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## 13.2 A Model of Consumption

### Learning Objectives

After you have read this section, you should be able to answer the following questions:

1. What is the life-cycle model of consumption?
2. What are the effects of changes in a pay-as-you-go system?

Chapter 13 "Social Security", Section 13.1 "Individual and Government Perspectives on Social Security" examined an explicit example of what Social Security implies for households and for the government. We can take away the following insights from this example:

• Households decide on consumption and saving taking into account their lifetime income.
• Lifetime income includes both taxes paid during working years and benefits received during retirement.
• From the government’s view, taxes received and benefits paid need to balance, at least over long periods of time.
• In the example, the Social Security program was irrelevant: individuals had the same lifetime income and thus consumption opportunities regardless of the Social Security taxes paid and benefits received.

We now go beyond our numerical example and give a more general analysis of how an individual’s lifetime consumption choices are influenced by Social Security.

## Household Budget Constraints

We first consider the budget constraints faced by an individual or household (remember that we are using the two terms interchangeably). There are two household budget constraints. The first applies in any given period: ultimately, you must either spend the income you receive or save it; there are no other choices. For example,

disposable income = consumption + household savings.

Households also face a lifetime budget constraint. They can save in some periods of their life and borrow/dissave in other periods, but over the course of any household’s lifetime, income and spending must balance. The simplest case is when real interest rates equal zero, which means that it is legitimate simply to add together income and consumption in different years. In this case the lifetime budget constraint says that

If real interest rates are not zero, then the budget constraint must be expressed in terms of discounted present values. The household’s lifetime budget constraint is then

discounted present value of lifetime consumption = discounted present value of lifetime income.

If the household begins its life with some assets (say a bequest), we count this as part of income. If the household leaves a bequest, we count this as part of consumption. As in our earlier numerical example, we can think about the lifetime budget constraint in terms of the household’s assets. Over the course of a lifetime, the household can save and build up its assets or dissave and run down its assets. It can even have negative assets because of borrowing. But the lifetime budget constraint says that the household’s consumption and saving must result in the household having zero assets at the end of its life.

You can review both the household’s intertemporal budget constraint and the concept of discounted present value in the toolkit.

To see how this budget constraint works, consider an individual who knows with certainty the exact number of years for which she will work (her working years) and the exact number of years for which she will be retired (her retirement years). While working, she receives her annual disposable income—the same amount each year. During retirement, she receives a Social Security payment that also does not change from year to year. As before, suppose that the real interest rate is zero.

Her budget constraint over her lifetime states that

total lifetime consumption = total lifetime income = working years × disposable income+ retirement years × Social Security payment.

Our numerical example earlier was a special case of this model, in which

disposable income = $34,000, working years = 45, retirement years = 15, and Social Security payment =$18,000.

Plugging these values into the equation, we reproduce our earlier calculation of lifetime income (and hence also lifetime consumption) as ($45 ×$34,000) + (15 × $18,000) =$1,800,000.

## The Life-Cycle Model of Consumption

Economists often use a consumption functionA relationship between current disposable income and current consumption. to describe an individual’s consumption/saving decision:

consumption = autonomous consumption + marginal propensity to consume × disposable income.

The marginal propensity to consumeThe amount consumption increases when disposable income increases by a dollar. measures the effect of current income on current consumption, while autonomous consumption captures everything else, including past or future income.

The life-cycle modelA model studying how an individual chooses a lifetime pattern of saving and consumption given a lifetime budget constraint. explains how households make consumption and saving choices over their lifetime. The model has two key ingredients: (1) the household budget constraint, which equates the discounted present value of lifetime consumption to the discounted present value of lifetime income, and (2) the desire of a household to smooth consumption over its lifetime.

You can review the consumption function, consumption smoothing, and the life-cycle model in the toolkit.

Let us see how this model works. According to the life-cycle model of consumption, the individual first calculates her lifetime resources as

working years × disposable income + retirement years × Social Security payment.

(We continue to suppose that the real interest rate is zero, so it is legitimate simply to add her income in different years of her life.) She then decides how much she wants to consume in every period. Consumption smoothing starts from the observation that people do not wish their consumption to vary a lot from month to month or from year to year. Instead, households use saving and borrowing to smooth out fluctuations in their income. They save when their income is high and dissave when their income is low.

Perfect consumption smoothing means that the household consumes exactly the same amount in each period of time (month or year). Going back to the consumption function, perfect consumption smoothing means that the marginal propensity to consume is (approximately) zero.With perfect consumption smoothing, changes in current income will lead to changes in consumption only if those changes in income lead the household to revise its estimate of its lifetime resources. If a household wants to have perfectly smooth consumption, we can easily determine this level of consumption by dividing lifetime resources by the number of years of life. Returning to our equations, this means that

This is the equation we used earlier to find Carlo’s consumption level. We took his lifetime income of $1,800,000, noted that lifetime income equals lifetime consumption, and divided by Carlo’s 60 remaining years of life, so that consumption each year was$30,000. That is really all there is to the life-cycle model of consumption. Provided that income during working years is larger than income in retirement years, individuals save during working years and dissave during retirement.

