Inter Temporal Choice: Budget Constraint & Consumer Preferences

12 months ago

Atiqur Rahman

16 minutes

Introduction

Inter-temporal choice represents the way consumers rearrange consumption over time by saving and borrowing. If an individual is a saver, then he is consuming less in the current period and consuming more in the future period.

On the other hand, if an individual is a borrower, then he is consuming more in the current period, but consumption in the future must fall below future income to repay the loan. The decision to save or borrow is, therefore, a decision to rearrange consumption between various time periods.

There exist certain vital decisions in the real world that have strong inter-temporal components. How much schooling to obtain, whom to marry, whether to have children, how much to save for retirement, how to invest, whether to buy a house and if so, which house to buy.

We are using the theory of consumer choice to examine the factors that influence decisions to save or borrow. We are first deriving the budget constraint, then the preference of the consumer and then the optimal choice of the consumer. This will be further followed by changes in the factors that affect the budget constraint.

Inter Temporal Choice: Budget Constraint

Let us start with a situation where a consumer has to decide how much of the good to consume in each of the two time periods. Let us also assume that the amount of consumption in period 1 is c1, and the amount of consumption in period 2 is c2. The amount of money that a consumer has to spend in period 1 is y1, and the money that a consumer has to spend in period 2 is y2.

We assume that there is no inflation in the price level such that the price of consumption in periods 1 and 2 is constant at 1. We are discussing the inter-temporal choice in the case of a composite good.

Suppose the consumer wishes to transfer money from period 1 to period 2 then this is possible only by saving without earning any interest. In period 1, there is an assumption that a consumer cannot borrow money. So the maximum amount that a consumer can spend in period 1 is y1. The budget constraint faced by the consumer is shown in figure1. The consumption in period 1 is shown on X-axis, and consumption in period 2 is shown on Y-axis.

In the figure, the budget line is drawn with a negative slope equal to -1. This budget line is drawn when the interest rate is zero, and consumers are not allowed to borrow any amount. The consumer is facing a tradeoff between consumption in two time periods. The more the consumer consumes in period 1, the less he can consume in period 2 or vice versa. The Endowment point in the figure shows the consumption mix available to the consumer if he is not allowed to borrow.

Here, the consumer has two possible choices to choose from. The first choice is to consume y1 income in period 1 and y2 income in period 2 i.e. consuming the entire income in each period and nothing is left over to use in the next period.

The second choice is to consume less than his income during the first period and consume the saving in the second period along with income. In other words, the consumer is reducing his current consumption and using the saving to increase consumption at a later date.

Consumption with Borrowing:

Let us assume now that consumers can borrow and lend money at some rate of interest equal to r. Let us derive the budget constraint. We have already assumed that there is no inflation in the price level such that the price of consumption in periods 1 and 2 is constant at 1.

Let us start with consumption in the first period. Suppose that the consumer decides to save in the first period, so his consumption in the first period is less than his income, that is, c1<y1. The rate of interest on the saving is r. So, the consumer in the first period is earning interest on the amount saved, i.e. y1-c1.

The consumer is saving in the first period so as to have more in the second period. The consumption in the second period would become-

𝑐₂ = 𝑦₂ + (𝑦₁−𝑐₁) + 𝑟 (𝑦₁−𝑐₁)

𝑐₂ = 𝑦₂ + (1+𝑟) (𝑦₁−𝑐₁)

The amount that the consumer can consume in period 2 is the income in period 2 plus the amount he saved in period 1 plus the interest earned on the amount saved in period 1. The consumer is facing a tradeoff between current consumption and future consumption.

Consumption with Lending:

Let us start with consumption in the first period. Suppose that the consumer decides to borrow in the first period, so his consumption in the first period is more than his income, that is, c1>y1. The rate of interest on the saving is r. So, the consumer has to pay interest in the second period will be 𝑟(𝑦1 − 𝑐1). The consumer is consuming more in the first period, so he would consume less in the second period. The consumption in the second period would become-

𝑐₂ = 𝑦₂ − (𝑦₁−𝑐₁) − 𝑟 (𝑦₁−𝑐₁)

𝑐₂ = 𝑦₂ + (1+𝑟) (𝑦₁−𝑐₁)

The amount that the consumer can consume in period 2 is the income in period 2, less the amount he borrowed in period 1, less the interest paid on the amount borrowed in period 1. The consumer is facing a tradeoff between current consumption and future consumption. Here, if (𝑦1−𝑐1) is positive, then the consumer is earning interest on this saving.

