Chapter 15 – Problem 4 – the intertemporal budget constraint with three periods.

Consider an economy that exists for three periods: period 1, period 2, and period 3. In each period, the government must satisfy the budget constraint

(a) Write this budget constraint for each period.

This problem is asking you to redo the work on pages 390-394, but with a three periods economy instead of two. Below are the three period’s budget constraints:

(b) What must be true about ?

With no period 4 and this type of model, we assume the world has ended in period 4.

Therefore we can assume

Put in an intuitive way; if you knew you were going to be dead in four days, that means whether you spend your savings or give it away, it doesn’t change the fact you’re still dead in four days. Similarly, if everyone knew you were going to die in four periods, who would lend any money to you for that period? Thus savings/deficit at period four must equal zero.

(c) Using the result from part (b), solve the period 3 budget constraint for and substitute this back into the period 2 budget constraint.

(d) Solve the new version of the periods 2 budget constraint for and substitute the result back into the period 1 budget constraint.

(e) At this point, you should have the intertemporal budget constraint for the three-period economy. Interpret this equation.

is initial debt

is the present discounted value of government spending.

is the present discounted value of taxes

Your initial debt plus the PDV of government spending must equal the PDV of tax revenue.

Chapter 15 – Problem 5 – debt-GDP ratio

Solution on page 408. It seems the big picture point of this problem is that for economies in debt, having a growing economy makes taking on & holding debt more sustainable. Ceteris Paribus, a lender would prefer to loan to a growing country than a non-growing one.

Problem 6 – Ricardian equivalence and the government budget constraint – Consider the intertemporal budget constraint in equation 15.5. Assume the interest rate is

(a) Suppose the government cuts taxes today by $100 million. Describe three possible ways for the government can change spending and taxes to satisfy its budget constraint.

Equation 15.5 says:

The equation above says that initial debt plus the PDV of government spending must equal the PDV of taxes.

If today by $100 Million,

Our three options to keep the budget constraint in balance,

and with and

- is initial debt, we have no control over it

(i.e. we assume that we can’t default our away out of it, ala Greece, Argentina, etc. )

Or, of course, some combination of the three.

(b) Suppose consumers obey the permanent-income hypothesis (discussed in Chapter 10). Would their consumption rise, fall, or stay the same for each of the alternatives considered in part a?

The permanent income hypothesis suggests that people will perfectly smooth their lifetime consumption. For example, even though one got the one-time decrease in taxes, if consumers assume Ricardian Equivalence (that the budget constraint must be kept balanced over all periods) then they will know that it must be offset by decreased spending or increased taxes eventually.

Thus making these assumptions, consumers will keep their consumption the same given the tax decrease.

(I suppose for completeness’ sake, we should also assume that tax spending is a ‘transfer’ and not government spending on something that leads to externalities that would otherwise increase or decrease people’s consumption)

(c) What happens to private saving, total saving, and investment in the three scenarios? What? (Assume foreign saving does not change)

Remember that we are assuming the permanent income hypothesis here, that consumers want to perfectly smooth their consumption. Thus if the 100mil period one tax decrease is offset by changes to government spending today or tomorrow, or tomorrow’s taxes, then consumers will need to save or borrow in some way to effectively smooth their consumption.

By how much?:

Remember a few identities (equation under “Deficits & Investments”);

We are assuming that all government spending is a transfer payment, so

1) :

Private savings () stays the same

Government saving stays the same.

Total savings thus stays the same.

2) :

Period 1:

Private savings in period one private savings goes up by 100mil. With the permanent income hypothesis holding, C is unchanged – consumers save their income over to their next period, knowing they will have to pay the tax cut back then.

Government saving in period one goes down by 100mill

Total Savings: net the same

Period 2:

Private Saving in period two goes down by 105mil since.

Government saving in period two goes up by 105mil.

Total savings in is net the same again.

3) :

Period 1:

Private savings in period one goes up by 100mil,

Government saving in period one goes down by 100mil

Period 2:

Private Saving in period two goes down by 105mil.

Consumers have saves their tax cut for period two.

Government saving in period two goes up by 105mil

Total savings in both periods is net the same.

Chapter 15 – Problem 8 – Deficits and investment – Suppose the government decides to reduce taxes today by 1% of GDP, finances by higher borrowing, with the borrowing to be repaid 10 years from now with higher taxes. Discuss the various arguments about what effect this will have on the investment rate today.

This question is very similar to question 6 part c, where we look at the impact on savings with a tax increase today and a tax hike tomorrow (just above). We found that, assuming the Permanent Income Hypothesis and that government spending was just a transfer, that consumers will smooth their consumption and therefore increase savings today in anticipation of the increase in future taxes. This also means that governments will increase borrowing today – increased government debt.

Remember a few identities (equation under “Deficits & Investments”);

This accounting identity tells us that if government decreases taxes and pays for the decrease with taxes several years in the future, then private savings should increase today. But how accurate is the model above?

Will consumers save the tax cut (good for investment) or will they consume part of it?

Evidence suggests (page 398) about 50% of the tax will go to investment, and therefore a increases in taxes like the one in this problem will increase investment.

Further, tax cuts may be structured in a way to encourage increased investment (by, say, a tax cut on returns from investment). It may be argued that government is borrowing to spend on investment projects that individuals are unable to coordinate (however, this is a tax cut question).

But:

Crowding out investment: Income can be spent on consumption or investment – consumption is obviously good, but investment spending can be linked to increased future consumption. An efficient economy should have a good balance between desirable levels of consumption and investment. If the government borrows, it sells government backed bonds that earn a guaranteed interest rate - an alternative to investment.

“The extent to which government deficit crowd out investment is unclear. The Ricardian equivalence argument says that private saving should rise to offset temporary deficits, holding spending constant. This offset seems to be incomplete in the short run, however, as government saving and the investment rate move together”

The crowding out interpretation also does not take into account foreign savers, who may step in to finance government dept.

Issues of Debt: “Very large debts are potentially problematic, leading to dangers of default and higher inflation” but it’s not clear at what level of debt a particular country will face such dangers.

Recall the “Flow version of the government budget constraint;

Although not discussed in the book explicitly, the interest rate i, that government pays debt on is presumably a function of the quantity of debt government holds. As debt increases the interest payment may increases.

Further, should government be unable to raise taxes in future or borrow money (), the government must pay its bills by printing money. Which will lead to inflation.