This is a stylized version of the life-cycle model, but the underlying idea is much more general. For example, we could extend this story and make it more realistic in the following ways:

• Households might have different income in different years. Most people’s incomes are not constant, as in our story, but increase over their lifetimes.
• Households might not want to keep their consumption exactly smooth. For example, if the household expects to have children, then it would probably anticipate higher consumption—paying for their food, clothing, and education—and it would expect to have lower consumption after the children have left home.
• The household might start with some assets and might also plan to leave a bequest.
• The real interest rate might not be zero.
• The household might contain more than one wage earner.

Working through the mathematics of these cases is more complicated—sometimes a lot more complicated—than the calculations we just did, and so is a topic for advanced courses in macroeconomics. In the end, though, the same key conclusions continue to hold even in the more sophisticated version of the life-cycle model:

• A household will examine its entire expected lifetime income when deciding how much to consume and save.
• Changes in expected future income will affect current consumption and saving.

## The Government Budget Constraint

The household’s budget constraints for different years are linked by the household’s choices about saving and borrowing. Over the household’s entire lifetime, these individual budget constraints can be combined to give us the household’s lifetime budget constraint. Similar accounting identities apply to the federal government (and for that matter, to state governments and local governments as well).

In any given year, money flows into the government sector, primarily from the taxes that it imposes on individuals and corporations. We call these government revenues. The government also spends money. Some of this spending goes to the purchase of goods and services, such as the building of roads and schools or payments to teachers and soldiers. Whenever the government actually buys something with the money it spends, we call these government purchases (or government expenditures). Some of the money that the government pays out is not used to buy things, however. It takes the form of transfers, such as welfare payments and Social Security payments. Transfers mean that dollars go from the hands of the government to the hands of an individual. They are like negative taxes. Social Security payments are perhaps the most important example of a government transfer.

Any difference between government revenues and government expenditures and transfers represents saving by the government. Government saving is usually referred to as a government surplus:

government surplus = government revenues − government transfers − government expenditures.

If, as is often the case, the government is borrowing rather than saving, then we instead talk about the government deficit, which is the negative of the government surplus:

government deficit = −government surplus = government transfers + government expenditures − government revenues.

You can review the government budget constraint in the toolkit.

## Applying the Tools to Social Security

The life-cycle model and government budget constraint can be directly applied to our analysis of Social Security. Let us go back to Carlo again. Carlo obtains pretax income and must pay Social Security taxes to the government. Carlo’s disposable income in any given year is given by the equation

disposable income = income − Social Security tax.

Imagine that he receives no retirement income other than Social Security. Carlo’s lifetime resources are given by the following equation:

lifetime resources = working years × income − working years × Social Security tax+ retirement years × Social Security income.

Now let us examine Social Security from the perspective of the government. To keep things simple, we suppose the only role of the government in this economy is to levy Social Security taxes and make Social Security payments. In other words, the government budget constraint is simply the Social Security budget constraint. The government collects the tax from each worker and pays out to each retiree. For the system to be in balance, the government surplus must be zero. In other words, government revenues must equal government transfers:

number of workers × Social Security tax = number of retirees × Social Security payment.

Now, here is the critical step. We suppose, as before, that all workers in the economy are like Carlo, and one worker is born every year. It follows that

number of workers = working years

and

number of retirees = retirement years.

But from the government budget constraint, this means that

working years × Social Security tax = retirement years × Social Security payment,

so the second and third terms cancel in the expression for Carlo’s lifetime resources. Carlo’s lifetime resources are just equal to the amount of income he earns over his lifetime before the deduction of Social Security taxes:

lifetime resources = income from working.

No matter what level of Social Security payment the government chooses to give Carlo, it ends up taking an equivalent amount away from Carlo when he is working. In this pay-as-you-go system, the government gives with one hand but takes away with the other, and the net effect is a complete wash. We came to this conclusion simply by examining Carlo’s lifetime budget constraint and the condition for Social Security balance. We did not even have to determine Carlo’s consumption and saving during each year. And—to reiterate—the assumption that there is just one person of each age makes no difference. If there were 4 million people of each age, then we would multiply both sides of the government budget constraint by 4 million. We would then cancel the 4 million on each side and get exactly the same result.

We have gained a remarkable insight into the Social Security system. The lifetime income of the individual is independent of the Social Security system. Whatever the government does to tax rates and benefit levels, provided that it balances its budget, there will be no effect on Carlo’s lifetime income. Since consumption decisions are made on the basis of lifetime income, it also follows that the level of consumption is independent of variations in the Social Security system. Any changes in the Social Security system result in changes in the level of saving by working households but nothing else. As we saw in our original numerical example, individuals adjust their saving in a manner that cancels out the effects of the changes in the Social Security system.

The model of consumption and saving we have specified leads to a very precise conclusion: the household neither gains nor loses from the existence of the Social Security system. The argument is direct. If the well-being of the household depends on the consumption level over its entire lifetime, then Social Security is irrelevant since lifetime income (and thus consumption) is independent of the Social Security system.

### Key Takeaways

1. The life-cycle model of consumption states that the household chooses its consumption during each period of life subject to a budget constraint that the discounted present value of lifetime income must equal the discounted present value of lifetime consumption.
2. If the household chooses to perfectly smooth consumption, then consumption during each period of life is equal to the discounted present value of income divided by the number of years in a lifetime.
3. In general, a household’s lifetime income and consumption are independent of the taxes and benefits of a pay-as-you-go Social Security system. Changes to the system lead to adjustments in saving rather than consumption.