Unformatted text preview: Econ 103: Intermediate Macroeconomics Homework 4 Solutions Topic 7a: Income and Spending Conceptual questions: 1. We call the model of income determination developed in this chapter a Keynesian
one. What makes it Keynesian, as opposed to classical?
In the Keynesian model of income determination, 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 in this chapter assumes that the price level is fixed, it is a Keynesian model.
2. What is an autonomous variable? What components of aggregate demand have
we specified, in this chapter, as being autonomous?
The value of an autonomous variable is determined outside of a given model. This chapter specifies the following components of aggregate demand as being autonomous: autonomous consumption (C*), autonomous investment (Io), government purchases (Go), lump sum taxes (TAo), transfer payments (TRo), and net exports (NXo). 3. Why do we call mechanisms such as proportional income taxes and the welfare
system automatic stabilizers? Choose one of these mechanisms and explain
carefully how and why it affects fluctuations in output.
Income taxes, unemployment benefits, and the welfare system are often called
automatic stabilizers since they automatically reduce the amount by which total
output will fluctuate as a result of a disturbance. These stabilizers are a part of
the economic mechanism and therefore work without any case-by-case
government intervention. For example, in a recession, when output declines and
unemployment increases, many people experience a decline in their income and
may have to rely on unemployment benefits or welfare. If we had no welfare
system or unemployment benefits, then aggregate consumption would drop
significantly and the recession would deepen. But since unemployed workers
receive unemployment compensation and people living in poverty are eligible for
welfare payments, consumption, and therefore aggregate demand, do not
decrease as much, making the recession less pronounced. See the note on Automatic
Stabilizer provided on iLearn for a formal treatment of this issue. 4. Show analytically what happens to the budget surplus when government increases its expenditures.
See the last three slides of Income and Spending.
5. Here we investigate a particular example of the model studied in Sections 9-2 and
9-3 with no government. Suppose the consumption function is given by C = 100 +
.8Y, while investment is given by I = 50.
a. What is the equilibrium level of income in this case?
b. What is the level of saving in equilibrium?
c. If, for some reason, output is at the level of 800, what will the level of
involuntary inventory (i.e. unplanned inventory investment) accumulation be?
d. If I rises to 100, what will the effect be on the equilibrium income?
e. What is the value of the multiplier here?
f. Draw a diagram indicating the equilibria in both (a) and (d).
a. b. 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. 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. As we can see S = I, which means that the equilibrium condition is fulfilled. 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. 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: ΔY = (multiplier)(ΔSp) = (multiplier)(ΔI) ==> ΔY = 5*50 = 250, e. f. that is, output increases from Yo = 750 to Y1 = 1,000. From 1.a. and 1.d. we can see that the multiplier is α = 5. AD Y = AD AD1 = 200 = (0.8)Y ADo = 150 + (0.8)Y 200 150 0 750 1,000 Y 6. Suppose the consumption behavior in problem 4 changes so that C = 100 + .9Y,
while I remains at 50. Notice that the MPC has increased.
a. Is the equilibrium level of income higher or lower than it was in problem 4(a)?
Calculate the new equilibrium level, Y’, to verify this?
b. Now suppose investment increases to I 100, just as in problem 1(d). What is
the new equilibrium income?
c. Does this change in investment spending have more or less of an effect on Y
than it did in problem 4? Why?
a. Since the mpc has increased from 0.8 to 0.9, the size of the multiplier is now larger.
Therefore we should expect a higher equilibrium income level than in 1.a. AD = C + I = 100 + (0.9)Y + 50 = 150 + (0.9)Y ==> Y = AD ==> Y = 150 + (0.9)Y ==> (0.1)Y = 150 ==> Y = 10*150 = 1,500. b. From ΔY = (multiplier)(ΔI) = 10*50 = 500 ==> Y1 = Yo + ΔY = 1,500 + 500 = 2,000. c.
Since the size of the multiplier has doubled from α = 5 to α1 = 10, the change in
output (Y) that results from a change in investment (I) now has also doubled from
250 to 500. d. Y = AD AD AD1 = 200 = (0.9)Y ADo = 150 + (0.9)Y 200 150 0 1,500 2,000 Y 7. Now we look at the role taxes play in determining equilibrium income. Suppose
we have an economy of the type in Sections 9-4 and 9-5 (or 10-4 and 10-5 in the
new edition of the book), described by the following functions:
C = 50 + .8YD = 70 = 200 = 100 = 0.20
a. Calculate the equilibrium level of income and the multiplier in this model.
b. Calculate also the budget surplus, BS.
c. Suppose that t increases to .25. What is the new equilibrium income? The new
d. Calculate the change in the budget surplus. Would you expect the change in
the surplus to be more or less if c = .9 rather than .8?
e. Can you explain why the multiplier is 1 when t= 1?
a. AD = C + I + G + NX = 50 + (0.8)YD + 70 + 200 + 0 =320 + (0.8)[Y -‐ TA + TR] = 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 α = 1/0.36) = 2.78. b. c. d. e. BS = tY -‐ TR -‐ G = (0.2)(1,111.11) -‐ 100 -‐ 200 = 222.22 -‐ 300 = -‐ 77.78 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 α1 = (1/0.4) = 2.5. The size of the multiplier and equilibrium output will both increase with an increase
in the marginal propensity to consume. Thus income tax revenue will also go up and
the budget surplus should increase. This can be seen as follows:
BS' = (0.25)(1,000) -‐ 100 -‐ 200 = -‐ 50 ==> BS' -‐ BS = -‐ 50 -‐ (-‐77.78) = + 27.78 If the income tax rate is t = 1, then all income is taxed. There is no induced spending
and equilibrium income always increases by exactly the change in autonomous
spending. In other words, the size of the expenditure multiplier is 1. We can see this
from Y = C + I + G ==> Y = Co + c(Y -‐ TA + TR) + Io + Go = Co + c(Y -‐ 1Y + TRo) + Io + Go ==> Y = Co + cTRo + Io + Go = Ao ==> ∆Y = ∆Ao It should be noted that when t = 1 and all income is taxed, it is unlikely that much economic activity will take place other than activity in the “underground economy,” as there are no economic incentives to earn income. As the above equation shows, all income comes from autonomous spending, that is, spending that is predetermined and thus not dependent on currently earned income. 8. Suppose Congress decides to reduce transfer payments (such as welfare) but to
increase government purchases of goods and services by an equal amount. That is,
it undertakes a change in fiscal policy such that ΔG=ΔTR.
a. Would you expect equilibrium income to rise or fall as a result of this change?
Why? Check your answer with the following example: Suppose that, initially,
c = .8, t = .25, and Y0 = 600. Now let ΔG= 10 and ΔTR= -10.
b. Find the change in equilibrium income, ΔY0.
c. What is the change in the budget surplus, ΔBS? Why has BS changed? Solution a. While an increase in government purchases by ΔG = 10 will change intended spending by ΔA = 10, a decrease in government transfers by ΔTR = -‐10 will change intended spending by a smaller amount, that is, by only ΔA = c(ΔTR) = c(-‐
10). Thus the total change in intended spending equals ΔA = 10 + c(-‐10)= (1 -‐ c)(10) and equilibrium income should therefore increase by b. c. ΔY = (multiplier)(1 -‐ c)10 If c = 0.8 and t = 0.25, then the size of the multiplier is α = 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 therefore ΔY = α(ΔAo) = α[ΔG + c(ΔTR)] = (2.5)[10 + (0.8)(-‐10)] = (2.5)2 = 5 The budget surplus should increase, since the level of equilibrium income has increased and therefore the level of tax revenues has increased, while the changes in government purchases and transfer payments cancel each other out. Numerically, this can be shown as follows: ΔBS = t(ΔY) -‐ ΔTR -‐ ΔG = (0.25)(5) -‐ (-‐10) -‐ 10 = 1.25 Topic 8: Aggregate Demand I: Building the IS-LM Model
9.a. Explain in words how and why the multiplier and the interest sensitivity of
aggregate demand affect the slope of the IS curve.
Solution: We know that ΔY0= αΔA. So if the expenditure multiplier (α) becomes larger, the increase in equilibrium income caused by a unit change in autonomous spending also becomes larger. Assume investment spending increases due to a change in the interest rate. If the multiplier α becomes larger, any increase in spending will cause a larger increase in equilibrium income. This means that the IS-‐curve will become flatter as the size of the expenditure multiplier becomes larger. If aggregate demand becomes more sensitive to interest rates, any change in the interest rate will cause the [C+I+G+NX]-‐line to shift up by a larger amount and, given a certain size of the expenditure multiplier α, this will increase the level of equilibrium income by a larger amount. As a result, the IS-‐curve will become flatter. Another way to explain this is by using the IS equation written for interest rate i: IS: Y = α(Ao -‐ bi) = [1/(1 -‐ c + ct)](Ao -‐ bi Rearrange it for i: i = (1/b)Ao -‐ (1/αb)Y = (1/b)Ao – [(1 -‐ c + ct)/b]Y. The last two are the equations for IS written for i. Here you can see that the slope of the IS curve is given by 1/αb. So higher value of α or b would decrease the slope and flatten the IS curve. α is the multiplier and “b” captures the sensitivity of investment to interest rate. b. Explain why the slope of the IS curve is a factor in determining the working of
Solution: Steepness of IS curve can determine to what extent output and interest rate
might change for a given shift in the LM curve caused by monetary policy change.
This question can be answered both graphically and in words. I guess a graphical
illustration might be easier at this point.
Case 1: Draw an IS-LM curves where the IS curve is very steep (i.e. slope is
high). Now shift the LM curve from LM1 to LM2. You will see that interest rate falls,
but output increases.
Case 2: Draw another set of IS-LM curves where the IS curve is very flat (i.e. slope is
small). Now shift the LM curve from LM1 to LM2. This time you will see that for the
same magnitude of shift in LM as in case 1, interest rate falls less than before and
output increases more than before.
The main idea is that, the size of the changes in equilibrium interest rate and output
following a shift in LM curve depends on the slope of the IS curve. On the other hand,
IS curve's slope is determined by the multiplier and sensitivity of investment to interest
9. Explain in words how and why the income and interest sensitivities of the demand
for real balances affect the slope of the LM curve.
Solution: The easiest way to see the impact of the slop of LM curve is by looking at
the LM curve equation: 1 ⎛
i = ⎜ kY − ⎟
The slope is given by k/h. So greater responsiveness of the demand for money to
income, as measured by k, leads to higher slope and steeper LM curve. Also the
lower the responsiveness of the demand for money to the interest rate, h is, the higher the slope k/h is => steeper LM curve.
11. It is possible that the interest rate might affect consumption spending. An increase
in the interest rate could, in principle, lead to increases in saving and therefore a
reduction in consumption, given the level of income. Suppose that consumption is, in
fact, reduced by an increase in the interest rate. How will the IS curve be affected?
Soluton: A short video lecture was provided to answer this question. You can read the following to accompany that lecture: An increase in the interest rate stimulates saving and thus reduces consumption. But even if saving is not affected by a change in the interest rates, most likely consumption on durable goods will be reduced if interest rates rise. This means that now not only investment spending but also consumption is negatively affected by an increase in the interest rate. In other words, the [C+I+G+NX]-‐line in the Keynesian cross diagram now shifts down further than previously and the level of equilibrium income decreases more than before. In other words, the IS-‐
curve becomes flatter. This can also be shown algebraically, since we can now write the consumption function in the following way: C = Co + cYD -‐ gi In a simple model of the expenditure sector without income taxes, the equation for aggregate demand will now be AD = Ao + cY -‐ (b + g)i. From Y = AD ==> Y = [1/(1 -‐ c)][Ao -‐ (b + g)i] ==> i = [1/(b + g)]Ao -‐ [(1 -‐ c)/(b + g)]Y Therefore, the IS-‐curve now becomes flatter as its slope has been reduced from (1 -‐ c)/b to (1 -‐ c)/(b + g). 12. Between January and December 1991, while the U.S. economy was falling deeper
into its recession, the interest rate on Treasury bills fell from 6.3 percent to 4.1
percent. Use the IS-LM model to explain this pattern of declining output and interest
rates. Which curve must have shifted? Can you think of a reason—historically valid
or simply imagined—that this shift might have occurred?
Solution: In the IS-‐LM model, a simultaneous decline in the interest rate and the level of output can only be caused by a shift of the IS-‐curve to the left. This shift could easily have been caused by a decrease in private spending due to negative business expectations or a decline in consumer confidence. In 1991, the economy was in a recession and firms did not want to invest in new machinery since they did not want to be left holding unwanted inventory. Since consumer confidence was very low and people feared lay-‐offs, consumer spending decreased also. In the IS-‐LM diagram below, the adjustment process can be described as follows: Io ↓ ==> Y ↓ (the IS-‐curve shifts left) ==> md ↓ ==> i ↓ ==> I ↑ ==> Y ↑. Effect: Y ↓ and i ↓ . i ISo IS1 i1 i2 0 Y2 Y1 LM Y Technical questions:
13. The following equations describe an economy. (Think of C, I, G, etc., as being
measured in billions and i as a percentage; a 5 percent interest rate implies i = 5.) A
correction has been made here in the function for C.
C = .8(1-t)Y = 900 − 50 = 800 = 0.25 − 62.5 = 0.25 = 500 a.
e. What is the equation that describes the IS curve?
What is the general definition of the IS curve?
What is the equation that describes the LM curve?
What is the general definition of the LM curve?
What are the equilibrium levels of income and the interest rate? Solution:
13.a. Each point on the IS-‐curve represents an equilibrium in the expenditure sector. (Note that this is a closed economy, that is, NX = 0). The IS-‐curve can be derived by setting actual income equal to intended spending, or Y = C + I + G = (0.8)[1 -‐ (0.25)]Y + 900 -‐ 50i + 800 = 1,700 + (0.6)Y -‐ 50i ==> (0.4)Y = 1,700 -‐ 50i ==> Y = (2.5)(1,700 -‐ 50i) ==> Y = 4,250 -‐ 125i. IS-‐curve 13.b. The IS-‐curve shows all combinations of the interest rate and the output level such that the expenditure sector (the goods market) is in equilibrium, that is, actual output equals intended spending. A decrease in the interest rate stimulates investment spending, making intended spending greater than actual output. The resulting unintended inventory decrease leads firms to increase their production until actual output is again equal to intended spending. This means that the IS-‐curve is downward sloping. 13.c. Each point on the LM-‐curve represents an equilibrium in the money sector. Therefore the LM-‐curve can be derived by setting real money supply equal to real money demand, that is, M/P = L ==> 500 = (0.25)Y -‐ 62.5i ==> Y = 4(500 + 62.5i) ==> Y = 2,000 + 250i. LM-‐curve 13.d. The LM-‐curve shows all combinations of the interest rate and level of output such that the money sector is in equilibrium, that is, the demand for real money balances is equal to the supply of real money balances. An increase in income will increase the demand for real money balances. Given a fixed real money supply, this will lead to an increase in interest rates, which will then reduce the quantity of real money balances demanded until the money sector is again in equilibrium. In other words, the LM-‐curve is upward sloping. 13.e. The equilibrium levels of income and the interest rate are determined by the intersection of the IS-‐curve with the LM-‐curve. At this point, the expenditure sector and the money sector are both in equilibrium simultaneously. From IS = LM ==> 4,250 -‐ 125i = 2,000 + 250i ==> 2,250 = 375i ==> i = 6 ==> Y = 4,250 -‐ 125*6 = 4,250 -‐ 750 ==> Y = 3,500 Check to verify (optional): Y = 2,000 + 250*6 = 2,000 + 1,500 = 3,500 Graph is in the next page. 14. Refer to question 13. a. What is the value of the multiplier which corresponds to the simple multiplier (with
taxes) of Chapter 10 (in the latest edition of the book)?
b. By how much does an increase in government spending of ∆ increase the level of
income in this model, which includes the money market?
c. By how much does a change in government spending of ΔG affect the equilibrium
d. Explain the difference between your answers to parts (a) and (b).
14.a. As we have seen in 13.a., the value of the expenditure multiplier is α = 2.5. This multiplier is derived in the same way as in Chapter 10. But now intended spending also depends on the interest rate, so we no longer have Y = αAo, but rather Y = α(Ao -‐ bi) = (1/[1 -‐ c + ct])(Ao -‐ bi) ==> Y = (2.5)(1,700 -‐ 50i) = 4,250 -‐ 125i. 14.b. In the IS-‐LM model, an increase in government purchases (G) will have a smaller effect on output than in the model of the expenditure sector used in Chapter 10 (new edition), in which interest rates are assumed to be fixed. This can be demonstrated most easily with a numerical example. If government purchases are increased by ΔG = 300, the IS-‐curve shifts parallel to the right by ΔIS = (2.5)(300) = 750. Therefore, the equation of the new IS-‐curve is: Y = 5,000 -‐ 125i. From IS' = LM ==> 5,000 -‐ 125i = 2,000 + 250i ==> 375i = 3,000 ==> i = 8 ==> Y = 2,000 + 250*8 ==> Y = 4,000 ==> Δ Y = 500 When interest rates are assumed to be fixed, the size of the expenditure multiplier is α = 2.5, that is, (ΔY)/(ΔG) = 750/300 = 2.5. However, when interest rates are allowed to vary, the size of the multiplier is reduced to α1 = (ΔY)/(ΔG) = 500/300 = 5/3 = 1.67. 14.c. An increase in government purchases by ΔG = 300 causes a change in the interest rate from io = 6 to i1 = 8, that is, by 2 percentage points. Therefore government spending has to change by ΔG = 150 to increase the interest rate by one percentage point. 14.d. The simple multip...
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