Intermediate Microeconomic Theory Intertemporal Choice

So far, we have considered: How an individual will allocate a given amount of money over different consumption goods. How an individual will allocate his time between enjoying leisure and earning money in the labor market to be used for consuming goods. Another thing to consider is how an individual will decide how much of his money should be consumed now, and how much he should save for consumption in the future (or how much to borrow for consumption in the present).

Intertemporal Choice To think about this, instead of considering how an individual trades off one good for another and vice versa, we can think about how an individual trades off consumption (of all goods) in the present for consumption (of all goods) in the future. i.e. two “goods” we will consider are: c 1 (dollars of consumption in the present period), and c 2 (dollars of consumption in a future period).

Intertemporal Choice So an intertemporal consumption bundle is just a pair {c 1, c 2 }. E.g. a bundle containing $50K worth of goods this year, and $30K next year is denoted {c 1 = 50K, c 2 = 30K}. Endowment now describes how much money an individual will have in each period, without saving or borrowing, denoted {m 1, m 2 }. For example, An individual who earns $50K each year in the labor market {m 1 = 50K, m 2 = 50K}. An individual who inherits $100K this year, but doesn’t expect to earn or receive any money next year {m 1 = 100K, m 2 = 0}.

Intertemporal Budget Constraint Consider an individual has an intertemporal endowment of {m 1, m 2 } and can borrow or lend at an interest rate r. What will be his intertemporal budget constraint? What is one bundle you know will be available for consumption? What else can he do?

Intertemporal Budget Constraint What is slope? Hint: How much more consumption will he have next period if he saves $x this period? To put another way, how much does consuming an extra $x this period “cost” in terms of consumption next period. What will intercepts be? x c2m2c2m2 m 1 c 1 ?

Intertemporal Budget Constraint Intercepts Vertical – What if you saved all of your period 1 endowment, how much would you have for consumption in period 2? Horizontal – How much could you borrow and consume today, if you have to pay it back next period with interest? What happens to budget constraint when interest rate r rises?

Intertemporal Budget Constraint Example: Suppose person is endowed with $20K/yr Interest rate r = 0.10 What will graph of BC look like? What if r falls to 0.05?

Writing the Intertemporal Budget Constraint Given this framework, we can write out the intertemporal budget constraint in the typical form p 1 c 1 + p 2 c 2 = p 1 m 1 + p 2 m 2 We know the interest rate r will determine relative prices, but like with goods, we have to determine our “numeraire”.

Writing the Intertemporal Budget Constraint So intertemporal budget constraint can be written in two equivalent ways: Future value: future consumption is numeraire, price of current consumption is relative to that. How much does another dollar of current consumption cost in terms of future consumption? BC: (1+r)c 1 + c 2 = (1+r)m 1 + m 2 Present value: present consumption is numeraire, price of future consumption is relative to that How much does another dollar of future consumption cost in terms of current consumption? BC: c 1 + c 2 /(1+r) = m 1 + m 2 /(1+r)

Intertemporal Preferences Do Indifference Curves make sense in this context? What does MRS refer to in this context? Do Indifference Curves with Diminishing MRS makes sense in this context? What Utility function might be appropriate to model decisions in this context?

Intertemporal Choice We can again think of analyzing optimal choice graphically. What does it mean when optimal choice is a bundle to the left of endowment bundle? How about to the right of the endowment bundle?

Intertemporal Choice Similarly, we can solve for each individual’s demand functions for consumption now and consumption in the future, given interest rate (i.e. relative price) and endowment. c 1 (r,m 1,m 2 ) c 2 (r,m 1,m 2 ) So if u(c 1, c 2 ) = c 1 a c 2 b, what would be the demand function for consumption in the present?

Intertemporal Choice As we showed graphically, If c 1 (r,m 1,m 2 ) > m 1 the individual is a borrower If c 1 (r,m 1,m 2 ) < m 1 the individual is a lender Equivalently, If c 2 (r,m 1,m 2 ) < m 2 the individual is a borrower If c 2 (r,m 1,m 2 ) > m 2 the individual is a lender

Analog to Buying and Selling So instead of being endowed with coconut milk and mangos, we can think of being endowed with money now and money in the future. Moreover, instead of being a buyer of coconut milk by selling mangos, we can think of being a buyer of consumption now (i.e. a borrower) by selling future consumption.

Comparative Statics in Intertemporal Choice Suppose the interest rate decreases. Will borrowers always remain borrowers? Will lenders always remain lenders?

Present Value and Discounting The intertemporal budget constraint reveals that timing of payments matter. Suppose you are negotiating a sale and 3 buyers offer you 3 different payments schemes: 1. Scheme 1 - Pay you $200 one year from today. 2. Scheme 2 - Pay you $100 one year from now and $100 today. 3. Scheme 3 - Pay you $200 today. Assuming buyers’ word’s are good, which payment scheme should you take? Why? (Hint: think graphically)

Present Value and Discounting This is idea of present value discounting. To compare different streams of payments, we have to have some way of evaluating them in a meaningful way. So we consider their present value, or the total amount of consumption each would buy today. Also called discounting. In terms of previous example, with r = 0.10 the present value of each stream is: 1. PV of Scheme 1 = $200/(1+0.10) = $ PV of Scheme 2 = $100 + $100/(1+0.10) = $ PV of Scheme 3 = $200 While you certainly might not want to consume the entire payment stream today, as we just saw, the higher the present value the bigger the budget set (assuming same interest rate applies to all schemes!)

Present Value and Discounting What about more than two periods? As we saw, if r is interest rate one period ahead, PV of payment of $x one period from now is $x/(1+r). What is intuition? If you were going to be paid $m two years from now, what is the most you could borrow now if you had to pay it back with interest in two years? So what is general form for present value of a payment of $x n periods from now?