Overview

• Some macroeconomic issues are explicitly dynamic.  This chapter studies one such issue, the economics of government debt and budget deficits.  The mathematical tool that we use is a difference equation that shows how a variable (government debt) changes over time.  More specifically, difference equations enable us to understand how deficits accumulated in U.S., and led to a crisis in the early 1990’s that caused the government to raise taxes to balance the budget.

 

• The primary deficit is defined as the value of government expenditures plus transfer payments minus government revenues.  Its relationship with the national debt is described by the government budget equation (14.1).  The reported deficit is defined as the primary deficit plus the value of interest payments on outstanding debt.  Figure 14.1 illustrates these deficits together with government debt from 1940 to 2000 in the U.S. economy.

• The government budget equation (14.1) does not tell whether the national debt is growing too fast relative to government’s ability to repay.  A more useful tool is transforming the government budget equation and measuring debt and deficit using GDP as a unit of measurement, as is shown in Figure 14.2.  This yields a difference equation (14.2) in which the debt to GDP ratio bt is the state variable.

 

• The solution to a difference equation is a list of values for the state variable, one for each date in the future.  To compute the solution to (14.2) with a given initial value of the debt to GDP ratio, we iterate the equation forward to generate successive future values of the state variable.  The values of bt could increase without bound, or grow for a while and then settle down to a fixed value.  Figure 14.3 illustrates how to solve equation (14.2) with a graph.

• A steady state solution is a value of the state variable, which satisfies the government budget equation (14.2) that is independent of time.  If the state variable starts at the steady state, then it will stay there forever.

 

• A steady state can be either stable or unstable, as is shown in Figure 14.4.  If the steady state is stable, the state variable moves closer towards the steady state over time, wherever it starts from.  If a steady state is unstable, the state variable moves further away from the steady state for any starting point other than the steady state itself.

• The government budget equation (14.2) has a stable steady state if the interest rate is lower than the growth rate of nominal GDP.  This is consistent with U.S. data before 1979.  In this case, the national debt would eventually have converged to a steady state level, and a small positive primary deficit can be sustainable forever.

 

• The government budget equation (14.2) has an unstable steady state if the interest rate is higher than the growth rate of nominal GDP.  This is consistent with U.S. data between 1980 and 1993.  In this case, the national debt would explode and eventually the U.S. government would become bankrupt.  Therefore, a small positive primary deficit can not be sustainable forever.  Finally, in the period since 1993, the average growth rate has once more exceeded the average interest rate.  This most recent period looks a lot like the immediate post-war situation.

• Under a balanced budget policy, the government would try to set its reported deficit equal to zero.  This implies that the government must run a negative primary deficit equal to the interest payment on its outstanding debt.

 

• The Ricardian Equivalence argues that it does not matter whether government spending is financed by debt or taxes.  When government spending is financed by debt, households will choose to hold all of this increased debt as extra wealth in order to pay higher future taxes.  In this case, households will not change the amount of saving that they devote to investment in new factories and machines.  Therefore, higher government spending has no impact on the economy since aggregate consumption and investment remain unchanged.

• There are two possible explanations for the differences between the pre- and post-1979 periods.  One is that productivity growth has slowed down due to a change in the pace at which new technologies are discovered.  The second is that higher government borrowing has raised interest rates in U.S.  Whatever explanation is correct, Figure 14.9 shows that the phenomena of slow productivity growth and high interest rate are worldwide.  All of the G7 countries have experienced similar difficulties.
•  

MULTIPLE ChOICE QUESTIONS

 

1.    The primary budget deficit refers to

a)    The debt of the federal government.
b)    Government expenditures plus transfers net of tax revenues.
c)    Interest payment on government debt.
d)    The reported budget deficit plus interest payments on government debt.

Answer: B

2.    If government expenditures are $454 billion, transfers $713 billion, tax receipts $1,490 billion, then the reported budget deficit is

a)    $323 billion.
b)    $1,167 billion.
c)    $1,749 billion.
d)    Cannot calculate, since we need figures for interest payments.

Answer: D

3.    Under a balanced budget policy, the government sets its reported deficit equal to zero.  This implies that the government must run

a)    A primary budget deficit equal to the interest payment on its outstanding debt.
b)    A primary budget surplus equal to the interest payment on its outstanding debt.
c)    A zero primary budget deficit.
d)    A reported budget surplus.

Answer: B

4.    The government budget deficit is

a)    A stock variable.
b)    A flow variable.
c)    Neither a flow nor a stock variable.
d)    Always increasing over time.

