1. Chapter 9
A Two-Period Model: The Consumption-Savings Decision and Credit Markets
Macroeconomics
6th Edition
Stephen D. Williamson
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2. Learning Objectives, Part I
9.1 Construct a consumer’s lifetime budget constraint and preferences in the two-period model, and solve his or her optimization problem.
9.2 Show how the consumer responds to changes in his or her current income, future income, and the market real interest rate.
9.3 Construct the government’s present-value budget constraint.
This chapter deals with credit market behavior – the consumption/savings behavior of individual consumers and the effects of government deficits that are due to changes in the timing of taxes.
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3. Learning Objectives, Part II
9.4 Show how a competitive equilibrium is constructed in the two-period model.
9.5 Explain the Ricardian Equivalence Theorem.
9.6 Discuss how the Ricardian Equivalence Theorem helps us understand the burden of the government debt.
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4. The consumer’s current-period budget constraint:
Budget Constraints
The consumer’s current-period budget constraint:
This is a model with may consumers, and we start by looking at the behavior of a single consumer, who lives in two periods, the current period and the future period. The slide shows the consumer’s budget constraint in the current period. Disposable income is on the right-hand side. The consumer divides his or her disposable income between current consumption and saving for the future.
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5. Budget Constraints The consumer’s future-period budget constraint:
In the future period, the consumer has his or her future disposable income and the interest earnings on savings. All of this is consumed, as the consumer does not live beyond the future period. Note that the consumer treats the market real interest rate r as given. He or she is a price taker.
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6. Solve the future-period budget constraint for s:
Simplify
Solve the future-period budget constraint for s:
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7. Next,
Substitute in the current-period budget constraint obtaining lifetime budget constraint:
We can collapse the two budget constraints into one, by substituting for s.
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8. Consumer’s Lifetime Budget Constraint
Substitute in the current-period budget constraint obtaining lifetime budget constraint:
This gives the consumer’s lifetime budget constraint. This states that the present value of consumption over the consumer’s lifetime, on the left hand side, must equal the present value of disposable income, on the right-hand side.
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9. Consumer’s Lifetime Wealth
We can define the consumer’s lifetime wealth, we, to be the present value of disposable income over the consumer’s lifetime.
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10. Simplified Lifetime Budget Constraint
Substituting for lifetime wealth, we obtain the simplified constraint.
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11. Simplified Lifetime Budget Constraint: Slope-Intercept
Re-arranging the simplified lifetime budget constraint, we obtain the equation on the slide, in slope-intercept form.
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12. Figure 9.1 Consumer’s Lifetime Budget Constraint
This figure diagrams the equation on the previous slide. The endowment point, if consumed, implies that the consumer would save zero in the first period, and consume his or her disposable income in each period. To the northwest of the endowment point the consumer is a lender, and to the southeast, he or she is a borrower.
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13. Figure 9.2 A Consumer’s Indifference Curves
The consumer’s indifference curves take the usual shapes, but now the two goods are current and future consumption.
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14. Marginal condition that holds when the consumer is optimizing:
Optimization
Marginal condition that holds when the consumer is optimizing:
When the consumer optimizes, a standard marginal condition holds, but here the relative price of current consumption in terms of future consumption is 1+r.
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15. Figure 9.3 A Consumer Who Is a Lender
The figure shows a consumer who chooses to be a lender, with positive savings. Current consumption is less than current disposable income.
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16. Figure 9.4 A Consumer Who Is a Borrower
For a borrower, savings is negative. Current consumption exceeds current disposable income.
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17. An Increase in Current Income for the Consumer
Current and future consumption increase.
Saving increases.
The consumer acts to smooth consumption over time.
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18. Figure 9.5 The Effects of an Increase in Current Income for a Lender
If current income increases, holding constant the real interest rate, this shifts out the budget constraint. The consumer wishes to smooth consumption over time relative to income, so current and future consumption increase, and savings increases. So, current consumption increases less than one-for-one with the increase in income.
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19. Observed Consumption-Smoothing Behavior
Aggregate consumption of non-durables and services is smooth relative to aggregate income, but the consumption of durables is more volatile than income.
This is because durables consumption is economically more like investment than consumption.
The model predicts that consumption should be smooth relative to income. But observed consumption is not as smooth as the theory predicts. In part this is because consumption expenditures are entirely what we would economically think of as consumption. Consumer durables expenditures (expenditures on appliances, cars, boats, etc.) is more like investment than consumption.
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20. Figure 9.6 Percentage Deviations from Trend in Consumption of Durables and Real GDP
The figure shows how durables consumption is actually more volatile than real GDP.
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21. Figure 9.7 Percentage Deviations from Trend in Consumption of Nondurables and Services and Real GDP
This figure shows how nondurables and services consumption is less variable than real GDP, exhibiting the effects of consumption smoothing.
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22. An Increase in Future Income for the Consumer
Aggregate consumption of non-durables and services is smooth relative to aggregate income, but the consumption of durables is more volatile than income.
This is because durables consumption is economically more like investment than consumption.
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23. Figure 9.8 An Increase in Future Income
If future income increases, then the consumer wishes to smooth consumption, so current and future consumption increases. For this to happen, current savings falls, i.e. the consumer effectively borrows against the increase in future income.
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24. Temporary and Permanent Increases in Income
As a permanent increase in income will have a larger effect on lifetime wealth than a temporary increase, there will be a larger effect on current consumption.
A consumer will tend to save most of a purely temporary income increase.
