 5 / 70 Preferences 6 / 70 Preferences We make three assumptions about preferences. If the rst two are met, an individual is considered to be rational. Transitivity Completeness Here we just mean that preferences are logically consistent. When facing bundles a and b, the consumer must be able to say either a % b, b % a, or a  b. If a % b and b % c, by the transitive property a % c In other words, the consumer can rank any two bundles. Rules out the possibility that the consumer can’t make a decision. 7 / 70 8 / 70 Preferences Utility Utility is a set of numerical values that re‡ect the ranking of various bundles. More is Better (Non-Satiation) Just like it sounds. We are talking about goods rather than bads and I will always prefer more to less. If basket a yields a higher utility than basket b, we strictly prefer a to b. We can summarize preferences using a utility function. we assign a numerical value to all possible bundles of goods re‡ecting the consumers rankings of the bundles. People prefer bundles with strictly more of both goods. This really is just here to simplify the thing. 9 / 70 Utility 10 / 70 Utility EXAMPLE For two goods q1 and q2 , lets say our utility function takes the form. In practice, it is impossible to quantify these things We almost always talk about ordinal rather than cardinal utility U = (q1 q2 ) 1 2 Do we prefer a basket with 10 q1 and 10 q2 or a basket of 48 q1 and 3 q2 ? 11 / 70 12 / 70 Utility Utility 1 Lets graph U = (q1 q2 ) 2 For ordinal, the ordering is all that matters. For cardinal, the order AND the magnitude matter. Bob did better on Sam on the test is an ordinal ranking Bob got a 97% and Sam got a 15% is a cardinal ranking. This is not incredibly useful. But we can turn this into two dimensions. 13 / 70 Utility 14 / 70 Utility An indi¤erence curve shows all the bundles that the consumer sees as equally desirable. Any combination on this indi¤erence curve will give us a utility of 10. We are indi¤erent between any basket on this curve. Lets x a level of utility at U and draw all the combinations of of q1 and q2 that give the same level of utility. There is an indi¤erence curve associated with each level of utility. For U = 10, the equation for the indi¤erence curve would be q2 = 100 q1 . As we move from the origin, utility increases. 15 / 70 An indi¤erence map shows all the indi¤erence curves. 16 / 70 Utility Utility EXAMPLE Suppose our utility function is U = 2q1 + 3q2 Here, baskets A and B give us the same level of utility because they are on the same curve. Lets draw an indi¤erence curve for U = 2 and U = 3. Basket C gives us a higher utility because it is on a higher indi¤erence curve. c  a, c  b, b  a. 17 / 70 Utility 18 / 70 Utility Properties of Indi¤erence Curves Properties of Indi¤erence Curves 2. There is an indi¤erence curve through every possible bundle 1. Bundles on indi¤erent curves farther from the origin are preferred This comes from the completeness property The consumer is able to rank any combination of bundles This comes from the more is better principle 19 / 70 20 / 70 Utility Utility Properties of Indi¤erence Curves Properties of Indi¤erence Curves 4. Indi¤erence curves slope downward 3. They cannot cross Otherwise the person would be indi¤erent between a bundle with more of both goods and less of both goods 21 / 70 Utility 22 / 70 Utility Properties of Indi¤erence Curves 5. They can’t be thick violates more is better Of course the bundles you choose between don’t have just two goods. You can treat one good as a composite good... or spending on everything else. 23 / 70 24 / 70 Utility Utility EXAMPLE It is useful to know how much of one good the consumer is willing to trade for another. What would an indi¤erence curve look like for two goods where you get satiated? That is, you only want to consume so much of the good then you start getting sick of it. This is captured by the slope of the indi¤erence curve... and this is called the marginal rate of substitution (MRS ) 25 / 70 Utility 26 / 70 Utility The MRS tells us the maximum amount of one good a consumer will sacri ce to get one more of the other If your MRS is -2 you would be willing to give up two burritos for one piece of pizza. It is simply the slope of the indi¤erence curve. 