Intertemporal choice

Imagine that a consumer can decide to consume in two periods: period 1 and period 2, the amount of consumption in period 1 is indicated with c1 while the amount of consumption in period 2 is denoted with c2.Suppose also that in each one of these periods the consumer receives some money that we are going to indicate with m1 and m2 respectively for period 1 and period 2. The consumer can transfer money from period 1 to period 2 and vice versa at an interest rate that we are going to denote with r. The consumer has therefore three choices:

• Consume m1 in the first period and m2 in the second

• Lend some money in the first period (m1 – c1), thus consuming less during it, and consuming more in the second. If this is the case, the amount saved in the first period will be multiplied by (1+r) thanks to the presence of the interest rate. This means that, if interest rate is present, by saving $2 in the first period you will have in the second period more than $2.

• Borrow some money from the second period (m2 – c2), thus consuming more in the first period but less in the second. If this is the case, the amount borrowed from the second period will be divided by (1+r) thanks to the presence of the interest rate. This means that, if interest rate is present, if you want to consume $2 in the first period, you have to give up more than $2 in the second.

Obviously, the consumer can lend money until he finishes the amount received in the first period and he can borrow until he finishes the amount received in the second period. The set of possible combinations of consumption in the two period make up the consumer’s budget constraint. It can be written in two ways depend on whether you want to look at this relationship from the point of view of a consumer in period 1 or in period 2.

(1+r)c1 + c2 = (1+r)m1 + m2

This is the form based on the second period consumption because, as you can see, period two consumption and income are not effected by the interest rate while what is referred to period 1 is multiplied for it. If you divide this formula by (1+r) you get:

This is the form based on the first period consumption because, this time, period 1 consumption is free from the interest rate while period two consumption and income are divided by it. Whether you like more one or the other is indifferent because the graphical representation and calculations related to them is the same regardless of the one you consider. When dealing with problems of intertemporal choice, in order to represent them, we use on the horizontal axis period 1 consumption and on the vertical axis period 2 consumption. The relationship depicted through the formulas above is a linear one and therefore is represented on the graph thanks to a straight line. The vertical intercept of this line is C2 = (1+r)m1 + m2 which represents the maximum amount that the person can consume in period 2 if he saves everything he has in period 1, while the horizontal intercept is c1 = m1 + m2/(1+r) which represents the maximum amount the person can consume in period 2 by borrowing everything from period 2. The slope of the budget line is – (1+r).

THE PREFERENCES OF CONSUMERS

We can represent the preferences of various consumers in the way we are used to (did you miss our lecture on consumer’s preferences?). We consider three cases of preferences:

• Linear preferences which are represented with a line and are related to consumers who always substitute in the same way among consumption in the first or second period.

• Convex preferences represented with a Cobb-Douglass function and are related to consumers who are willing to substitute consumption across the two periods but not always in the same way.

• Fixed proportion preferences represented with L-shaped functions and are related to consumers who are not willing to substitute consumption across the two periods.

WHAT THE CHOICE OF A CONSUMER SAYS ABOUT HIM

By using together the budget constraint and the indifference curves we can find the optimal choice of the consumer on the usual way: by finding the tangency condition between the two. If the consumer ends up consuming in the first period less than his income, therefore if c1 < m1, we define him as a saver (or lender). If instead, the consumer ends up consuming more in the first period than his income, therefore if c1 > m1, we define him as a borrower.

WHAT HAPPENS WHEN THE INTEREST RATE CHANGES

When the interest rate changes the indifference curves will not move but the budget constraint will. This is because the equation of the budget constraint depends on it. Remember that we said that the consumer possesses some income in period 1 and some in period 2. This combination of consumption is always available to him no matter the interest rate and therefore it represents the consumer’s endowment. This means that, when the interest rate changes, the budget line will pivot around it. We can find the vertical intercept with the equation C2= m2 + (1+r)(m1 – c1), because of that, an increase in the interest rate will make that number bigger and therefore the vertical intercept of the line will be higher. The horizontal one instead was defined as c1 = m1 + m2/(1+r) and, because of that, the horizontal intercept of the line, when the interest rate rises, becomes smaller and graphically closer to the origin. The line will become steeper when the interest rate increases and becomes flatter when the interest rate becomes lower.

WHAT WE KNOW THANKS TO PREFERENCES

If the consumer we are considering is a saver and then we increase the interest rate we know for sure that he will remain a saver. Because he is a saver he is consuming in the first period less than he could thanks to his income (c1 < m1,) and therefore the consumption bundle he will choose is to the left of the endowment. When the interest rate increases, as we saw, the budget line becomes steeper. Notice that now the part of the budget line to the left of the endowment has become higher while the part of the budget line to the right of the endowment has become lower. Because the consumer chose to move to the left of the endowment when the line was lower now he has more incentive to do so because the combinations of consumptions he can now afford are higher. Because the consumer did not choose in the beginning the combinations to the right of the endowment, why should he do so when the combinations he can afford are even worse? The same exact reasoning can be done with a consumer that is initially a borrower and the interest rate declines because this time he will surely continue to consume points on the right of the endowment given that the budget line has become flatter. We can not be so sure about what a consumer will do when he is initially a borrower and the interest rate increases or when he is initially a saver and the interest rate decreases. This is because, after the decline in the interest rate, a saver who chooses to remain a saver will surely be worse off given the fact that to the left of the endowment the budget line becomes lower. The same happens with a borrower who chooses to remain a borrower even after an increase in interest rate.

HOW TO COMPARE MOENEY CORRESPONDING TO DIFFERENT TIME PERIODS

As we said, thanks to the interest rate, we can transform $1 of consumption in the first period to $(1+r) for the following period or transform $1 of consumption in the second period in $1/(1+r) of consumption in the first one. When you want to compare money in the second period with money in the first period you have to divide it by (1+r) because otherwise, thanks to the presence of interest, the money in the second period will have an “advantage”. If you consider all the amounts of money in terms of the first period consumption, you can truly compare them and determine which one is really larger. Because the consumer obviously prefers to have more consumption possibilities consumption having a higher first period value will dominate a combination of consumption having a lower first period value. This is useful when comparing investments of money. Because not all the investments last the same amount of time, in order to compare them and, therefore, decide which one is in the end better it is sufficient to transform all of them in terms of present (first period) value and then pick the one which is the highest. When you consider multiple time periods, in order to set the future consumption in present terms you have just to modify slightly the formula in:

Where t is a number indicating the time period you are considering. The present corresponds to t=1 in fact, by setting t=1, the denominator becomes 1 and the expression gives $1 as a result.

HOW TO EXPLAIN THE BAHAVIOUR OF CONSUMERS

In order to explain why a consumer behaves in a certain way let’s express the utility of that consumer as

In that formula U(C) represents a general utility function of our consumer which depends on his consumption. As you can see the utility the consumer gets from the second period consumption is influenced by a factor p which represents his impatience. The more he is impatient the higher p and the lower will be the utility from period 2 consumption. If we calculate the marginal rate of substitution we get:

In that formula U’(C) denotes the derivative of the utility function of the consumer. The consumer will borrow money if the utility he gets from consuming in period 1 compared to period 2 is higher than the time impatience. Meaning that the consumer will borrow money if the MRS in the endowment (M1 , M2) is higher than 1+r which represents the absolute value of the slope of the budget line. So the consumer will borrow if:

MRS > (1+r)

The consumer will instead be a saver if: