# 6 Government Debt

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## 6.1 Ricardian equivalence theorem

Suppose government plans to spend $1 millions. It is pondering two ways of financing–through tax or debt. Through debt, households immediately lose$1 millions. Through debt, we consider two scenarios: a short-term debt where debt matures by one year, and a long-term debt where debt matures by two years.

For short-term debts,

1. Suppose government pay back its debt through taxation after one year. How much should government tax? What is its present discounted value?

2. Suppose government pay back its debt through another round of debt issuance after one year. How much debt should it issue? To clear that debt in year three by taxation, how much tax should government collect? What is its present discounted value?

For long-term debts,

1. Suppose government pay its interest through tax every year and clear the debt in year three. How much tax should government collect in year two and three? What is the present discounted value of those taxes?

2. If short-term debt bears an interest rate of $$i_s$$ and long-term debt bears an interest rate of $$i_l$$, will your answer regarding the present discounted value in (c) change?

## 6.2 Imperfect credit market

In reality the interest rate ($$i^{g}$$) on government bond is different from the interest rate (i) on private bond.

1. Write down household’s current budget constraint.

2. In reality, $$i$$ and $$i^{g}$$ are different. Which one do you expect to be larger? Why?

Let $$\bar{B}$$ be the total amount of bond holdings. Then when there are $$B^{g}$$ amount of government bond in circulation, there are ($$\bar{B}-B^{g}$$) amount of private bonds in circulation. In other words, the financial portfolio of household consists of B^{g} dollars of government bond and $$\bar{B}-B^{g}$$ dollars of private bond.

1. What is the rate of return of such a portfolio?

Suppose $$i>i^{g}$$.

1. Suppose government increases its purchase by $100m today and increases tax by$100m correspondingly. What is its impact on household’s present value of life-time wealth?

2. Suppose government increases its purchase by $100m today and issues government bonds of$100m correspondingly today. If it is to retire this debt tomorrow, how much tax should it collect tomorrow? What is its impact on today’s household’s present value of life-time wealth? Does Ricardian Equivalence hold in this case?

## 6.3 Uneven tax burdens

When government increases its expenditure through either government purchase or transfer payment, it creates tax burdens on households. A lot of time the burden does not fall on households evenly.

1. If government increases transfer payment to the poor, at the same time increasing tax on the rich, what would happen to the present value of both parties’ life-time wealth? What would happen to aggregate consumption demand?

2. If government increases transfer payment to the elderly, at the same time increasing tax on the youth, what would happen to the present value of both parties’ life-time wealth? What would happen to aggregate consumption demand?