Microeconomics Corso E John Hey

Notation Intertemporal choice. Two periods: 1 and 2. Notation: m 1 and m 2 : incomes in the two periods. c 1 and c 2 : consumption in the two periods. r: the rate of interest. 10% r = 0.1, 20% r = 0.2. Hence the rate of return = (1+r)

The Budget Line c 1 (1+r) + c 2 = m 2 + m 1 (1+r). In the space (c 1,c 2 ) a line with slope -(1+r). The intercept on the horizontal axis = m 1 + m 2 /(1+r)... the present value of the income stream (note that m 2 is discounted at the rate r). The intercept on the vertical axis = m 1 (1+r) + m 2... the future value of the income stream.

Preferences? If I offer you a choice between 10 CDs today and 10 CDs in a year, which do you prefer? 10 CDs today and 11 CDs in a year? 10 CDs today and 13 CDs in a year? 10 CDs today and 16 CDs in a year? 10 CDs today and 20 CDs in a year? 10 CDs today and 25 CDs in a year? Implications? Individuals discount future … … and the discount rate varies from individual to individual.

The Discounted Utility Model Consumption c gives utility u(c) and the utility of a bundle (c 1,c 2 ) is given by: U(c 1,c 2 )=u(c 1 ) + u(c 2 )/(1+ρ) where ρ is the discount rate of the individual. u(c 2 )/(1+ρ) is the discounted value of the income of period 2 discounted at the rate ρ – which is individual-dependent. (Recall that m 2 /(1+r) is the discounted value of the income of the second period Discounted at the rate of interest r.)

The Discounted Utility Model U(c 1,c 2 ) = u(c 1 ) + u(c 2 )/(1+ρ) There are two components: The utility function of the individual: u(c) The individual’s discount factor: ρ Usually u(c) is concave in the space (c,u(c)) (Why?) Usually ρ > 0 (Why?)

Indifference curves in the space (c 1,c 2 ) An indifference curve is given by: Utility = constant …. … U(c 1,c 2 ) = constant … … u(c 1 ) + u(c 2 )/(1+ρ) = constant Note the difference U(c 1,c 2 ), the utility of the basket (c 1,c 2 ), and u(c), the utility of consumption c.

Indifference curves in the space (c 1,c 2 ) u(c 1 ) + u(c 2 )/(1+ρ) = constant If u(c) is linear, we have c 1 + c 2 /(1+ρ) = constant Hence c 2 = constant - c 1 (1+ρ) A line with slope (1+ρ).

An example: u(c) = √c and ρ = 0 U(c 1, c 2 ) = u(c 1 ) + u(c 2 ) = √c 1 + √c 2 The indifference curve through the point (9,9) is given by: √c 1 + √c 2 = 6 Other points on this curve are: (0,36), (1,25), (4,16), (16,4), (25,1), (36,0) Note that at every point √c 1 + √c 2 = 6 The equation is : c 2 = (6 - √c 1 ) 2 See the next graph ….

The indifference curves If u(c) is concave the indifference curves are convex. If u(c) is linear the indifference curves are linear. If u(c) is convex the indifference curves are concave. The slope along the equal consumption line are –(1+ρ)

The Discounted Utility Model If u(.) is concave (linear, convex), the indifference curves in the space (c 1,c 2 ) convex (linear, concave). The slope of every indifference curve on the equal consumption line in (c 1,c 2 ) space is equal to -(1+ ρ). There is a proof in the text. Let’s go to Maple …

Past Exam Questions In the next two questions you will be asked to consider an individual, taking intertemporal decisions and having Discounted Utility preferences and utility function u(x) = x^0.5 (that is, the utility of x is the square root of x). Suppose the individual is faced with a choice of two intertemporal streams of consumption P and Q. Such a stream is denoted by (c1,c2) where c1 is the consumption in period 1 and c2 the consumption in period 2. His discount factor is specified below. The consumption streams are: P = (16,9) Q = (25,4). The individual's discount factor is 0. Question 10: Does the individual prefer stream P or stream Q? P The individual is indifferent We cannot tell from the information given Q Question 11: Suppose the individual could have the same consumption c in both periods. What would c have to be to make him indifferent to stream P?

Summary The budget line has slope = -(1+r) The indifference curves given by the Discounted Utility Model along the equal consumption line have slope = -(1+ρ)

Chapter 21 Goodbye!