# Consumption: Part III

## 2.12 Intertemporal budget constraint

Assume price level P=1 for every period. Consider a household with an initial asset balance A. Suppose the income source for the houshold comes from: non-asset income I, and asset income iA where i is the rate of return on the asset. Each period the household can use its income only for consumption C and saving \(\Delta A\) (which is the change of asset balance).

- Write down one-period budget constraint.

For period t, the initial asset balance is \(A_{t−1}\) (that is the asset balance inherited from period t-1), non-asset income is \(I_t\) , consumption is \(C_t\) , and saving is \(\Delta A_t\) which represents \(A_t-A_{t-1}\). If interest rate i is constant and the price level P=1,

- Write down the intertemporal budget constraint for periods 1 and 2.

## 2.13 Wealth/Income Effect

Consider a two-period intertemporal budget constraint: \[(1+i)A_{0}+I_{1}+\frac{I_{2}}{1+i}=C_{1}+\frac{C_{2}}{1+i}+\frac{A_{2}}{1+i}.\] Here we assume P=1.

- Suppose only \(C_{1}\) and \(C_{2}\) can be decided. Other variables are fixed. In a figure where x-axis is \(C_{1}\) unit and y-axis is \(C_{2}\) unit, draw the budget constraint line. Please mark the slope.

Assume \(A_{0}=0,A_{2}=0\).

If i increases, under what circumstance will there be a positive income effect?

If \(I_{2}=0\), argue that when i increases there must be a positive income effect.

## 2.14 Permanent income

We define permanent income level as the level of consumption under perfect consumption smoothing. Suppose a consumer only lives for three periods. He has no initial wealth, and the incomes are \(I_{1}=100,\ I_{2}=200,\ \) and \(I_{3}=50\) for three periods.

Suppose the interest rate \(i=5\%\). What is his permanent income level?

Argue that “So long as the present discounted value of total life income is the same, income variation across periods does not affect consumption level under perfect consumption smoothing.”

Suppose consumers like perfect consumption smoothing under the current interest rate.

If \(I_{1}\) increases to 120, how much is MPC in period 1 (\(MPC_{1}\))?

If all incomes increas by 20, how much is \(MPC_{1}\)?

Continue from a. If in period 1 the consumer expects \(I_{2}\) to increase by 20, how much is \(\Delta C_{1}\)?

Continue from e. However, in period 2 he finds out that such an expectation is not true; \(I_{2}\) does not change at all. How much is \(\Delta C_{2}\) in period 2?

## 2.15 Borrowing and income effect

Suppose consumers adopt consumption smoothing under current interest rate. Now government decides to give each consumer \(\$3000\) temporarily.

If the source of such a fiscal gift comes from the donation from other countries (so no need to pay it back later), how much will consumption today increase?

Suppose the source of such a fiscal gift comes from the borrowing from consumers and pay them back by \(\$3000(1+i)\) tomorrow. This is equivalent to moving your tomorrow’s \(\$3000(1+i)\) to today and it is worth only \(\$3000\) today (due to present value discounting), how much will consumption today increase?

## 2.16 the Economist

“Your money and your life Financing longevity”, Jul 8th 2017, the Economist.

Life-cycle theory of consumption pictures people with three stages in their lives: young, middle-aged, and old. At young stage, they have almost no income but still have to consume. At middle-aged stage, they earn most of their life incomes; however, they consume less than their incomes so as to either payback their debt from youth or to accumulate asset to spend in their old time. At old age, they have less income than their consumption.

In the article, it mentions that

[Pension plan] has to update the rigid three-stage life-cycle model on which most of its products are based.

- What does it mean?

Under the current pension model,

it needs to resolve two opposite but equally troubling problems: undersaving during working life and oversaving during retirement.

- What are the reasons that create those problems?

a more creative approach is needed to the range of assets that pensioners can draw on, including their homes, which have so far played little part in provision for old age.

- What can homes do to mitigate the problems?

In a multi-stage life, the idea of hitting a cliff-edge retirement at 65 and then living off an annuity is outdated

- What does that mean?

“Often people just need the confidence that we’ve run the numbers and that they really can afford to make that donation to a charity, or spend a little more on themselves,”

- What flaw of the current pension plan does it highlight?