Microeconomics MSc. Utility. Todd R. Kaplan. November University of Haifa. Kaplan (Haifa) micro November / 30

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1 Microeconomics MSc Utility Todd R. Kaplan University of Haifa November 2010 Kaplan (Haifa) micro November / 30

2 My favorite utility functions Perfect substitutes: u(x 1, x 2 ) = x 1 + x 2 Perfect complements: u(x 1, x 2 ) = minfx 1, x 2 g Quasilinear: u(x 1, x 2 ) = g(x 1 ) + x 2 Cobb-Douglas: u(x 1, x 2 ) = x1 ax 2 b where a > 0 and b > 0. CES: u(x 1, x 2 ) = (x1 r + x 2 r )/r. When r =, 0, 1,, CES equals which of the following functions: min, max, perfect substitutes, Cobb-Douglas? Guess. What do the indi erence curves look like. Find a function that changes C-D into x1 c x 1 2 c Kaplan (Haifa) micro November / 30

3 Marginal Utility The marginal utility of commodity i is the rate-of-change of total utility as the quantity of commodity i consumed changes. MU i = U x i. What are the marginal utilities of perfect substitutes, perfect complements, quasi-linear, and Cobb-Douglas, CES? Kaplan (Haifa) micro November / 30

4 Marginal Rate of Substitution. MRS = MU 1 /MU 2 (note some use negative of this). This is also the slope of the indi erence curve. Why? MRS is not a ected by a transformation V (x) = f (U(x)). Why? (hint: chain rule). What is MRS of Cobb-Douglas, Quasi-Linear, Perfect Substitutes, Perfect Complements, CES? Kaplan (Haifa) micro November / 30

5 Choice Draw the budget set. Draw the indi erence curves. Best choice is at the highest indi erence curve that is still in the budget set. This is equivalent to solve the problem: max x1,x 2 U(x 1, x 2 ) s.t. p 1 x 1 + p 2 x 2 m, and x 1, x 2 0 We maximize utility subject to the budget constraint. Kaplan (Haifa) micro November / 30

6 Rational Constrained Choice The most preferred a ordable bundle is called the consumer s MARSHALLIAN DEMAND at the given prices and budget. Marshallian demands will be denoted by x1 (p 1, p 2, m) and x2 (p 1, p 2, m). This is the solution to the previous problem. Note that xi (t p 1, t p 2, t m) = xi (p 1, p 2, m). Indirect utility v(p 1, p 2, m) is the utility achieved in the previous problem: v(p 1, p 2, m) = u(x1 (p 1, p 2, m), x2 (p 1, p 2, m)). Note that v(t p 1, t p 2, t m) = v(p 1, p 2, m). Kaplan (Haifa) micro November / 30

7 To solve the consumer problem To solve the consumer problem Check to see what type of preferences. Smooth preferences such as Cobb-Douglas can be solved in one of 3 ways. 1 Substitution. 2 MRS=Slope of Budget constraint. 3 Lagrangian. Kaplan (Haifa) micro November / 30

8 Substitution method. Substitution method. 1 Solve b.c. for one var. p 1 x 1 + p 2x 2 = m 2 Plug into utility u(x 1, m/p 2 p 1 x 1 /p 2) 3 Take the derivative with respect to x 1 and set this equal to zero. 4 Use this and original b.c to solve for x 1 and x 2 Try this for x 1 x 2 Kaplan (Haifa) micro November / 30

9 MRS method. MRS method. (x1, x 2 ) satis es two conditions: 1 The budget is exhausted; p 1 x 1 + p 2x 2 = m 2 The slope of the budget constraint, p 1 /p 2, and the slope of the indi erence curve containing (x 1, x 2 ) are equal at (x 1, x 2 ). Try this for CES What is solution when r! 0. Kaplan (Haifa) micro November / 30

10 Homework 4. 1 For u(x 1, x 2 ) = (x1 r + x 2 r ) /r. Show how when r!, r! 1, the Marshallian demand goes to that of perfect complements, x i = m/(p 1 + p 2 ), and perfect substitutes, x i = m/p i if p i < p j, respectively.. Solve for Marshallian Demand and Indirect Utility for the following utility functions and methods. 2. u(x 1, x 2 ) = p x 1 + x 2 by substitution. 3. u(x 1, x 2 ) = e x 1 e 2x 2 by MRS. Kaplan (Haifa) micro November / 30

11 Lagrangian method. Lagrangian method. Steps. 1 Set up Langrangian:L = U(x 1, x 2 ) + λ(m p 1 x 1 p 2 x 2 ) 2 Take derivatives w.r.t. x 1, x 2, and λ. 3 Set them equal to zero and solve. Note: All methods basically are the same. Solve for Cobb-Douglas x a 1 x b 2. Kaplan (Haifa) micro November / 30