Answer: B

5.    Suppose the government has outstanding debt obligations of $400 billion from last year, and it runs a deficit of $85 billion this year.  If the nominal interest rate is 5%, then the reported deficit this year is

a)    $65 billion.
b)    $85 billion.
c)    $105 billion.
d)    Cannot calculate, need growth rate of nominal GDP.

Answer: C

6.    The steady-state value of b for the difference equation bt = 0.2 + 1.10 bt-1 is

a)    -2.
b)    2.
c)    1.3.
d)    Does not exist since it is unstable.

Answer: A

7.    The steady-state of the difference equation 3xt = 6 + 1.5 xt-1 is

a)    4/3, which is stable.
b)    4/3, which is unstable.
c)    4, which is stable.
d)    4, which is unstable.

Answer: C

8.    Consider the difference equation yt = 4 + 0.5 yt-1.  Which of the following statements about the long-run behavior of y is true, if it starts from 0?

a)    yt diverges away from the steady-state value.
b)    yt does not change over time.
c)    yt converges to a steady-state value of 8/3.
d)    yt converges to a steady-state value of 8.

Answer: D

9.    Let Bt denote the nominal value of government debt in the year t, i denote the interest rate, and Dt denote the primary budget deficit that the government runs in the year t.  Which of the following equations describes the law of motion of government debt accurately?

a)    .
b)    .
c)    .
d)    .

Answer: C

10.  The steady state of a difference equation is unstable, if

a)    The state variable always increases over time irrespective of the starting value.
b)    The state variable always decreases over time irrespective of the starting value.
c)    The state variable always moves towards the steady state, no matter where it starts out from.
d)    The state variable always moves away from the steady state if it does not start exactly at the steady state.

Answer: D

11.  If the nominal rate of interest is 5%, the growth rate of GDP 7%, and the government runs a budget deficit equal 3% of GDP every year, what is the value of the steady-state debt-to-GDP ratio?

a)    0.031.
b)    0.04.
c)    1.605.
d)    53.5.

Answer: C

12.  Since 1979, the debt-to-GDP ratio started to increase in U.S. because

a)    The GDP growth rate was bigger than the nominal interest rate.
b)    The nominal interest rate was bigger than the GDP growth rate.
c)    The government cut down on its budget deficits.
d)    The U.S. started to run trade surpluses.

Answer: B

13.  According to the proposition of Ricardian Equivalence,

a)    It does not matter whether the government finances its spending by raising taxes or by borrowing.
b)    The government has to run a budget surplus or else raise taxes.
c)    Higher government borrowing must be paid by increasing current taxes.
d)    Running a budget deficit is equivalent to running a negative budget surplus.

Answer: A

14.  If the difference equation describing the law of motion of the government debt-to-GDP ratio is unstable, it means that the debt-to-GDP ratio will keep on increasing if the government runs a constant budget deficit every year.  This situation is unsustainable because

a)    The government has to cut down on its expenditures.
b)    The tax base will be shrinking so that the government will be have to increase the amount of deficit it is running every year.
c)    The debt will eventually exceed the value of GDP, and hence the government will go bankrupt.
d)    Eventually the GDP growth rate will increase to equal the interest rate.

Answer: C

15.  One reason why the interest rates in the G-7 countries increased in the 1980's could be that

a)    The G-7 countries experienced lower growth in the 1980's.
b)    The growth rates in all the G-7 countries increased significantly during the 1980's.
c)    Countries other than the U.S. started building up large budget surpluses during the 1980's.
d)    Higher borrowing by the U.S. government drove up interest rates, not only in the U.S. but in the other countries as well, because their currencies are in practice tied to the U.S. dollar.

Answer: D

Problems

 

1.    What is meant by a stock variable?  What is meant by a flow variable?  Which of the following are stocks and which are flows -- GDP, income, the money supply, and employment in 2000?

2     Is government debt a stock or a flow variable?  How about budget deficit?  Explain your answer.

3.    Explain why Figure 14.1 overstates the importance of the U.S. debt/deficit problem in recent years.

4.    Consider the following difference equations:

       (1)    xt = 1.5 + 0.5xt-1
(2)    xt =  0.5 xt-1
(3)    xt =  3 + 2 xt-1
(4)    xt = 2xt-1  
(5)    xt  =  2 - 0.5xt-1
(6)    xt  =  - 0.5xt-1
(7)    xt  =  1 - 1.5xt-1
(8)    xt  = - 1.5xt-1
(9)    xt  =  1 + xt-1
(10)   xt  =  1 - xt-1

(a)   Draw a graph of xt against xt-1.

(b)  Find the value of the steady state.

(c)   Determine whether the steady state is stable or unstable.

5.    Explain what economists mean by the “steady state” of a model.  Use the model  as an example.

6.    This question refers to the model in question 5.  If you know that the parameter b was equal to 1.5, what would you predict about the long run behavior of yt?  Use a diagram to explain your answer.