On the other hand, if (𝑦1−𝑐1) is negative, then the consumer pays interest on the amount he borrowed. And if 𝑦1=𝑐1, then the consumer is neither a borrower nor a lender. In this situation, consumption is at the “Polonius point”.

The budget constraint of the consumer can be written in an alternative form as follows:

(i) Budget Constraint in Terms of Future Value

(1+𝑟) 𝑐₁ + 𝑐₂ = (1+𝑟) 𝑦₁ + 𝑦₂ …….(1)

The budget constraint in terms of future value makes the price of future consumption equal to 1. The budget constraint measures the price of period 1 relative to the price of period 2.

(ii) Budget Constraint in Terms of Present Value

𝑐₁ + 𝑐₂/(1+𝑟) = 𝑦₁ + 𝑦₂/(1+𝑟) …… (2)

The budget constraint in terms of present value makes the price of present consumption equal to 1. The budget constraint measures the price of period 2 relative to the price of period 1. Let us explain it graphically.

Here both equations have the form

𝑝₁𝑥₁ + 𝑝₂𝑥₂ = 𝑝₁𝑚₁ + 𝑝₂𝑚₂

In equation (1), 𝑝₁=(1+𝑟) and 𝑝₂=1. And in equation (2), 𝑝₁=1 and 𝑝₂=1/(1+𝑟).

In Figure 2, the consumption in period 1 is shown on X-axis and consumption in period 2 is shown on Y-axis. In the figure, the budget line is drawn with a negative slope equal to –(1+r). The horizontal intercept of the budget constraint measures the maximum amount of the first-period consumption. This amount is the present value of the endowment, which is equal to this 𝑦₁+𝑦₂/(1+𝑟).

The present value of an endowment of money in two periods is the amount of money in period 1 that would generate the same budget set as the endowment. The vertical intercept of the budget constraint measures the maximum amount of the second-period consumption. This amount is the future value of the endowment, which is equal to this (1+𝑟)𝑦1+𝑦2.

The present value form is more useful to express the inter-temporal budget constraint as it measures the future relative to the present. The budget line passes through an affordable consumption bundle, i.e. (𝑦1, 𝑦2). The budget line is negatively sloped, showing the tradeoff between present consumption and future consumption, and the slope is equal to –(1+𝑟).

Budget Constraint in Terms of Present and Future Value

Inter Temporal Choice: Consumer Preferences

The preferences of consumers are shown with the help of the indifference curve. An indifference curve is a curve which shows a different combination of two goods yielding the same level of satisfaction to the consumer.

In other words, it identifies the various combinations of goods among which the consumer is indifferent. It is also known as Iso- Utility curves. It is downward-sloping and convex to the origin.

Properties of Indifference Curve:

The properties of the indifference curve are as follows.

(a) An Indifference Curve Slopes Downward: An indifference curve slopes downward from left to right. This downward slope property is based on the assumption that consumption of both goods gives positive satisfaction to the consumer (i.e. the non-satiety). It is shown in Fig 3.

In Fig 3, IC1 is an indifference curve. If we start from bundle B and move down to the left, then bundle A is worse than bundle B, and if we move from bundle B to anywhere up and to the right, then bundle C is preferred over bundle B.

So the relationship between point E and point D is an indifference curve because while moving from point E to point D, present consumption is increasing and future consumption is decreasing.

(b) Convex to the Origin: An indifference curve is generally convex to the origin. Convexity implies that it bows inward to the origin. The convexity of the indifference curve is due to the diminishing marginal rate of substitution. The marginal rate of substitution is the rate at which one good can be substituted for another.

When we move down along an indifference curve, then marginal rate of substitution diminishes. In Fig 4, in order to have a CD amount of more present consumption, the consumer has to reduce AB amount of future consumption.

(c)Two indifference Curves never intersect each other: Each indifference curve shows a different level of satisfaction, so two indifference curves never intersect each other. In Fig 5, IC1 and IC2 are the two indifference curves showing different levels of consumption preferences.

On the indifference curve, IC1 satisfaction from point A is equal to satisfaction from point C because all the points on an indifference curve show the same level of utility. On the other hand, on the indifference curve, IC2 satisfaction from point A is equal to satisfaction from point B.

It shows that point B and point C have the same level of satisfaction. But this is incorrect because point B shows more future consumption than point C though the amount of present consumption is equal in both combinations.

