# Consumption smoothing

## What is consumption smoothing?

Individuals and households often carry out **consumption smoothing** by maintaining a relatively stable level of consumption over time. Most people are averse to drastic fluctuations in their consumption and try to ‘smooth’ out changes in their income by strategically borrowing and saving.

For example, a person may choose to save their bonus cheque (instead of spending it all) during good times to prepare for an unexpected change in their economic condition, such as unemployment.

### How do people smooth their consumption?

No one can determine with absolute certainity if and when their socioeconomic situation will change, and for how long. Although no one can predict the future, individuals can make an educated guess. Long-term financial plans allow individuals and households to change their consumption in response to their outlook of the future. For instance, an individual may choose to borrow money during college years and cut down on expenses in this period, with the assumption that they can pay off this debt once they have their college degree. Similarly, many people save for their retirement because they expect their income to decrease during their old age. The level of smoothing an individual chooses depends on many factors, such as risk preferences and credit constraints.

## Understanding income and consumption paths

First, let’s observe how income and consumption interact over time. The graph below presents an individual’s income and consumption over three time periods. Their income fluctates but their consumption is perfectly smooth; they consume an equal amount of their total income each period.

Assume that they earn $3,500 in the first period, $6,000 in the second period and $4,000 in the third period. Observe that the consumption line is **above** the income line in periods one and three. This indicates that their expenses are higher than their income. How is that possible?

Recall what we learned about borrowing and lending behavior with respect to current and future income in the previous tabs. It is possible to consume more than you earn by borrowing at market interest rates. Similarly, by saving money when income is higher than consumption in period two, you meet expenses in period three when income falls again.

Try plugging in different values of income and observe the difference between income and consumption at different periods of time. How would an individual with perfect smoothing behavior react to each scenario? This figure is simplified because it does not incorporate interest rates. Next, we will introduce interest rates into the discussion of consumption smoothing by plotting a budget constraint.

### Income and consumption paths

## Consumption smoothing point

Now, let’s add to a budget constraint to identify the point where consumption is smoothed. In the figure below, you can adjust the value of current income, future income, the borrowing interest rate, and the lending interest rate. We find the consumption smoothing point by equating consumption now (the \(x\)-axis) to consumption later (the \(y\)-axis). This forms the equation of a line (the dashed line), where \(y = x\). The consumption smoothing point is the instersection of this line with the budget constraint.

### Calculating the smoothing point

When you have different levels of income across periods, you must lend or borrow to reach your smoothing point and make your consumption equal in each period.

Lenders and borrowers receive different interest rates. Therefore, the first step in calculating the smoothing point is determining if you should lend or borrow. If you have more income in the future, you should borrow to translate some of your future earnings into money you can spend now. If you have more income now, you should lend your money to equalize consumption between periods. The choice to lend or borrow determines how you calculate the smoothing point.

#### Case One: Borrowing

Suppose you make more money in the future. This means you must borrow an amount \(B\) at the borrowing interest rate (\(r_{B}\)) to smooth your consumption. Your consumption now becomes the sum of your income now (\(y_{1}\)) and the amount you borrow (\(B\)). Your future consumption is your future income (\(y_{2}\)) minus the size of your loan and the interest paid. We can represent this with the following equation:

\[y_{1} + B = y_{2} - B(1 + r_{B})\]Solve for \(B\) to determine how much money you should borrow to smooth consumption.

#### Case Two: Lending

You lend the amount \(L\) if you make more income in the current period than in the future. You can solve for \(L\) with another system of equations that utilizes the lending interest rate (\(r_{L}\)). When you lend money, your consumption becomes the difference between your current income and the amount you lend. Your future consumption becomes the sum of your future income, the amount you lent, and the extra income you earned in interest:

\[y_{1} - L = y_{2} + L(1 + r_{L})\]Solve for \(L\) to calculate how much money you should lend to smooth consumption.

Try shifting the inputs in the figure below to explore how the consumption smoothing point changes as interest rates and income shift.

### Visualizing the consumption smoothing point

### Exercise

Suppose an individual’s current income is $4 and their expected future income is $47. The lending rate of interest is 10% and the borrowing rate is 15%. If they want to have perfectly smooth consumption, how much can they consume in each period?

- Enter the income and interest rate data into the graph above and check the midpoint.
- Enter the income and interest rate data into the graph above and check the midpoint.
- Enter the income and interest rate data into the graph above and check the midpoint.