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25. Figure 9.9 Temporary Versus Permanent Increases in Income
This figure illustrates permanent vs. temporary increases in income for the consumer. Permanent increases in income imply larger increases in current consumption than do temporary increases in income. This matters for tax policy, for example. A permanent tax cut will in principle imply a larger effect on spending than a temporary tax cut.
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26. Figure 9.12 An Increase in the Real Interest Rate
The next step is to see how the consumer responds to a price change, which is somewhat nonstandard here. An increase in the real interest rate, holding income and taxes constant, causes the budget constraint to pivot around the endowment point. The budget constraint shifts in for borrowers, and out for lenders.
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27. An Increase in the Market Real Interest Rate
An increase in the market real interest rate decreases the relative price of future consumption goods in terms of current consumption goods – this has income and substitution effects for the consumer.
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28. Figure 9.13 An Increase in the Real Interest Rate for a Lender
The effects of the increase in r are different for lenders and borrowers. For lenders, the income effect is positive. The substitution effect increases future consumption and decreases current consumption. So, current consumption can increase or decrease, savings could increase or decrease, and future consumption must increase.
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29. Figure 9.14 An Increase in the Real Interest Rate for a Borrower
For a borrower, the income effect of the increase in r is negative. So, current consumption must fall, savings must fall, and future consumption could rise or fall.
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30. Effects of an Increase in the Real Interest Rate for a Lender
The next two tables summarize the results on the two previous slides, respectively.
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31. Effects of an Increase in the Real Interest Rate for a Borrower
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32. Perfect Complements Example
With perfect complements, the ratio of future consumption to current consumption is constant.
The consumer’s budget constraint must hold.
Perfect complements is convenient, as we can solve the consumer’s problem explicitly. There are some problems with perfect complements at the end of the chapter. With perfect complements, there are no substitution effects, and the consumer always consumes in fixed proportions. So the two equations on the slide determine current and future consumption.
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33. Perfect Complements Example
With perfect complements we can solve explicitly for current and future consumption:
Using substitution, we can solve for current and future consumption. Consumption in each period is proportional to lifetime wealth.
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34. Perfect Complements Example
Substituting for lifetime wealth gives:
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35. Figure 9.15 Example with Perfect Complements Preferences
This figure depicts the solution for consumption in the perfect complements example.
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36. Government Budget Constraints
The government’s current-period budget constraint:
The last step in the model is to add the government, which spends and taxes in both the current and future periods. In the current period, the government finances G by taxing and borrowing.
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37. Government Budget Constraints
The government’s future-period budget constraint:
In the future period, the government spends and pays off its debt by taxing. There is no debt issued, as this is the last period.
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38. Government Budget Constraints
The government’s present-value budget constraint:
Just as for the consumer, we can determine a present-value budget constraint by substituting out for B using the two government budget constraints. The present value of government spending must equal the present value of taxes, assuming the government never defaults on its debt.
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39. Credit Market Equilibrium Condition
Total private savings is equal to the quantity of government bonds issued in the current period.
In equilibrium total net saving by the private sector consumers must equal the quantity of government debt issued, since there is no investment in this model.
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40. Income-Expenditure Identity
Credit market equilibrium implies that the income-expenditure identity holds.
As usual, the income-expenditure identity must hold in equilibrium.
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41. Ricardian Equivalence
The Ricardian Equivalence Theorem is illustrated algebraically, numerically, and in two graphs.
We will now work through the Ricardian equivalence theorem. This theorem helps us understand why government deficits matter – though what it shows is that, under ideal conditions, deficits are irrelevant, i.e. the timing of taxes is irrelevant.
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42. Ricardian Equivalence
Key equation: The consumer’s lifetime tax burden is equal to the consumer’s share of the present value of government spending – the timing of taxation does not matter for the consumer.
There are N consumers and, assuming each consumer pays the same taxes in the present and the future, in equilibrium the consumer’s present value share of taxes is just his or her share of the present value of government spending. This tells us that it does not matter how the taxes are timed – present vs. future. All that matters for the consumer is the implications of the government’s budget for his or her lifetime wealth, which is unaffected by whether current taxes are large or small.
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43. Ricardian Equivalence
Then, substitute in the consumer’s budget constraint – taxes do not matter in equilibrium for the consumer’s lifetime wealth, just the present value of government spending.
For a given consumer, in equilibrium the present value of consumption depends only on the present value of government spending – not on taxes.
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44. Figure 9.16 Ricardian Equivalence with a Cut in Current Taxes for a Borrower
This illustrates what is going on. A current tax cut moves the endowment point, but does not change the budget constraint for the consumer. The consumer consumes the same amounts. But, he or she saves the tax cut in order to pay the higher taxes that are coming in the future. These higher future taxes are required to pay off the higher government debt.
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45. Figure 9.17 Ricardian Equivalence and Credit Market Equilibrium
In the aggregate, the real interest rate is determined by the supply and demand for government debt. If there is a current tax cut, this shifts the supply of government debt to the right, but the demand for government debt shifts to the right by the same amount, so the real interest rate is unchanged.
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46. Why Might Ricardian Equivalence Fail in Practice?
Redistributional effects of taxes: tax changes affect the wealth of different consumers differently.
Intergenerational redistribution: debt issued by the government today is paid off by future generations.
Taxes are not lump sum; they cause distortions.
Credit market frictions: Chapter 10.
Ricardian equivalence gives us a starting point for understanding why government deficits matter. In practice, there are four key reasons why Ricardian equivalence can fail. We are particularly interested in credit market frictions, which will be studied in detail in Chapter 10.
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