CHAPTER 10

                                                        INCOME AND SPENDING

Chapter Outline

• Aggregate Demand and Equilibrium Income
• The Consumption Function
• The Consumption Function and Aggregate Demand
• Equilibrium Income
• The Multiplier
• Fiscal Policy in the Very Short Run
• Automatic and Discretionary Fiscal Policy
• The Government Budget in the Very Short Run
• Effects of Government Purchases and Tax Changes on the Budget Surplus
• The Full Employment Budget Surplus
• The Foreign Sector
• Working With Data

 

Changes from the Previous Edition

This chapter has been rewritten, reorganized and streamlined.  It is now shorter, and much less repetitive.  It serves as an introduction to the idea behind the IS curve which is introduced in Chapter 11, and also gives a basic review of material that, although it should have been learned in principles, it is needed to proceed to the IS/LM model in the next chapter.

Learning Objectives

• Students should understand that the consumption function is a relationship between consumption spending and disposable income.
• Students should understand that aggregate demand is composed of the consumption function plus the other component of demand – investment, government spending and net exports.
• Students should understand that the equilibrium condition Y=AD is really a statement about supply equals demand in equilibrium.  The aggregate supply curve in this model is horizontal in P,Y space, so that if demand changes, supply changes automatically.
• Students should learn that the multiplier is a dynamic process which drives the model to a new equilibrium after an autonomous change.
• Students should be aware that the size of the multiplier is determined by the model under discussion.
• Students should understand that any change in autonomous spending affects the equilibrium output by a larger amount, while the multiplier is unaffected. However, any change in the marginal propensity to consume, the income tax rate, or other automatic stabilizers will affect the size of the multiplier as well as the equilibrium output level.
• Students should be able to differentiate between active fiscal stabilization policy and automatic stability.
• Students should be able to assess the effects of changes in government purchases or taxes on the budget surplus and equilibrium income.
• Students should be familiar with the concept of the full-employment budget surplus and with recent trends in the actual, cyclical, and structural budget surpluses.
•  

Achieving the Objectives

This chapter presents a basic model of an adjustment to a macroeconomic equilibrium under the assumption that interest rates and prices are fixed. A simple model of national income determination follows a brief analysis of the consumption function. The concept of the expenditure multiplier is introduced, using the standard Keynesian cross diagram for the graphical analysis. As the government sector is introduced, the role of income taxes as automatic stabilizers and the effects of tax changes and changes in government purchases on the budget surplus are assessed. The importance of recent budgetary trends and the distinctions between the actual, cyclical, and structural budget surpluses are also presented.
Instructors may find it helpful to interpret the GDP identity

Y = C + I + G + NX.

in the following manner.  The left hand side is output, or supply, and the right hand side is demand.  The Keynesian model developed in this chapter has theory on the demand side through the consumption function, but there is no really good explanation for aggregate supply.  The AS curve is horizontal, so supply is always accommodating to demand changes.
The determination of the equilibrium output level introduces the concept of the expenditure multiplier, which is used to answer the question of how much equilibrium income changes following a $1 increase in autonomous spending. In this simplified model, the multiplier is necessarily greater than 1, which implies that the change in equilibrium output is greater than the initial change in autonomous spending. If we omit the government sector and foreign trade, we can define the simple expenditure multiplier as

a = 1/(1 - c).

In other words, any increase in autonomous spending by DA will increase national income by

DY = [1/(1 - c)](DA) = a(DA).

When income taxes are introduced, the value of the expenditure multiplier is reduced, thus providing automatic stability. Students should know that, although automatic stabilizers (income taxes, unemployment benefits, etc.) successfully reduce the effect of external disturbances by reducing the size of the multiplier, built-in stability cannot replace active fiscal policy when the economy is headed for a major recession.
Finally, recent trends in federal revenues and spending deserve some attention. In the late 1990s, the Canadian government had a federal budget surplus for the first time in three decades. Figure 10-6 shows the variations in the federal budget deficit as a percentage of GDP. Students tend to focus their attention on the federal budget and neglect government activity on the state and local level. However, the national income accounts include all levels of government. The notion of a full-employment budget surplus (or structural budget surplus) as a measure for fiscal policy also deserves some attention.  Students should know that the actual budget surplus has a cyclical and a structural component. The actual budget surplus is defined as

BS = TA - TR - G = tY - TR - G,

whereas the full-employment budget deficit is defined as

BS* = tY* - TR - G,   where Y* = full-employment income.
The full-employment (or structural) budget surplus (BS*) only changes with changes in fiscal policy. But the actual budget surplus (BS) can change without any government action, due to economic disturbances that affect income and thus income tax revenues. The difference between the full-employment budget surplus and the actual budget surplus is the cyclical component of the budget surplus, that is,

BS* - BS = t(Y* - Y).

It should be pointed out that estimates of the true value of the full-employment budget surplus largely depend on the assumptions that lead to the calculation of full-employment income. Furthermore, the full-employment budget surplus does not describe the true thrust of fiscal policy in all cases.

Suggestions and Pitfalls

The textbook uses the term aggregate demand (AD) for the [C+I+G+NX]-line in Figure 10-2 and thereafter. However, some instructors may prefer to use the term aggregate expenditure to distinguish more clearly this upward-sloping line from the downward-sloping aggregate demand curve that has been used in the AD-AS analysis in earlier chapters. Such a distinction will help to reduce confusion and stresses the point that a shift in spending (the [C+I+G+NX]-line in a Keynesian cross diagram) is of a different magnitude than a shift in aggregate demand in an AD-AS framework. In the Keynesian cross diagram, prices and interest rates are assumed to be fixed; however, in an AD-AS model, interest rates and prices are allowed to change. This will become clearer in Chapter 11, when the IS-LM framework has been introduced and the AD-curve in the AD-AS framework has been formally derived.
The usefulness of the Keynesian cross diagram in showing the effects of spending changes on equilibrium income is limited. Therefore it may not be advisable to spend much time on this simple model in which prices and the interest rate are fixed. However, since the diagram is used to graphically derive the IS-curve, students need at least a basic understanding of the adjustment processes in this model. Students tend to intuitively understand the chain reaction process of the multiplier when it is explained in plain English, but many find the graphical and mathematical analysis difficult to understand. Since many students see the multiplier process as an immediate change from one equilibrium to the next, it is important to point out that the multiplier effect is actually a dynamic process.
Values of the multiplier that exist in students' minds after dealing with this chapter often far exceed the values established by reputable economic studies. It is therefore very important to point out that the value of the multiplier is not always 1/(1 - c), a formula they may remember from their introductory courses, or even 1/[1 - c(1 - t)], the formula used in this chapter. Instead, the expenditure multiplier is always determined by the particular model of the expenditure sector that is being used. To avoid misconceptions, instructors may prefer to introduce a somewhat more complicated model of the expenditure sector than that discussed in this chapter (for specific examples, look under "Additional Problems" below).
A simple model without income taxes can be used to derive the balanced budget theorem. This theorem states that, in a simple Keynesian cross model, if government purchases and taxes are increased by an equal amount, income increases by the same amount, while the budget surplus or deficit is not affected. In this case, the size of the balanced budget multiplier is 1, implying that output expands precisely by the amount of the increased government purchases with no induced increase in consumption spending. Consumption remains constant, since the effect of higher taxes exactly offsets the effect of the income expansion, leaving disposable income unchanged.
When introduced to the balanced budget theorem, students' first inclination is often to assume that an equal increase in both government purchases (G) and taxes (TA) has no effect on equilibrium output. It is therefore important to show them that part of a (permanent) tax increase is financed through a reduction in saving. Since

 

a lump-sum change in taxes by DTAo will change consumption by

DC = - c(DTAo) with c < 1, so saving changes by DS = (1 - c)(DTAo).

In a simple model without income taxes, equilibrium output is defined as

Y* = [1/(1 - c)]A  ==>  DY = [1/(1 - c)](DA).

Thus if government purchases (G) and taxes (TA) are both increased by the same lump sum, let's say (DG) = (DTAo) = 100, then autonomous spending (A) increases by

DA = DG - c(DTAo) = 100 - c(100) = (1 - c)(100), and

DY = [1/(1 - c)][(DA) = [1/(1 - c)](1 - c)(100) = 100 = DG = DTAo.

Another way of showing that the size of the balanced budget multiplier is 1 is to consider the successive rounds of spending changes caused by such a government policy change. This is shown in the following table:

spending rounds     1         2         3         4         5       ......n-1          n

effects of DG          1         c         c2        c3       c4      .....cn-2        cn-1

effects of DTAo     -c       -c2      -c3       -c4      -c5      ....-cn-1      -cn
_______________________________________________________
total effect              1  -  cn

The first row shows the successive changes in spending resulting from a $1 increase in government purchases (G). The second row shows the successive changes in spending from a $1 increase in taxes (TA). If we add the two rows, all but two items cancel each other out and only 1 - cn remains as the total change. Since cn becomes extremely small as the number of spending rounds (n) increases, the final change in spending is $1.
This will help students recognize that the multiplier process is not a static but a dynamic process and that it will take many periods of adjustment to get to a new equilibrium. The preceding discussion also shows that an increase in government purchases has a stronger impact on intended spending than the same increase in taxes and results in an increase in the equilibrium output level.
At this point students should also be told that as soon as income taxes are introduced, an equal change in lump sum taxes (TAo) and government purchases (G) will no longer lead to a change in equilibrium output (Y) of equal magnitude. This is often very confusing, so it is important to go through yet another numerical example. The multiplier in this expanded model is now

1/(1 - c'),   where    c' = c(1 - t)

and the effect on equilibrium output is

DY = [1/(1 - c')](1 - c)(100),

which is less than 100 since c > c' = c(1 - t), that is,

            (1 - c)/(1 - c') < 1.

