# 4 Government Purchase

For this topic, there are two videos to watch:

video 1

video 2

## 4.1 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.

1. Write down government’s budget constraint.

2. Write down household budget constraint with the existence of such a government.

Now consider multiple periods of government budgetting 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.

1. Consider a two-periods ($$t=1,2$$) government budget constraint, with an initial debt burden $$B_0^g$$ and an end of period 2 debt burden $$B_2^g$$. Write down the corresponding two-periods government budget constraint.

2. Write down the two-periods ($$t=1,2$$) household intertemporal budget constraint.

3. 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?

## 4.2 The role of public services

Assume that when government takes $1 from you, he spend $$\\lambda$$ to buy things you intend to buy for you and the rest $$\(1-\lambda)$$ to buy things that government wants but useless for you. In other words, each unit of government purchases, $$G$$, was equivalent to $$\lambda$$ units of private consumption, $$C$$. 1. Consider various categories of government expenditure, such as military spending, police, highways, public transit, and research and development. How do you think the coefficient $$\lambda$$ varies across these categories? 2. If $$G$$ increases by one unit, how much consumption spending will you reduce in response to that? 3. Suppose that $$G$$ rises permanently by one unit. What are the responses of real GDP, consumption and investment? How do the results depend on the size of the coefficient $$\lambda$$? 1. Another role of public services.Assume that when government takes$1 from you, he will be able to provide some public service so that private sector final output production can increase by $$\\beta$$ value. Assume 0<$$\beta$$<1.
1. Other things being equal, by one unit of $$G$$ increase how much can final output supply, $$Y^{s}$$, increase?

2. Do you think it will induce a wealth effect on consumption? If so, for how much?

For the following analysis, we assume that labor supply is fixed. Therefore, today’s output can only be affected by public service variation.

1. Analyze the effect of a temporary increase in $$G$$.

2. Analyze the effect of a permanent increase in $$G$$.

## 4.3 A prospective change in government purchases

Suppose that people learn in the current year that government purchases will increase in some future year, but not now.

1. What happens in the current year to real GDP, consumption and investment?

2. Can you think of some real-world cases to which this question applies?