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Albert and Franco both follow the life-cycle hypothesis: they smooth consumption as much as possible
Albert and Franco both follow the life-cycle hypothesis: they smooth consumption as much as possible. They each live for five periods, the last two of which are retirement. Here are their incomes earned during each period:
| Period | Albert | Franco |
| 1 | $100,000 | $40,000 |
| 2 | 100,000 | 100,000 |
| 3 | 100,000 | 160,000 |
| 4 | 0 | 0 |
| 5 | 0 | 0 |
To keep things simple, assume that the interest rate is zero for both saving and borrowing and that the life span is perfectly predictable.
a. For each individual, compute consumption and saving in each period of life.
b. Compute each individual’s wealth (i.e., their accumulated saving) at the beginning of each period, including period six.
c. Graph consumption, income, and wealth for each of them, with the period on the horizontal axis.
d. Now suppose consumers cannot borrow, so wealth cannot be negative. How does that change your answers? Draw a new graph for part (c) if necessary.
Expert Solution
a) According to the life cycle hypothesis, individuals will try to smooth out there consumption over their lifetime. This would involve saving when the income is high and dissaving when the income is low.
| Period | Albert's income | Franco's income |
| 1 | $100,000 | $40,000 |
| 2 | 100,000 | 100,000 |
| 3 | 100,000 | 160,000 |
| 4 | 0 | 0 |
| 5 | 0 | 0 |
| Total | $300,000 | $300,000 |
Smoothening out of consumption would involve consuming the same amount of money during each period. This amount of consumption if given by the average income of all periods ie it is given by divinng the total income in all periods by the number of periods.
For Albert, consumption in each period = Total income in all periods/ number of periods.
= $300,000/5
= $60,000
For Franco, consumption in each period = Total income in all periods/ number of periods.
= $300,000/5
= $60,000
Savings in a particular period are given by subtracting the consumpton in that period from the income earned in that period. The following table shows Albert's and Franco's savings:
| Period | Albert's income | Albert's Consumption | Albert's Savings | Franco's Income | Franco's Consumption | Franco's Savings |
| 1 | 100,000 | 60,000 | 40,000 | 40,000 | 60,000 | -20,000 |
| 2 | 100,000 | 60,000 | 40,000 | 100,000 | 60,000 | 40,000 |
| 3 | 100,000 | 60,000 | 40,000 | 160,000 | 60,000 | 100,000 |
| 4 | 0 | 60,000 | -60,000 | 0 | 60,000 | -60,000 |
| 5 | 0 | 60,000 | -60,000 | 0 | 60,000 | -60,000 |
Negative savings represent dissavings.
b) Wealth is defined as accumulated savings. It is given by the summation of savings in each period preceeding the period in question.
Albert's and Franco's wealth is shown in the table below.
| Albert | Franco | |||||||
| Period | Income | Consumption | Savings | Wealth at the beginning of period | Income | Consumption | Savings | Wealth at the beginning of period |
| 1 | 100,000 | 60,000 | 40,000 | 0 | 40,000 | 60,000 | -20,000 | 0 |
| 2 | 100,000 | 60,000 | 40,000 | 40,000 | 100,000 | 60,000 | 40,000 | -20,000 |
| 3 | 100,000 | 60,000 | 40,000 | 80,000 | 160,000 | 60,000 | 100,000 | 20,000 |
| 4 | 0 | 60,000 | -60,000 | 120,000 | 0 | 60,000 | -60,000 | 120,000 |
| 5 | 0 | 60,000 | -60,000 | 60,000 | 0 | 60,000 | -60,000 | 60,000 |
| 6 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
c) The requisite graphs are shown below:
d) Wealth is negative in a period t if in the preceeding period, t-1, the person consumes more than they have (income in the period t-1 + wealth in the period t-1).
Since Albert's wealth is non-negative in every period, his graph remains the same.
Franco's wealth in period 2 is negative because in period 1, he consumes more than he earns. If negative wealth is not allowed then Franco will spend only as much as his income in period 1. He will no consume an amount less than that because he would try to smooth out his consumption over all periods as much as possible. Consuming a quantity less than his income in period 1 will lead to some scope for further smoothing out the consumption. Franco will not spend more than his income in period 1 because that would lead to borrowing (his wealth at the beginning of period 1 is zero) - which is not allowed.
Franco will then try to smooth out his future income over the remaining periods by consuming an average of his total income from future periods, in the remaining periods.
That is, Franco's consumption in periods 2-5 is given by
= (100,000+160,000)/4
= $ 65,000
Thus, his Franco's consumption, wealth and savings change as follows:
| Franco | |||
| Income | Consumption | Savings | Wealth at the beginning of period |
| 40,000 | 40,000 | 0 | 0 |
| 100,000 | 65,000 | 35,000 | 0 |
| 160,000 | 65,000 | 95,000 | 35,000 |
| 0 | 65,000 | -65,000 | 130,000 |
| 0 | 65,000 | -65,000 | 65,000 |
| 0 | 0 | 0 | 0 |
please see the attached file for the complete solution.
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