            It also helps to mention that the increase in equilibrium output will be even less if interest rates and prices are allowed to vary..
Not all students, especially those who may have a tough time with the Keynesian cross diagram anyway, should be exposed to such complications at this point. Often it is enough to briefly point out that the actual effect of an intended spending change on national income (the multiplier effect) is significantly reduced if prices and interest rates are assumed to be flexible rather than fixed.
While the previous edition stressed concerns over budget deficits, now students' attention should be drawn to the discussion of what to do with the budget surplus. Some policy makers favor tax cuts, while others favor increased spending..
When discussing the difference between an actual, cyclical, and structural budget surplus, it is worthwhile to point out that it is possible for an actual budget deficit to co-exist with a structural budget surplus. If we have an actual budget deficit but a structural budget surplus, then the negative cyclical component of the budget surplus outweighs the positive structural component. In other words, the economy is in a recession--maybe because fiscal policy was so restrictive that the full-employment level of output cannot be reached.
Since there is no agreement among economists on just what constitutes full employment, there is no perfect measure for the full-employment budget surplus. It is also important to stress that the full-employment budget surplus cannot be used as a perfect measure of the effectiveness of fiscal policy. Fiscal policy can achieve changes in national income without any change in the actual or the full-employment budget surplus, as we have seen in the discussion of the balanced budget theorem.

 

                                             Solutions to the Problems in the Textbook

Discussion Questions:

1.   In the Keynesian model, the price level is assumed to be fixed, that is, the AS-curve is horizontal and the level of output is determined solely by aggregate demand. The classical model, on the other hand, assumes that prices always fully adjust to maintain a full-employment level of output, that is, the AS-curve is vertical. Since the model of income determination in this chapter assumes that the price level is fixed, it is a Keynesian model.

2.   An autonomous variable’s value is determined outside of a given model. In this chapter the following components of aggregate demand have been specified as being autonomous: autonomous consumption (C*), autonomous investment (Io), government purchases (Go), lump sum taxes (TAo), transfer payments (TRo), and net exports (NXo).

• Since it often takes a long time for policy makers to agree on a specific fiscal policy measure, it is quite possible that economic conditions may drastically change before a fiscal policy measure is implemented. In these circumstances a policy measure can actually be destabilizing. Maybe the economy has already begun to move out of a recession before policy makers have agreed to implement a tax cut. If the tax cut is enacted at a time when the economy is already beginning to experience strong growth, inflationary pressure can be created.

      While such internal lags are absent with automatic stabilizers (income taxes, unemployment benefits, welfare), these automatic stabilizers are not sufficient to replace active fiscal policy when the economy enters a deep recession.

4.   Income taxes, unemployment benefits, and the welfare system are often called automatic stabilizers since they automatically reduce the amount by which output changes as a result of a change in aggregate demand. These stabilizers are a part of the economic mechanism and therefore work without any case-by-case government intervention. For example, when output declines and unemployment increases, there may be an increase in the number of people who fall below the poverty line. If we had no welfare system or unemployment benefits, then consumption would drop significantly. But since unemployed workers get unemployment compensation and people living in poverty are eligible for welfare payments, consumption will not decrease as much. Therefore, aggregate demand may not be reduced by as much as it would have without these automatic stabilizers.

5.   The full-employment budget surplus is the budget surplus that would exist if the economy were at the full-employment level of output, given the current spending or tax structure. Since the size of the full-employment budget surplus does not depend on the position in the business cycle and only changes when the government implements a fiscal policy change, the full-employment budget surplus can be used as a measure of fiscal policy. Other names for the full-employment budget surplus are the structural budget surplus, the cyclically adjusted surplus, the high-employment surplus, and the standardized employment surplus. These names may be preferable, since they do not suggest that there is a specific full-employment level of output that we were unable to maintain.

Application Questions:

1.a.  AD = C + I = 100 + (0.8)Y + 50 = 150 + (0.8)Y

       The equilibrium condition is Y = AD  ==>

       Y = 150 + (0.8)Y  ==>  (0.2)Y = 150  ==> Y = 5*150 = 750.

1.b. Since TA = TR = 0, it follows that S = YD - C = Y - C.  Therefore

S = Y - [100 + (0.8)Y] = - 100 + (0.2)Y  ==>  S = - 100 + (0.2)750 = - 100 + 150 = 50.

1.c. If the level of output is Y = 800, then  AD = 150 + (0.8)800 = 150 + 640 = 790.

       Therefore the amount of involuntary inventory accumulation is   UI = Y - AD = 800 - 790 = 10.