12 Cobb Douglas Solve for Marshallian Demand and Indirect Utility for the following utility functions and methods. Cobb Douglas x1 2x 2 1 using the MRS method. Cobb Douglas x1 2x 2 3 using the substitution method. Cobb Douglas x1 ax 2 b by Langrangian. Which prices does x 1 depend upon? What does this mean? Kaplan (Haifa) micro November / 30

13 Constrained Choice Problems If preferences are well behaved, then we can usually obtain the ordinary demands are obtained by solving those 3 methods. Problems (IMPORTANT!!) 1 Preferences are not convex. 2 Corner Solutions. (x 1 = 0 or x 2 = 0) 3 Kinky I.C s such as minfax 1, x 2 g To stop 1, one needs 2nd-order conditions. Puzzle. Does x1 2x 2 2 satisfy this? Try to solve a case of 2 (perfect substitutes) and 3 (perfect complements). Kaplan (Haifa) micro November / 30

14 Roy s Identity Theorem Roy s identity: x i (p, m) = v (p,m) p i v (p,m) m. Example Show that Roy s identity holds for u(x 1, x 2 ) = x 1 x 2. Kaplan (Haifa) micro November / 30

15 Envelope Theorem As part of the proof we need to know the handy Envelope Theorem. Theorem Envelope Theorem:If M(a) = max x f (x, a) = f (x(a), a) then dm(a) f (x, a) = da a Proof. dm (a) da = However, Why??. f (x,a) a f (x,a) x + x =x (a) = 0. x =x (a) f (x,a) x x =x (a) x 0 (a). x =x (a) Kaplan (Haifa) micro November / 30

16 Envelope Theorem Example Example Try: f (x, a) = a x x 2 What is x(a)? What is and x =x (a) f (x,a) x What is f (x(a), a)? What is d da f (x(a), a)? f (x,a) a? x =x (a) Kaplan (Haifa) micro November / 30

17 Roy s Identity Proof. x i (p, m) = Proof. v (p,m) p i v (p,m) m v(p, m) = max x1 u(x 1, m p 1x 1 p 2 ) We have v (p,m) p 1 = d max x 1 u(x 1, dp 1 = du dx 1 x 1 p 1 + u 2 ( since du dx 1 We have x 1 p 2 ) = u 2 ( m p 1 x 1 p 2 ) x 1 p 2 ) = 0 by the Envelope Theorem. v (p,m) m = d max x 1 u(x 1, dm m p 1 x 1 p 2 ) = du(x m 1(p,m), p 1 x 1 (p,m) p ) 2 dp 1 = = du dx 1 x 1 m + u 2 p 2 = u 2 p 2 Kaplan (Haifa) micro November / 30

18 Homework 5. 1 People in Smallsville take pictures and eat pizza. They also like to smile. Rational Ralph has utility u(s, x 1, x 2 ) = x 1 x 2 + sx 1 + x 2 where s is the amount of smiling he does and x 1 is the amount of pictures he takes and x 2 is the amount of pizza he consumes. Assume he maximizes the amount of money he spends on pictures and pizza (he doesn t have to pay to smile). Assume the price of pictures and pizza are $1 each and Ralph has $3. (i) Assume Rational Ralph is rational and hence maximizes his utility subject to his budget constraint. As a function of s, how much will Ralph spend on pictures and pizza. Using the envelope theorem, how much would his utility go up by smiling more (in terms of x 1 and x 2 )? (ii) Crazy Eddie also has utility u(s, x 1, x 2 ) = x 1 x 2 + sx 1 + x 2. However, Eddie feels compelled take a picture for each smile he has. This results in x 1 = s for all s 3 and x 1 = 3 for all s > 3. How much would his utility go up by smiling more (in terms of x 1 and x 2 )? Kaplan (Haifa) micro November / 30

19 Homework Indirect utility is given by v(p 1, p 2, m) = (m+p 1+p 2 ) 2 4p1 2p. What are the 2 demands for x 1 and x 2? When is the demand for good x 1 greater than that for good x 2? Kaplan (Haifa) micro November / 30

20 Problems Taxes. How do taxes a ect choice? The budget constraint with taxes is p 1 (1 + vat)x 1 + p 2 (1 + vat)x 2 m(1 t) Should there be a di erence between (vat, t) = (0.25, 0) and (vat, t) = (0, 0.2)? Blumkin and Ru e found there is. People work harder if there is a consumption tax rather than an income tax. Form of Money Illusion. Kaplan (Haifa) micro November / 30