7.    If the steady state of a difference equation is stable, then all solutions for the state variable must be negative.  True, false or uncertain?  Explain your answer.

8.    Can the government maintain a permanent budget deficit?  What would you need to know about the future path of interest rates and growth rates to be able to answer this question?

9.   
(a)   Consider the difference equation: , where a and b are positive numbers and b ¹ 1.  Find an expression for the steady state of this equation in terms of the parameters a and b.

(b)  Explain, using diagrams, what will happen to the state variable yt as t gets bigger assuming that yt does not start out at the steady state.  How does your answer depend on the value of b?

(c)   Write down the equation of the government budget constraint in the same form as the equation in part (a) using the following definitions:

“bt” the debt to GDP ratio at date t
“d” the deficit to GDP ratio (assumed constant)
“i”  the interest rate (assumed constant)
“n” the growth rate of GDP (assumed constant)

(d)  Derive an equation for the steady state budget deficit.  Under what conditions will the steady state be positive and stable?

(e)   Are there conditions under which the government can run a positive deficit forever?  If so, what are these conditions?

10.  Consider the difference equation:  with the initial condition X0 = 1.

(a)   Find the steady state value of X.

(b)  Explain what will happen to X over time if its path is described by the above equation.

(c)   Let X represent the ratio of government debt to GDP, explain what happened in 1979 that caused a problem for the U.S. government budget.

11.  This question is about the government budget equation.  Let Bt be nominal government debt at date t.  Let Yt be the nominal value of GDP.  Let i ( > 0) be the nominal interest rate and assume that it is constant for all time.  Let Dt be the nominal primary government budget deficit.  Let bt be the debt to GDP ratio.  Let d be the deficit to GDP ratio, which is assumed to be a constant.  Let n ( > 0) be the constant growth rate of nominal GDP.

(a)   Define what is meant by the primary budget deficit.

(b)  Write down the government budget constraint as a difference equation, using Bt as the state variable and taking i, Dt and Yt as exogenous variables. 

(c)   Transform your answer to (b) into a difference equation, using bt as the state variable and taking i, d and g as exogenous variables.

(d)  Use your answer to part (c) to obtain a set of condition(s) under which the government could sustain a positive deficit to GDP ratio forever.  Explain the economic intuition.

ANSWER
(a) The value of government expenditures plus transfer payments minus government revenues.
(b) Bt = Dt + Bt-1(1+i).
(c) From  we obtain
(d) n > i, which ensures that the budget equation exhibits a stable steady state.

12.  Suppose that the nominal interest rate is 5% and the growth rate of nominal GDP is also 5%.  The government runs a primary deficit to GDP ratio that is positive.  What will happen to the debt to GDP ratio over time?  How would your answer differ if the government ran a primary deficit that was negative?

13.  Let Bt-1 be the nominal value of existing government debt, carried over from year t-1.  Let Dt be the nominal primary budget deficit.  Let RDt be the nominal reported budget deficit.  Assume that all government bonds are of one year maturity and let i ( > 0) be the constant nominal interest rate.

(a)   Write an equation to explain the difference between the primary deficit and reported deficit.

(b)  Based on your answer to part (a), explain why the reported deficit is always at least as large as the primary deficit.

(c)   Use your answer to part (a) to explain what is meant by a balanced budget policy.

14.  Suppose that the government runs a primary budget deficit of 2% of GDP, the nominal interest rate is equal to 5% per year and GDP grows at 3% per year. 

(a)   Find the steady state debt to GDP ratio. 

(b)  Is this steady state stable or unstable?  Explain your answer.

(c)   Explain the difference between the primary deficit and the reported deficit.  Assume that in 2000, the government had a debt to GDP ratio of 25%.  How big was the reported deficit to GDP ratio?

(d)  Using the numbers from part (c), calculate the size of the primary deficit to GDP ratio necessary to balance the budget in 2000 and fix the debt to GDP ratio at 25% forever.

15.  In Econoland, the government budget deficit in 2000 was equal to 2% of GDP.  The nominal interest rate was 3% and the growth rate of GDP was 5%.  If the economy is in a steady state equilibrium,

(a)   What is the ratio of government debt to GDP?  Is this equilibrium stable?  [HINT: you may use the government budget equation  where b is the ratio of debt to GDP, d is the ratio of the deficit to GDP, i is the nominal interest rate and n is the growth rate of GDP.]

       President Spendit of Econoland was elected to power at the end of 2000 and he decided to raise government spending on defense.  As a result, the budget deficit increases to 4% of GDP.