1.d.  AD' = C + I' = 100 + (0.8)Y + 100 = 200 + (0.8)Y

        From Y = AD'  ==> Y = 200 + (0.8)Y  ==> (0.2)Y = 200 ==> Y = 5*200 = 1,000

       Note: This result can also be achieved by using the multiplier formula:
DY = (multiplier)(DSp) = (multiplier)(DI)    ==>   DY = 5*50 = 250,

       that is, output increases from Yo = 750 to Y1 = 1,000.

1.e. From 1.a. and 1.d. we can see that the multiplier is 5.

2.c. Since the size of the multiplier has doubled from 5 to 10, the change in output (Y) that results from a change in investment (I) now has also doubled from 250 to 500.

3.a. AD = C + I + G + NX = 50 + (0.8)YD + 70 + 200 = 320 + (0.8)[Y - (0.2)Y + 100]

             = 400 + (0.8)(0.8)Y = 400 + (0.64)Y

       From Y = AD ==> Y = 400 + (0.64)Y ==> (0.36)Y = 400

        ==> Y = (1/0.36)400 = (2.78)400 = 1,111.11

        The size of the multiplier is (1/0.36) = 2.78.

3.b. BS = tY - TR - G = (0.2)(1,111.11) - 100 - 200 = 222.22 - 300 = - 77.78

3.c. AD' = 320 + (0.8)[Y - (0.25)Y + 100] = 400 + (0.8)(0.75)Y = 400 + (0.6)Y

       From Y = AD' ==> Y = 400 + (0.6)Y ==> (0.4)Y = 400 ==> Y = (2.5)400 = 1,000

       The size of the multiplier is now reduced to 2.5.

3.d. BS' = (0.25)(1,000) - 100 - 200 = - 50

       BS' - BS = - 50 - (-77.78) = + 27.78

       The size of the multiplier and equilibrium output will both increase with an increase in the marginal propensity to consume. Therefore income tax revenue will also go up and the budget surplus should increase.

3.e. If the income tax rate is t = 1, then all income is taxed. There is no induced spending and equilibrium income only increases by the change in autonomous spending, that is, the size of the multiplier is 1.

       From Y = C + I + G ==> Y = Co + c(Y - 1Y + TRo) + Io + Go

       ==>  Y = Co + cTRo + Io + Go = Ao

4. In Problem 3.d. we had a situation where the following was given:

    Y = 1,000,   t = 0.25,  G = 200   and  BS = - 50.

     Assume now that t = 0.3 and G = 250  ==>

     AD' = 50 + (0.8)[Y - (0.3)Y + 100] + 70 + 250 = 370 + (0.8)(0.7)Y + 80 = 450 + (0.56)Y.

      From Y = AD' ==> Y = 450 + (0.56)Y ==> (0.44)Y = 450

      ==> Y = (1/0.44)450 = 1,022.73

      BS' = (0.3)(1,022.73) - 100 - 250 = 306.82 - 350 = - 43.18

      BS' - BS = -43.18 - (-50) =  + 6.82

      The budget surplus has increased, since the increase in tax revenue is larger than the increase in government purchases.

5.a. While an increase in government purchases by DG = 10 will change intended spending by DSp = 10, a decrease in government transfers by DTR = -10 will change intended spending by a smaller amount, that is, by only DSp = c(DTR) = c(-10). The change in intended spending equals DSp = (1 - c)(10) and equilibrium income should therefore increase by

DY = (multiplier)(1 - c)10.

5.b. If c = 0.8 and t = 0.25, then the size of the multiplier is

a = 1/[1 - c(1 - t)] = 1/[1 - (0.8)(1 - 0.25)] = 1/[1 - (0.6)]  = 1/(0.4) = 2.5.

      The change in equilibrium income is

      DY  =  a(DAo) = a[DG + c(DTR)] = (2.5)[10 + (0.8)(-10)] = (2.5)2 = 5

5.c. DBS = t(DY) - DTR - DG = (0.25)(5) - (-10) - 10 = 1.25

Additional Problems:

1.  "An increase in the marginal propensity to save increases the impact of one additional dollar in income on consumption." Comment on this statement. In your answer discuss the effect of such a change in the mps on the size of the expenditure multiplier.

The fact that the marginal propensity to save (1 - c) has risen implies that the marginal propensity to consume (c) has fallen. This means that now one extra dollar in income earned will affect consumption by less than before the reduction in the mpc. When the mpc is high, one extra dollar in income raises consumption by more than when the mpc is low. If the mps is larger, then the expenditure multiplier will be larger, since the expenditure multiplier is defined as 1/(1-c).

2.   Using a simple model of the expenditure sector without any government involvement, explain the paradox of thrift that asserts that a desire to save may not lead to an increase in actual saving.

The paradox of thrift occurs because the desire to increase saving leads to a lower consumption level. But a lower level of spending sends the economy into a recession and we get a new equilibrium at a lower level of output. In the end, the increase in autonomous saving is exactly offset by the decrease in induced saving due to the lower income level. In other words, the economy is in equilibrium when S = Io. Since the level of autonomous investment (Io) has not changed, the level of saving at the new equilibrium income level must also equal Io.
This can also be derived mathematically. Since an increase in desired saving is equivalent to a decrease in desired consumption, that is, DCo = -DSo, the effect on equilibrium income is

      DY = [1/(1 - c)](DCo) = [1/(1 - c)](-DSo).