21 Mental Accounting and Relativity Where the money comes from matters: mental accounting. Have a $200 ticket to the superbowl and lost it. Do you buy another one? How about you are just about to buy a ticket to the superbowl and discover that you are missing $200. Do you still buy it? Do you go across town when you nd that you can pay 985 instead of 999 on a computer? How about if you can pay 14 instead of 28 on a lamp? How about free instead of 14? Kaplan (Haifa) micro November / 30

22 Dan Ariely found price a ects utility! This happened in painkillers. They also found this with wine and fmri studies. Book vouchers. A. $10 amazon voucher for free. B. $20 amazon voucher for $7. C. $10 amazon voucher for $1. D. $20 amazon voucher for $8. Does this violate utility maximization? Say your budget was $8 and there was an $8 reading light you wanted to get. The remaining money was spent on doughnuts at $1 each. You might enjoy jeans more if you pay more for them. Could also be signalling/game theory. Kaplan (Haifa) micro November / 30

23 Utility over time. Say there are two time periods t=0 and 1. We can have a utility function over consuming a good at time 0 or time 1. u(c 0, c 1 ) We like people to have time impatience: u(x + c, x) > u(x, x + c) for all x and c. Kaplan (Haifa) micro November / 30

24 Common form In economics, the most common form of utility is (where β < 1) V (c) = Does this satisfy time impatience? β t U(c t ) t=0 Stationary means that if (c 0, c 1, c 2, c 3,...) (c 0, d 1, d 2, d 3...), then (c 1, c 2, c 3...) (d 1, d 2, d 3...). Is the above utility stationary? Kaplan (Haifa) micro November / 30

25 Hyperbolic Discounting Hyperbolic Discounting Use whatever means possible to remove a set amount of money from your bank account each month before you have a chance to spend it. Advice in New York Times: Your Money column [1993] People perhaps do best to commit not to spend money: pension funds, not having credit cards. Can our tools model this? Kaplan (Haifa) micro November / 30

26 Hyperbolic Discounting Question Do you prefer $100 today vs. $200 two years from now? Do you prefer $100 six years from now vs. $200 eight years from now? What does this violate? Kaplan (Haifa) micro November / 30

27 Hyperbolic Discounting Our typical discounting the future: At time 0, u = t=0 β t U(c t ). At time 1, u = t=1 β t 1 U(c t ) Instead here we could use: At time 0, u = t=0 1 1+t U(c t) At time 1, u = t=1 1 t U(c t) Why does this work in explaining behaviour? Utility is Ln c t for t = 1, 2. (This is Cobb Douglas). Interest rate is 0% real (Japan/US). We receive income of m at time 1. At time zero we decide between consuming at time 1 and time 2. Instead say at time 1 we make the same decision Kaplan (Haifa) micro November / 30

28 Hyperbolic Discounting At time 0, we want to save At time 1, we want to save = 2 5 = 1 3 of income. of income. Conclusion: Ripping up credit cards or enrolling in automatic pension plans (opt out pension plans) is a good idea Kaplan (Haifa) micro November / 30

29 References The classic paper that reintroduced hyperbolic discounting into economics. D. Laibson, Golden Eggs and Hyperbolic Discounting, Quarterly Journal of Economics 112 (1997), A paper that strongly critiques the use of such utility functions. A. Rubinstein, Economics and Psychology? The Case of Hyperbolic Discounting. International Economic Review, 44 (2003), Kaplan (Haifa) micro November / 30

30 Homework 5b. 1 We have utility of ln x for t = 1, 3 and utiliy of 1.5 at time 2 (independent of how much we consume). We receive income of m at time 1. Interest rate is 0% real (Japan/US). Discounting is hyperbolic (1/(1 + t)). At time zero we decide between consuming at time 1 and time 3. (Notice the change.) How much income would we save? Now say instead, at time 1 we make the same decision. How much would we save then? 2 Normal discounting is β t for t periods in the future (where 0 < β < 1). The hyperbolic discounting example we had was 1/(1 + t) for t periods in the future. Let us say that the discount is quasi-hyperbolic: β δ t for t periods in the future where (0 < δ < 1 and β > 0). Assume utility u(x t ) = ln(x t ) and a consumer has income m and consumes in periods 1 and 2. If the consumer decides in period 0 what to consume in periods 1 and 2, what is x 1 and x 2? If the consumer decides in period 1 what to consume in periods 1 and 2, what is x 1 and x 2? For what range of β and δ does the consumer s desire Kaplan to (Haifa) consume early increase? micro November / 30