For each question, you need to explain in detail.

Incentive to save

Suppose households increase their incentive to save. They decide to increase their saving and cut down their consumption. In other words, they value their current consumption more now.

What would happen to aggregate demand $Y^d$?

Consumption will decrease which will shift $Y^d$ to the left.

What would happen to labor market?

Higher incentive to save could imply a higher incentive to work since working can generate a higher saving; labor supply will shift to the right.
On the other hand, the decrease of aggregate demand (i.e. the left-shift of $Y^d$) implies a higher national saving that lead to an interest rate drop, which triggers an intertemporal substitution effect (ISE) on labor supply (i.e. lower interest rate makes working to save less enticing); labor supply decrease.
The ultimate result depends on which effect is stronger. In reality, labor supply is not very sensitive to interest rate change. Therefore, an increase in labor supply is more likely to happen, which leads to a lower real wage rate and a higher employment level.

What would happend to interest rate? What about capital market today and tomorrow?

In (a) we know that aggregate demand will increase. In (b) higher labor supply increases aggregate supply. Both will work to set an excess national saving that leads interest rates to fall. When it comes to capital market, today and tomorrow behave differently. Today’s capital market will see an increase in capital demand since higher labor employment enhances capital’s productivity. As a result, today’s capital real rental price will increases. But for tomorrow, it is today’s interest rate that governs its behavior. As interest rate is the cost of investment, with today’s falling interest rates, investment increases, which builds up a higher capital supply tomorrow. With abundant capital supply tomorrow, the real rental price of capital will decrease.

Government Budget Constraint

Government spending can be divided into two parts: government purchase $PG$ and transfer payment $PV$. In order to finance these spendings, government relies on two income sources: tax revenue $PT$ and issuance of new debt $\Delta B^g$. The existence of government debt means that there is interest payment $iB^g$ that government has to pay in each period.

Now consider multiple periods of government budgeting with time-varying $G_t$, $V_t$, $T_t$, $\Delta B_t^g$ and $B_{t-1}^g$. The last one represents government’s initial debt burdent at the beginning of period t.

An insightful household understand that give government’s fiscal plan regarding its spending (i.e. its plan on $G$ and $V$), he has to live within the constraints of both his own household budget constraint and the government budget constraint that he is required to help the government follow. What does this mean to an insightful household’s budget constraints?

It means that the household’s intertemporal budget constraint, take two periods as an example, will look like \[ WL_1+\frac{WL_2}{1+i}+(1+i)(PK_0+B_0)-PG_1-\frac{PG_2}{1+i}=PC_1+\frac{PC_2}{1+i}+\frac{PK_2+B_2}{1+i}\] None of V,T and $B^g$ matters for the household’s wealth. It is government purchase G that really affects household’s life-time wealth directly.

Analyze the effect of a permanent government purchase increase on today’s economy regarding consumption, investment, final output, interest rate, employment, real wage rate and real rental price of capital.

A permanent increase of government purchase will create a strong negative wealth effect on households, which leads to a one-to-one reponse on consumption; consumption drops as much as the change of government purchase. Therefore, if there is nothing change in input markets, national saving (Y-C-G) does not change, which leaves interest rate intact–investment remains the same. However, it might be the case that such a strong wealth effect pushes households to work more. Labor supply increases, leading to a higher employment and a lower real wage rate. As labor is a productivity complement to capital, higher labor employment pushes the demand for capital higher, generating a higher real rental price of capital. Final output will increase given labor employment increases.

Analyze the effect of a temporary government purchase increase on today’s economy regarding consumption, investment, final output, interest rate, employment, real wage rate and real rental price of capital.

A temporary increase of government purchase will create a small and negligible negative wealth effect on households, which leaves consumption almost intact. Therefore, if there is nothing change in input markets, national saving (Y-C-G) drops, which pushes up interest rate–the cost of investment becomes higher; investment drops. In labor market, it is the increase of interest rate that motivates workers to work harder as saving gives a better return (i.e. Intertemporal Substitution Effect, ISE)–labor supply increases. It leads to a higher employment and a lower real wage rate. As labor is a productivity complement to capital, higher labor employment pushes the demand for capital higher, generating a higher real rental price of capital. Final output will increase given labor employment increases.