Therefore the overall effect on total saving is

DS = s(DY) + DSo = [s/(1 - c)](-DSo) + DSo = 0, since   s = 1 - c.

3.   "When aggregate demand falls below the current output level, an unintended inventory accumulation occurs and the economy is no longer in an equilibrium." Comment on this statement.

If aggregate demand falls below the equilibrium output level, production exceeds desired spending. When firms see an unwanted accumulation in their inventories, they respond by reducing production. The level of output falls and eventually reaches a level at which total output equals desired spending. In other words, the economy eventually reaches a new equilibrium at a lower value of output.

4.   For a simple model of the expenditure sector without any government involvement, derive the multiplier in terms of the marginal propensity to save (s) rather than the marginal propensity to consume (c). Does this formula still hold when the government enters the picture and levies an income tax?

In the text, the expenditure multiplier for a model without any government involvement was derived as

      a = 1/(1 - c).

But since the marginal propensity to save is s = 1 - c, the multiplier now becomes a = 1/s = 1/(1-c).

In the text, we have also seen that if the government enters the picture and levies an income tax, then the simple expenditure multiplier changes to

      a = 1/[1 - c(1 - t)] = 1/(1 - c').

By substituting s = 1 - c, this equation can be easily manipulated, to get

      a’ = 1/[1 - c + ct] = 1/[s + (1 - s)t] = 1/s'.

Just as s = 1 – c, we can say that s' = 1 - c', since 

      s' = 1 - c'  =  1 - c(1 - t) = 1 - c + ct = s + (1 - s)t.

This can also be derived in another way:
S = YD - C = YD - (C* + cYD) = - C* + (1 - c)YD = - C* + sYD

If we assume for simplicity that TR = 0 and NX = 0, then

      S + TA  = I + G   ==> - C* + sYD + TA = I* + G*   ==>

      s(Y - tY - TA*) + tY + TA* = C* + I* + G*   ==>

      [s + (1 - s)t]Y = C* + I* + G* - (1 - s)TA* = A*  ==>

      Y = (1/[s + (1 - s)t])A* = (1/s')A*.

5.  The balanced budget theorem states that the government can stimulate the economy without increasing the budget deficit if an increase in government purchases (G) is financed by an equivalent increase in taxes (TA). Show that this is true for a simple model of the expenditure sector without any income taxes.  

If taxes and government purchases are increased by the same amount, then the change in the budget surplus can be calculated as 

      DBS = DTAo - DG = 0,   since DTAo = DG.

The resulting change in national income is

      DY = DC + DG = c(DYD) + DG = c(DY - DTAo) + DG

            = c(DY) - c(DTAo) + DG = c(DY) + (1 - c)(DG) since DTAo = DG.

       ==>  (1 - c)(DY) = (1 - c)(DG)  ==>  DY = DG

In this case, the increase in output (Y) is exactly of the same magnitude as the increase in government purchases (G). This occurs since the decrease in the level of consumption due to the higher lump sum tax has exactly been offset by the increase in the level of consumption caused by the increase in income.

6.   Assume a model without income taxes and in which the only two components of aggregate demand are consumption and investment. Show that, in this case, the two equilibrium conditions Y = C + I and S = I are equivalent.

We can derive the equilibrium value of output by setting actual income equal to intended spending, that is,

Y = C + I ==> Y = C* + cY + I* ==> (1 - c)Y = C* + I* ==> Y = [1/(1 - c)](C* + I*) = [1/(1 - c)]A*.

But since S = YD - C = Y - [C* + cY] = - C* + (1 - c)Y,

we can derive the same result from
S = I*  ==> S = - C* + (1 - c)Y = I* 

      ==>  (1 - c)Y = C* + I*  ==> Y = [1/(1 - c)](C* + I*) = [1/(1 - c)]A* .

7.   Suppose that, in an effort to stimulate the economy, the federal government proposed a $20 billion tax cut in combination with a $20 billion cut in government purchases. Do you consider this a good policy proposal? Why or why not?

This is not a good policy proposal. According to the balanced budget theorem, equal decreases in government purchases and taxes will decrease rather than increase income. Therefore the intended result would not be achieved.

8.  Assume the following model of the expenditure sector:
Sp   = C + I + G + NX        C  = 420 + (4/5)YD    YD = Y - TA + TR       TA   = (1/6)Y
      TRo = 180                            Io = 160                        Go = 100                      NXo  = - 40
(a) Assume the government would like to increase the equilibrium level of income (Y) to the full-employment level Y* = 2,700. By how much should government purchases (G) be changed?
(b) Assume we want to reach Y* = 2,700 by changing government transfer payments (TR) instead. By how much should TR be changed?
(c)  Assume you increase both government purchases (G) and taxes (TA) by the same lump sum of DG = DTAo = + 300. Would this change in fiscal policy be sufficient to reach the full-employment level of output at Y* = 2,700? Why or why not?
(d) Briefly explain how a decrease in the marginal propensity to save would affect the size of the expenditure multiplier.

a.   Sp = C + I + G + NX = 420 + (4/5)[Y - (1/6)Y + 100] + 160 + 180 - 40

            = 720 + (4/5)(5/6)Y + 80 = 800 + (2/3)Y

      From Y = Sp ==> Y = 800 + (2/3)Y ==> (1/3)Y = 800 ==>Y = 3*800 = 2,400   

      ==> the expenditure multiplier is a = 3

       From DY = a(DAo) ==>  300 = 3(DAo) ==> (DAo) = 100

       Thus government purchases should be changed by DG = DAo = 100.

b.   Since DAo = 100  and DAo = c(DTRo)  ==>100 = (4/5)(DTRo) ==> DTRo = 125.

c.   This is a model with income taxes, so the balanced budget theorem does not apply in its strictest form, which states that an increase in government purchases and taxes by a certain amount increases national income by that same amount, leaving the budget surplus unchanged. Here total tax revenue actually increases by more than 100, since taxes are initially increased by a lump sum of 100, but then income taxes also change due to the change in income. Thus income does not increase by DY = 300, as we can see below.

            DY = a(DG) + a[(-c)(aTAo)] = 3*300 + 3*[-(4/5)300] = 900 - 720 = 180

       This change in fiscal policy will increase income by only DY = 180, from Y0 = 2,400 to Y1 = 2,580, and we will be unable to reach Y* = 2,700.

d.   If the marginal propensity to save decreases, people spend a larger portion of their additional disposable income, that is, the mpc and the slope of the [C+I+G+NX]-line increase. This will lead to an increase in the expenditure multiplier and equilibrium income.

9.   Assume a model with income taxes similar to the model in Problem 8 above. This time, however, you have only limited information about the model, that is, you only know that the marginal propensity to consume out of disposable income is c = 0.75, and that total autonomous spending is Ao = 900, such that Sp = Ao + c'Y = 900 + c'Y. You also know that you can reach the full-employment level of output at Y* = 3,150 by increasing government transfers by a lump sum of DTR = 200.
(a)  What is your current equilibrium level?
(b)  Is it possible to determine the size of the expenditure multiplier with the information you have?

• Assume you want to change the income tax rate (t) in order to reach the full-employment level of income Y* = 3,150. How would this change in the income tax rate affect the size of the expenditure multiplier?

a.    Since DA = c(DTR) = (0.75)200 = 150,

       the new [C+I+G+NX]-line is of the form  Sp1 = 1,050 + c1Y.

       For each model of the expenditure sector we can derive the equilibrium level of income by using the following equation:

            Y* = aAo = 1/(1-c’) ==> 3,150 = a1,050  ==> the expenditure multiplier is a = 3.

       If we now change autonomous spending by DA = 150, then income will have to change by

            DY = a(DA)  ==> DY = 3*150 = 450.

       Therefore the old equilibrium level of income must have been Y = 3,150 - 450 = 2,700.

b.    From our work above we can see that the size of the multiplier is a = 3.

c.   The new [C+I+G+NX]-line is of the form Sp2 = 900 + c2Y. This new intended spending line intersects the 45-degree line at Y = 3,150. Thus the slope of the new intended spending line can be derived as

            c2 = (3,150 - 900)/(3,150) = 5/7.
From Y = Sp2  ==> Y = 900 + (5/7)Y  ==>  (2/7)Y = 900  ==>

            Y = (7/2)900 = (3.5)900 = 3,150.

      The new value of the multiplier is 3.5.

10. Assume you have the following model of the expenditure sector:
      Sp   = C + I + G + NX            C   = 400 + (0.8)YD  Io   = 200              Go    = 300 + (0.1)(Y* - Y)  
      YD  = Y - TA + TR              NXo  = - 40                   TA  = (0.25)Y        TRo = 50
(a)  What is the size of the output gap if potential output is at Y* = 3,000?
      (b)  By how much would investment (I) have to change to reach equilibrium at Y* = 3,000, and how does this change affect the budget surplus?
(c) From the model above you can see that government purchases (G) are counter-cyclical, that is they are increased as national income decreases. If you compare this specification of G with a constant level of G, how is the value of the expenditure multiplier affected?
   (d)  Assume the equation for net exports is changed such that NXo = - 40 is now  NX1 = - 40 - mY, with 0 < m < 1. How would this affect the expenditure multiplier?