27 / 70 28 / 70 Utility Utility The marginal utility of q1 is just ∂U ∂q 1 . The marginal rate of substitution is the negative ratio of marginal utilities To nd the MRS, we need to understand marginal utility. The marginal utility is the extra utility you get from increasing something by an in nitesimally small amount. MRS = ∂U ∂q 1 ∂U ∂q 2 What is the intuition behind this? 29 / 70 Utility 30 / 70 Utility EXAMPLE Most indi¤erence curves are convex to the origin. This means that people prefer averages to extremes What is the MRS for this utility function? For example... U = (q1 q2 ) 1 2 if you have 100 pairs of trousers and no shirts, you would be willing to give up a lot of trousers for one shirt. If you have 100 shirts and no trousers, you would be willing to give up a lot of shirts for one pair of trousers. What does this mean? If you eat 6 units of q1 and 3 units of q2 , how many q1 would you give up for a q2 ? 31 / 70 32 / 70 Utility Utility But they can be other shapes. For example, if goods are perfect substitutes they have a constant MRS You have no desire to consume perfect substitutes together... They do not compliment each other in any way. If goods are perfect complements, the indi¤erence curves are right angles You only want the goods in xed proportions For right shoes and left shoes it might look like U = minfq1 , q2 g For Coke and Pepsi our utility function might look like U = 2q1 + 2q2 If I have 1,000,000 right shoes and 1 left shoe, I am no better o¤ than having 1 right shoe and 1 left shoe They are just straight parallel lines in this case. 33 / 70 Utility 34 / 70 Utility They can also be quasi-linear If they are linear in one argument A quasi-linear utility function would look something like 1 U = (q1 ) 2 + q2 35 / 70 36 / 70 Utility Budget Constraint EXAMPLE Preferences make zero reference to income constraints. Draw the indi¤erence curves for the following utility functions = U = U = U = U Given our assumptions on preferences, consumers would like an in nite amount of both goods. Obviously, consumers are constrained by the amount of money they have. x1 + x2 x 1 x2 minfx1 , 3x2 g maxfx1 , x2 g 37 / 70 Budget Constraint 38 / 70 Budget Constraint If Y is the consumer’s income and p1 and p2 are the prices of q1 and q2 respectively, the budget line will take the form. We can depict this constraint using a budget line. Y The budget line tells us all the combinations of goods a consumer can buy if she spends all her money. q2 = p1 q1 + p2 q2 Y p1 q1 = p2 Of course people save and borrow, but we can easily incorporate this by talking about consumption today and consumption tomorrow as two di¤erent goods. 39 / 70 40 / 70 Budget Constraint Budget Constraint We call the opportunity set the set of all goods the consumer can possibly buy. Y  p1 q1 + p2 q2 The slope of the budget line is the marginal rate of transformation (MRT ) This is how many units of one good the consumer must give up in order to purchase on more of the other. This is just equal to the price ratio The budget line is the frontier of the opportunity set. MRT = dq2 = dq1 p1 p2 41 / 70 Budget Constraint 42 / 70 Budget Constraint EXAMPLE You make $ 100,000 a year The price of pickles is $ 5 and the price of cell phones is $ 100. Draw the budget set, what is the marginal rate of transformation? Can you think of a case in which the MRT is not constant? 43 / 70 44 / 70 Budget Constraint Constrained Consumer Choice We can rank baskets by the indi¤erence curves, and we know what we can a¤ord from the opportunity set EXAMPLE What happens if the price of pickles doubles to $ 10? All that is left to do is to nd the basket the consumer likes the most that is contained within the budget set. What happens if your income doubles to $ 200,000? We will see how this is done graphically rst, then use math. 45 / 70 Constrained Consumer Choice 46 / 70 Constrained Consumer Choice First lets show that the optimal bundle lies on the budget constraint. If it is outside the budget constraint, the consumer cannot a¤ord it. If it is within the opportunity set, the consumer can be made better o¤ by consuming more of both goods (more is better). The optimal choice is the bundle on the indi¤erence curve just touching the budget constraint. 47 / 70 48 / 70 Constrained Consumer Choice Constrained Consumer Choice If there is an interior solution, the indi¤erence curve is tangent to the budget constraint at the optimal bundle. It is possible to have an interior solution or a corner solution An interior solution is one in which the optimal bundle has positive quantities of both goods A corner solution is when the optimal bundle contains none of one of the goods This means that the indi¤erence curve and the budget line have the same slope. The slope of the budget constraint is the price ratio, and the slope of the indi¤erence curve is the marginal rate of substitution. The optimal bundle is the one such that. MRS = MU1 = MU2 p1 = MRT p2 49 / 70 Constrained Consumer Choice 50 / 70 Constrained Consumer Choice To get some intuition for why this is the optimal bundle, lets rearrange the thing