(b)  Assuming that the growth rate and the interest rate remain unchanged, what would you expect to be the new steady state value of the debt to GDP ratio?  Is the new steady state stable?

16.  In Econoland, the primary budget deficit is $20,000 in 2000 and the administration plans to maintain this primary deficit as a fixed fraction of GDP.  The interest rate is 5% and nominal GDP is growing at 6%.  If the stock of debt outstanding is $400,000 and GDP in 2000 is $500,000.

(a)   What was the reported budget deficit in 2000?

(b)  What is ratio of debt to GDP in 2000?

(c)   Is this ratio going to grow or shrink over the next few years?

(d)  Is the administration’s plan sustainable?  Explain your answer.

17.  In Italy, the government is running a budget deficit equal to 8% of GDP. 

(a)   If the Italian interest rate is 5% and the growth rate of nominal GDP is 6%, calculate the steady state ratio of debt to GDP.

(b)  Suppose that the maximum that the government can possibly raise in taxes is equal to 50% of GDP.  Assuming that the interest rate equals 5% and the nominal growth rate of GDP equals 6%, at what ratio of the deficit to GDP will the interest payments on the steady state debt exceed the government’s ability to finance these payments through taxes?

18.  Explain why a Ricardian believes that budget deficits don’t matter?

19.  Briefly explain why the interest rate and the growth rate in all of the G7 countries have displayed similar patterns.  What’s the implication of such patterns on the sustainability of a positive deficit in each country?

20.  Between 1929 and 1979, in each of the G7 countries, the interest rate was less than the nominal GDP growth rate on average.  True, false or uncertain?  Explain your answer.

ANSWER KEY TO PROBLEMS

1.    A stock is used to measure wealth at a particular point in time and a flow is the rate of change of a stock.  The money supply is a stock.  GDP, income and employment in 2000 are flows.
2.    Government debt is a stock and budget deficit is a flow.
3.    It does not take into account the country’s ability to pay the interest on its debt and eventually repay the debt itself.
4.    (a)

 

       (b) Setting xt = xt-1, we find that = (1) 0.75, (2) 0, (3) -3, (4) 0, (5) 4/3, (6) 0, (7) 0.4, (8) 0, (9) does not exist, (10) 0.5.  (c) (1) stable, (2) stable, (3) unstable, (4) unstable, (5) stable, (6) stable, (7) unstable, (8) unstable, (9) not available, (10) cannot tell.
5.    A value of the state variable, which satisfies the difference equation that is independent of time.  Setting yt = yt-1, we find that   If the state variable yt starts at the steady state, then it will stay there forever.
6.    It will explode.
7.    False, consider xt = 1.5 + 0.5xt-1 and x0 = 1.
8.    It depends.  The government can (cannot) maintain a permanent budget deficit if the interest rate is lower (higher) than the GDP growth rate.
9.    (a) Setting yt = yt-1, we find that   (b) yt will converge to the steady state if  and yt will diverge away from the steady state if .  (c)   (d) Setting bt = bt-1, we find that .  n > i.  (e) n > i.
10.  (a) Setting Xt+1 = Xt, we find that   (b) Xt will explode.  (c) The interest rate became higher than the GDP growth rate.
11.  See the above.
12.  In this case, the budget equation becomes   The debt to GDP ratio bt will explode (grow forever) regardless of whether the government runs a positive or a negative primary deficit.
13.  (a) , that is, the reported deficit is defined as the primary deficit plus the value of interest payments on outstanding debt.  (b) Since .  (c) , that is, setting the reported deficit to zero.
14.  (a) In this case, the budget equation becomes   Setting bt = bt-1, we find that .  (b) Unstable, since the nominal interest rate is greater than the GDP growth rate.  (c) The reported deficit is defined as the primary deficit plus the value of interest payments on outstanding debt.  0.02 + 0.05*0.25 = 0.0325.  (d) d = -0.05*0.25 = -0.0125.
15.  (a) In this case, the budget equation becomes   Setting bt = bt-1, we find that .  Yes.  (b)   Yes.
16.  (a)  (b)   (c) Shrink.  (d) Yes, since the GDP growth rate is higher than the nominal interest rate.
17.  (a)   (b) Setting  we find that d = 0.0943.
18.  When government spending is financed by debt, households will choose to hold all of this increased debt as extra wealth in order to pay higher future taxes.  In this case, households will not change the amount of saving that they devote to investment in new factories and machines.
19.  Productivity growth has slowed down or higher government borrowing has raised interest rate in the U.S.  As a result, the interest rate in all of the G7 countries became higher than the GDP growth rate after 1979, which made a positive deficit in each country less sustainable.
20.  False, the interest rate is lower than the GDP growth rate between 1929 and 1979.