a.    Sp = 400 + (0.8)YD + 200 + 300 + (0.1)(3,000 - Y) - 40

            = 1,160 + (0.8)(Y - (0.25)Y + 50) - (0.1)Y = 1,200 + [(0.8)(0.75) - (0.1)]Y    = 1,200 + (0.5)Y

       Y = Sp ==> Y = 1,200 + (0.5)Y ==> (0.5)Y = 1,200  ==>Y = 2*1,200 = 2,400

      The output gap is  Y* - Y = 3,000 - 2,400 = 600.

 b.  From DY = (mult.)(DA) ==> 600 = 2(DI) ==> DI = 300

      BuS = TA - TR - G = (0.25)(2,400) - 50 - [300 + (0.1)(600)] = 600 - 50 - 300 - 60 =  190

       BuS* = (0.25)(3,000) - 50 - 300 = 400, so the budget surplus increases by DBuS = 210.

c.   If government purchases are used as a stabilization tool, the size of the multiplier should be lower than if the level of government spending is fixed. In the model of the expenditure sector above, the slope of the [C+I+G+NX]-line is c' = 0.5 compared to c" = 0.6, when government purchases were defined as G = 300.

• With this change, net exports decrease as national income increases. This additional leakage implies that the size of the multiplier will decrease. In the model above, the slope of the [C+I+G+NX]-line decreases from c' = (0.5) to c" = (0.5) - m. Therefore the expenditure multiplier will decrease from 1/[1 - (0.5)] to 1/[1 - (0.5) + m].

 

11.  Assume you have the following model of the expenditure sector:
       Sp  = C + I + G + NX          C = Co + cYD            YD = Y - TA + TR          TA  = TAo
      TR  = TRo                              I = Io                                      G = Go                            NX = NXo
      (a)  If a decrease in income (Y) by 800 leads to a decrease in savings (S) by 160, what is the size of the expenditure multiplier?
      (b)  If a decrease in taxes (TA) by 400 leads to an increase in income (Y) by 1,200, how large is the marginal propensity to save?
      (c)  If an increase in imports by 200 (DNX = - 200) leads to a decrease in consumption (C) by 800, what is the size of the expenditure multiplier?

Recall that the expenditure multiplier for such a simple model can be calculated as:

      a = 1/(1 - c)

a.    (DS)/(DY) = 1 - c = (-160)/(-800) = .2 ==> 1/(1 - c) = 1/(.2) = 5   ==> the multiplier is a = 5.

b.   From (DY) = a[-c(DTAo)] ==> (DY)/(DTAo) = (-c)a =  (-c)/(1 - c)  ==>

      (1,200)/(-400) = - 3 = (-c)/(1 - c) ==>  -3(1 - c) = -c  ==> c = 3/4 

      ==> mps = 1 - c = 1/4 = 0.25.

c.    DY = DC + DNX = -800 + (-200) = - 1,000

       ==> c = (DC)/(DY) = (-800)/(-1,000) = .8 ==>  multiplier = a = 1/(1 - c) = 1/(.2) = 5

12. Explain why income taxation, the Social Security system, and unemployment insurance are considered automatic stabilizers.

Income taxes, unemployment benefits, and the Social Security system are often called automatic stabilizers because they reduce the amount by which output changes as a result of a change in aggregate demand. These stabilizers are a part of the structure of the economy and therefore work without any actual government intervention. For example, when output declines and unemployment increases. If we had no unemployment insurance, people out of work would not receive any disposable income and then consumption would drop significantly. But since unemployed workers get unemployment compensation, consumption will not decrease as much. Therefore, aggregate demand may not be reduced by as much as it would have without these automatic stabilizers.

13. Assume a simple model of the expenditure sector with a positive income tax rate (t). Show mathematically how an increase in lump sum taxes (TAo ) would affect the budget surplus.

From BS = TA - G - TR = tY + TAo - G – TR

 ==>  DBS = t(DY) + DTAo = t(mult.)(-c)(DTAo) + DTAo

                 = t[1/(1 - c + ct)](-c)(DTAo) + DTAo = ([ -(ct) + 1 - c + (ct)]/[1 - c + (ct)])(DTAo)

                 = (1 - c) /[1 - c + (ct)])(DTAo)  > 0,   since c < 1

In other words, a lump sum tax increase would increase the budget surplus.

14. True or false? Why?
     "A tax cut will increase national income and will therefore always increase the budget surplus."

False. Although a tax cut raises national income, not all of the increase in income is spent, nor is it completely taxed away. Income tax revenues fall and the budget deficit rises. Assume the following model of the expenditure sector:

      Sp  = C + I + G + NX               I    = Io                             
C    = Co + cYD                        G  = Go
YD = Y ‑ TA +TR                   NX = NXo
TA = TAo + tY                        BS  = TA - G - TR
TR = TRo                            

From Y = Sp ==> Y = Co + c(Y - TAo - tY + TRo) + Io + Go + NXo  ==>

      Y = Co - cTAo + cTRo + Io + Go + NXo + c(1 - t)Y = Ao + c'Y ==> 

      Y = [1/(1 - c')]Ao     with c' = c (1- t)

Thus DY = [1/(1 - c')][(-c)(DTAo)]

and  DBS = t(DY) + (DTAo) = {[t(-c)]/(1 - c') + 1}(DTAo) ==>

                = {[-(ct) + 1 - c + (ct)]/[1 - c + (ct)]}(DTAo) = {(1 - c)/[1 - c + (ct)]}(DTAo) > 0 if DTA > 0.