Permanent tax rate changes

Analyze the effect of a permanent increase of labor income tax rate on today’s consumption, investment, output, employment, real wage rate and real rental price of capital.

Increase of labor income tax rate means that working generates less consumption. This triggers a substituion effect (SE) that will decreases labor supply and consumption. The real wage rate in labor market will increase and the equilibrium employment will drop. Today’s capital productivity deccreases; $K^d$ decreases and $R/P$ decreases. Final output supply ($Y^s$) will decrase in corresponding to lower labor employment. In the meanwhile, consumption decrease pushes down final output demand ($Y^d$). The result will for sure push down equilibrium output level. Since the change is permanent, it is very likely that consumption will decrease further to accommodate this equilibrium output loss–due to Permanent Income Hypothesis. It is the falling of $Y^d$ that dominates. Interest rates will drop, which enhances investment.

Asset income tax

If we consider household’s investment as money allocated among bond and capital assets, for the following three types of asset income tax, how would they affect household’s investment amount and its portfolio weight on each asset, as well as its effect on capital accumulation.

A tax rate $\tau$ on interest income, but not on capital rental income.

At the prevailing equilibrium, with such tax after-tax rate of bond return $(1-\tau) i < R/P-\delta$ the rate of capitall return. The weight on bond of household’s portfolio will reduce, and at the same time the weight on capital will increase. The later will increase investment. However, it is the rate of portfolio return that affects household’s saving decision. The rate of portfolio return will drop, which will increase today’s consumption–hence, less investment. As a result, the overall change of investment is uncertain. The impact on capital accumulation is uncertain.

A tax rate $\tau$ on capital rental income, but not on interest income.

At the prevailing equilibrium, with such tax the rate of bond return $ i > (1-)(R/P-)$ the after-tax rate of capitall return. The weight on bond of household’s portfolio will increase, and at the same time the weight on capital will decrease. The later will decrease investment. Furthermore, it is the rate of portfolio return that affects household’s saving decision. The rate of portfolio return will drop, which will increase today’s consumption–hence, less investment. As a result, the overall change of investment will drop, which harms capital accumulation.

A tax rate $\tau$ on both capital rental income and interest income.

At the prevailing equilibrium, with such tax the after-tax rate of bond return $ (1-tau) i = (1-)(R/P-)$ the after-tax rate of capitall return. There is no need to portfolio weights reshuffling. However, the rate of portfolio return will drop, which will increase today’s consumption–hence, less investment. As a result, there will be less capital accumulated.

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,

Suppose government pays 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?

After one year, it needs to issue $\$(1+i) \times 1$m debt. In year three, it needs to collect tax total of $ $(1+i)^21 $m. In this question, discount rate is i. Therefore, the present value $\$(1+i)^2 \times 1$m needs to be divided by $(1+i)^2$, which is $1m.

For long-term debts,

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?

It needs to collect $\$i\times 1m$ in year two, and $\$(1+i)\times 1m$ in year three. The present value is still $1m.

What is Ricardian’s equivalence theorem?

To finance government purchase, there is no difference between tax financing and bond financing. Their wealth effects on households are equivalent.

Suppose short-term debt bears an interest rate of $i_s$ and long-term debt bears an interest rate of $i_l$. All private bonds are short-term, and households hold a portfolio of both short-term private bonds and long-term government bonds. Consider government finances its purchase through long-term bonds as in (b). Will Ricardian equivalence hold? Why? Or why not?

The present value of wealth impact is a decrease in wealth of
\[ i_l\times 1m/(1+i^*)+(1+i_l) \times 1m/(1+i^*)^2 \]
when $i_s<i_l$ (which is the general case) household’s portfolio rate of returns $i^*$ , which defines its discount rate, is less than $i_l$. Therefore, the wealth impact is larger than 1m, higher than taxing household immediately in year 1. Ricaridan equivalence does not hold.