Microeconomics 2 John Hey

Next two lectures The lectures tomorrow (Tuesday the 22 nd of January) and next Monday (the 28 nd of January) will be on examination preparation. You should look at the first 16 questions on the first specimen paper (on the site). I will go through the questions. I might ask you for comments and suggestions. So do come prepared.

Intertemporal Choice Two periods 1 and 2; the individual receives income in each period and re-arranges it to give optimal consumption in each period. Special case of a very important economic model: the life- cycle model. Lecture 20: the general problem with general preferences. Lecture 21: the Discounted Utility model and individual behaviour; some comparative statics. Lecture 22: Intertemporal exchange; capital markets. Both lecture 21 and 22 today (latter just PowerPoint). We have done it all before (starting with lecture 6)...

Notation 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 (given by the market). 10% r = 0.1, 20% r = 0.2. Hence the rate of return = (1+r) which can be interpreted as a relative price (of consumption in period 1 relative to consumption in period 2). We are going to show the problem in (c 1, c 2 ) space with both the budget line and preferences illustrated.

The Budget Line in (c 1,c 2 ) space We did this in Lecture 20. 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 the 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 that is, U(c 1,c 2 ) = constant that is, 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).

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

The slope along the equal consumption line c 1 =c 2 Indifference curves given by: u(c 1 ) + u(c 1 )/(1+ρ) = constant Differentiating we get u’(c 1 )dc 1 + u’(c 2 )dc 2 /(1+ρ) = 0 Re-arranging we get dc 2 /dc 1 = – u’(c 1 )(1+ρ)/u’(c 2 ) So when c 1 = c 2,slope, dc 2 /dc 1 = – (1+ρ) Let us go to Maple...

The Generalised Discounted Utility Model Here the individual is not just concerned with two periods but with many, possibly (?) an infinite number. Call them 1,2,3,... Consumption in period t is c t. Lifetime utility according to this model is U(c 1,c 2, c 3,...) = u(c 1 ) + u(c 2 )/(1+ρ) + u(c 3 )/(1+ρ) This is the Generalised Discounted Utility Model – used, for example, in the (in/famous) Life Cycle model of Friedman. Do you discount the future at a constant rate?

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!