MU1 MU2 MU1 p1 = = Suppose the marginal utility from pizza is 5 and pizza costs $1. The marginal utility from burritos is 6 and the price of a burrito is $3 p1 p2 MU2 p2 MU 2 We are not at an optimum because p1 1 > MU p 2 . If I buy 1 less burrito and 3 more pizzas, I will lose 6 units of utility, but I will gain 15 units. Think of this as "bang for your buck." The utility you get per dollar spent is the same for both goods 51 / 70 52 / 70 Constrained Consumer Choice Constrained Consumer Choice Sometimes, the highest indi¤erence curve attainable does not occur when the budget line and indi¤erence curve are tangent. We have a corner solution in this case, and the consumer will only buy one of the goods and none of the other. EXAMPLES Graphically, what will happen to your optimal bundle if the price of a good changes or income changes? Graphically show the optimal consumption bundle when the goods are perfect substitutes and when they are perfect complements. 53 / 70 Constrained Consumer Choice 54 / 70 Constrained Consumer Choice We can also use math, and we will get the same result. Just trying to maximize something subject to a constraint. The problem we are trying to solve is. If we know that there is an interior solution, we can use two di¤erent methods. maxU (q1 , q2 ) the substitution method and the Lagrangian method. q 1 ,q 2 s.t. Y = p1 q1 + p2 q2 That is, we need to pick the q1 and q2 that give use the biggest U while not violating the constraint 55 / 70 56 / 70 Constrained Consumer Choice Constrained Consumer Choice The substitution method entails just plugging the constraint into our objective function. We can do this by rearranging the constraint for one of our control variables Y q1 = p1 q1 + p2 q2 Y p2 q2 = p1 Plug this into the utility function. This is now an unconstrained maximization problem   Y p2 q2 , q2 maxU = q2 p1   dU ∂U dq1 ∂U p2 ∂U ∂U + = + =0 = dq2 ∂q1 dq2 ∂q2 p1 ∂q1 ∂q2 57 / 70 Constrained Consumer Choice Constrained Consumer Choice Rearranging we get the same condition that ∂U ∂q 1 ∂U ∂q 2 58 / 70 = The Langrangian method is similar, but works in a wider variety of cases and this is how you should do things p1 p2 We once again turn the constrained maximization problem into an unconstrained one To be thorough you would also check the second order condition 59 / 70 60 / 70 Constrained Consumer Choice Constrained Consumer Choice The rst step is to multiply the constraint by a constant λ and then add it or subtract it from the objective function max L = U (q1 , q2 ) λ (q1 p1 + q2 p2 q 1 ,q 2 ,λ Y) Take the rst order condition with respect to our controls, and then solve the system to get the solution ∂L ∂q1 ∂L ∂q2 ∂L ∂λ If we were minimizing, we would add the constraint rather than subtract. λ is the Lagrangian multiplier - it penalizes us for breaking the constraint = = ∂U ∂q1 ∂U ∂q2 = Y λp1 = 0 (1) λp2 = 0 (2) q1 p1 + q2 p2 = 0 61 / 70 Constrained Consumer Choice Solve equation 1 for λ λ= 62 / 70 Constrained Consumer Choice If we rearrange terms we will get the exact same solution ∂U 1  ∂q1 p1 ∂U ∂q 1 Plug this into equation two ∂U ∂q2 (3) p1 ∂U 1  p2 = 0 ∂q1 p1 = ∂U ∂q 2 p2 This is just the MRS = MRT 63 / 70 64 / 70 Constrained Consumer Choice Constrained Consumer Choice EXAMPLE Use the Langrangian method to nd the optimal consumption bundles for the following utility functions = (q1 q2 ) 2 U = (q1 )ρ + (q2 )ρ Rather than xing the budget line and nding the highest indi¤erence curve, we x the indi¤erence curve and nd the lowest budget line. This is the exact same thing... two sides of the same coin 1 U What if we want to minimize expenditure for some xed level of utility? It can be useful, though because we can observe expenditure but not utility
1ρ (In practice, we know the tangency condition so we can nd it right away and plug it into the constraint) 65 / 70 Constrained Consumer Choice 66 / 70 Behavioral Economics EXAMPLE We made a number of assumptions on how people behave. Show that expenditure minimization gives you the same solution as utility maximization for U (q1 , q2 ), and income Y and prices p1 and p2 . 67 / 70 Behavioral economics looks at some of the departures from our rationality assumptions. Experimental Economics (how we see people behave) + Economic Theory = Behavioral Economics 68 / 70 Behavioral Economics Summary What are the assumptions we make about preferences? Transitivity What is utility? What is a utility function? Endowment E¤ect What do indi¤erence curves tell us? Salience What is the marginal rate of substitution? Bounded Rationality What is the slope of the budget line? How do we nd the optimal bundle in general? What is behavioral economics? 69 / 70 70 / 70