Two Indifference Curves Never Intersect Each Other

(d) Higher Indifference Curve Higher is the Satisfaction: This is another most important property of an indifference curve. The higher (Lower) the indifference curve higher is (the lower) the level of satisfaction. It is shown in Fig 6

IC1 and IC2 are two indifference curves showing different levels of satisfaction. The indifference curve IC2 is higher than the indifference curve IC1. Point P on indifference curve IC1 represents less units of apples than point Q on indifference curve IC2, though the same unit of mangoes. So point Q on indifference curve IC2 will give more satisfaction than point P on indifference curve IC1.

Optimal Consumer Choice

In the last section, we have seen the consumer’s budget constraint and the preferences in each of the two periods. Let us find out the optimal consumption choice with the help of budget constraints and the indifference curve faced by the consumer.

The left panel shows the case of a borrower, and the right panel shows the case of a lender. If the consumer is a borrower, then consumption in the first period is greater than income, so c1>m1. In panel (a), the consumption in period 1 is shown on X-axis and consumption in period 2 is shown on Y-axis. IC1 is the indifference curve. E is the endowment point showing the various consumption mix available to the individual if no saving and borrowing take place.

The consumer’s optimal choice would occur where the indifference curve is tangent to budget constraint. This point is shown by choice in the left panel. The optimal choice in period 1 is Oc1, and the optimal choice in period 2 is Oc2. The optimal choice in period 1 is more than its current income, so the consumer is a borrower.

On the other hand, in the right panel consumer is a lender.

Consumer Optimal Choice- Borrow & Lender

In panel (b), the consumption in period 1 is shown on X-axis and consumption in period 2 is shown on Y-axis. IC1 is the indifference curve. E is the endowment point showing the various consumption mix available to the individual if no saving and borrowing take place.

The optimal consumer choice would occur where the indifference curve is tangent to budget constraint. This point is shown by choice in the right panel. The optimal choice in period 1 is Oc1, and the optimal choice in period 2 is Oc2. The optimal choice in period 1 is less than its current income, so the consumer is a lender.

The consumer always faces a tradeoff between current consumption and future consumption. If present consumption decreases, then future consumption increases and if present consumption increases, then future consumption decreases.

Effect of Change in Endowment on Inter Temporal Choice

We have seen that the budget line depends upon three factors- current income (income in period 1), future income (income in period 2) and the rate of interest. Any change in one or more of these variables will cause the budget line to shift and change the optimal choice of the consumer.

Let us start with a situation where there is a change in the future income or a change in the income in period 2. How does this change in income in period 2 affect the consumption and saving of an individual?

Suppose that the income of an individual in period 2 falls down to zero. This change in the income in period 2 changes the endowment.

Change in Income and Inter-temporal Choice

The consumption in period 1 is shown on X-axis, and consumption in period 2 is shown on Y-axis. In the figure, the budget line is drawn with a negative slope.MN is the initial budget line, IC1 is the initial difference curve. E is the endowment point. The optimal consumption occurs at point A where the indifference curve IC1 is tangent to budget line MN.

Suppose now the income in period 2 reduces to zero, then the endowment point moves from point E to M1 on the y-axis. If an individual is neither a borrower nor a saver, i.e. consumption is equal to income in each period, then in period 2, consumption would be zero as income falls down to zero. The new budget line would be M1N1 which is parallel to the old budget line MN.

This reduction in income in period 2 would not change the relative cost of present and future consumption. But this reduction in future income will influence consumer behaviour. If consumption in both periods is a normal good, then a shift in the budget line will lead to a reduction in consumption in both periods.

The consumer is just spreading the loss of income over both years. This is done by cutting back on consumption in periods 1 and period 2. The new equilibrium would now occur at point B where the indifference curve IC2 is tangent to the new budget line M1N1. The equilibrium point B shows lesser consumption in both periods as compared to point A.

This reduction in the expected income in period 2 causes saving to increase in period 1. We have seen that income is equal to saving plus consumption. Saving is the difference between income and consumption. As income decreases, so does the savings.

Now suppose an individual is a borrower rather than a saver in period 1. An individual being a borrower wants to consume more in the first period and pay a loan in the second period. Let us use the same figure to explain the concept of a borrower. If an individual is a borrower then his current consumption is more than future consumption. The same can be shown with the budget line MN but the endowment point would be E1 rather than E. The equilibrium consumer optimum choice would occur at point E where the consumer is borrowing in period 1 and then repaying the loan in period 2.

Changes in Interest Rate and Inter Temporal Choice

We have seen that changes in present and future income affect the budget line. Any change in interest rate changes the relative cost of present and future consumption. These changes in present and future consumption can be reflected in the change in the slope of the budget line.

Inter Temporal Choice: Budget Constraint & Consumer Preferences

Introduction