Therefore, if taxes fall, that is, if DTA < 0, the budget surplus decreases.  

15. Assume a simple model of the expenditure sector with a positive income tax rate (t). Show mathematically how a decrease in autonomous investment (Io ) would affect the budget surplus.

A decrease in autonomous investment (Io) will have a multiplier effect and will therefore decrease national income and tax revenue. The budget surplus will decrease as shown below:

       DBS = t(DY) = ta(DIo) < 0

16. "An increase in government purchases will always pay for itself, as it raises national income and hence the government's tax revenues." Comment on this statement.

An increase in government purchases will increase the budget deficit. If we assume a model of the expenditure sector with income taxes, then the multiplier equals [1/(1 - c')] with c' = c (1- t). The change in the budget surplus that arises from a change in government purchases can be calculated as

      DBS = t(DY) - DG = t[1/(1 - c')](DG) - DG = {[t - 1 + c - (ct)]/[1 - c + (ct)]}(DG)

               = - {[(1 - c)(1 - t)]/(1 - c + (ct))}(DG) < 0, sine DG > 0.

Therefore, if government purchases are increased, the budget surplus will decrease.

17. Is the size of the actual budget surplus always a good measure for determining fiscal policy? What about the size of the full-employment budget surplus?

The actual budget surplus has a cyclical and a structural component. The cyclical component of the budget surplus changes with changes in the level of income whether or not any fiscal policy measure has been implemented. This implies that the actual budget surplus also changes with changes in income and is therefore not a very good measure for assessing fiscal policy. The structural (full-employment) budget surplus is calculated under the assumption that the economy is at full-employment. It therefore changes only with a change in fiscal policy and is a much better measure for fiscal policy than the actual budget surplus. One should keep in mind, however, that the balanced budget theorem implies that the government can stimulate national income by an equivalent and simultaneous increase in taxes and government purchases, thereby affecting the actual or the full-employment budget surplus.

18. Assume a model of the expenditure sector with income taxes, in which people who pay taxes, have a higher marginal propensity to consume than people who receive government transfers, and the consumption function is of the following form: C = Co + c(Y - TA) + dTR, with c < d.
(a) What will happen to the equilibrium level of income and the budget surplus if government purchases are reduced by the same lump sum amount as taxes?
       (b) What will happen to the equilibrium level of income and the budget surplus if, now, government transfers are reduced by the same lump sum amount as taxes?

a.    Assume that DTAo = DG = - 100 ==>

            DY = [(-c)/(1 - c')(DTAo) + [1/(1 - c')](DG) = [(1 - c)/(1 - c')](-100) < 0       c' = c(1 - t)
National income would decrease.

            DBS = t(DY) + DTAo - DG = t(DY) < 0

      The budget surplus would decrease by the loss in income tax revenue.

b.   Assume that DTAo = DTRo = - 100 ==>

            DY = [(-c)/(1 - c')](DTAo) + [d/(1 - c')](DTRo) = [(d - c)/(1 - c')](-100) < 0     c' = c(1 - t)

       National income would increase.
DBS = t(DY) + DTAo - DTRo = t(DY) < 0

      The budget surplus would decrease.

19.  True or false? Why?
       "The higher the marginal propensity to import, the lower the size of the multiplier."

True. Imports represent a leakage out of the income flow. An increase in autonomous spending will raise income and we will see the usual multiplier effect. However, if imports are positively related to income, this effect is reduced since higher imports reduce the level of domestic demand.

      Closed Economy Model                   Open Economy Model

      Sp  = C + I + G                                Sp   = C + I + G + NX
C   = Co + cY                                   C     = Co + cY
G   = Go                                            G    = Go
I     = Io                                              I     = Io
NX  = NXo ‑ mY     with m > 0

                                            From Y = Sp ==>

Y = (Co + Io + Go) + cY                    Y = (Co + Io + Go + NXo) + (c - m)Y

      Y = Ao + cY                                     Y = Ao + (c - m)Y

      Y = [1/(1 - c)]Ao                               Y = [1/(1 - c + m)]Ao

                                  Therefore the multiplier is defined as

      [1/(1 ‑ c)]                                            [1/(1 ‑ c + m)]

Clearly the open economy multiplier falls short of the closed economy multiplier. This is because leakages reduce demand. If income taxes were included in these models, they too would reduce the multipliers, as income taxes represent another leakage from the income flow.