Answers to Assignment 4
Chapter 11:

Problem 1:

  mpc= DC/DDI = change in consumption spending divided by the change in real disposable income.

                     C - consumption spending
                     DI - real disposable income

         For this problemm, note in the table that every change in disposable income of 10 leads to a change
           in   consumption of 5.  Therefore, the mpc = 5/10 = 0.50

                   Realizing that C is linear, we know the slope and intercept.  So, using the linear relation

                          C = a + mpcDI  we get the equation   C = 5 + 0.5DI

                                 a - autonomous consumption or consumption that is independent of DI.
                                 mpc- marginal propensity to consume.

  Remember, DI = C+S.  Therefore, S = DI-C  .  The book asks you to create a table like the one below:

DI  ($millions)     C ($millions)      S  ($millions)
         We can also use DI = C+S to generate the savings equation:

                      S = -a+mpsDI   where  a - autonomous consumption, and -a = autonomous savings

         or           S = -5+0.5DI for this data set.

                        mps = marginal propensity to save = DS/DDI

                        Note:    1 = [DC/DDI] + [DS/DDI]

                                          1 = mpc+mps

                  Therefore, for this problem, mps = 1- 0.5 = 0.5.

                 Note, we can also read this from the table.

Problem 2:

     Remember that AE = A0 + mpcY

                 where A = the intercept of the AE line =  all autonomous spending.  In this case, I and G are
                    autonomous.      mpc = slope of AE line.    Also, with no taxes, DI = Y.  All in $billions.

 A.    From the graph, A0 = $2.0 billion.

 B.      From the graph, slope of AE line is 3.6/6.0 = 0.6

 C.      From the graph, when GDP is $6 billion, AE = $5.6 billion.

 D.     From this information, we know the equation for AE:  AE = $2+0.6Y

          Therefore, if Y = $4, then AE = $4.4  [since AE(4) = $2+0.6($4) = $2+$2.4 = $4.4]

                    This means that AE is greater than Y when Y = $4.  So, there is an unplanned inventory reduction.

 E.  If Y = $6, then AE = $2+0.6($6) = $5.6, as we saw.  There is an unplanned increase in the inventory stock.

 F.  The multiplier:  where k = [DY/DA0].

The expenditure multiplier measures the effect on equilibrium Y of a change in autonomous spending.

The numerical value of k = 1/[1-mpc] = 1/mps.

For our problem, k = 1/[1-0.6]  = 1/0.4 = 2.5

Problem 3
     We have the following information, ($billions):

      autonomous consumption = a = $100
      mpc = 0.9
      I = $460
      G = $400
      Tx = $400  (we are told these are autonomous taxes, since they do not vary with income.)

A.     Therefore, C = $100+0.9DI   where DI = Y-Tx      DI - disposable income, Y - real GDP

             subsituting for DI and then Tx,   C = $100+0.9(Y-Tx) = $100-0.9($400)+0.9Y

                                                                        or    C = -$260+0.9Y

B.    We also know that AE = C+I+G, or in this case, AE = -$260+0.9Y+$460+$400

                        simplifying:                   AE = $860-$260 + 0.9Y

                          subtracting                  AE =  $600 +0.9Y

C.  At equilibrium,  AE = Y.  So, we get:

                     $600+0.9Y = Y

                     Equilibrium Y = $600/(1-0.1) = $600(10) = $6000

         Alternative approach:   Calculate total autonomous spending:  a-mpcTx+I+G = $100-0.9($400)+$460+$400 = $600

                                                 Calculate the multiplier value:  k = 1/(1-mpc).  For this problem, k = 10.

                                                 Equilibrium Y = kA = 10($600) = $6000

D.    If I falls from $460 to $360, but nothing else changes, we can re-do the AE function, and then solve for equilibrium:

               AE = C+I+G = -$260+0.9Y+$360+$400 = $500+0.9Y

              Solve for equilibrium:  AE = Y

                                             $500+0.9Y = Y

                                               Equilibrium Y = $500/(1-0.9) = 10($500) = $5000.  Therefore, the $100 drop in I lead to a $1000 drop in Equilibrium Y.

             Alternative approach:   The $100 drop in I leads to a drop in total autonomous spending of $100.  Therefore, since we know that
                  the multiplier k = 10, we know:

                                           DEquilibrium Y=k DA = k(-$100)=10(-$100)= -$1000 .  Therefore, Equilibrium Y falls from $6000 to $5000.

Problem 4

                The answer to this question will relate the AE analysis in Problem 3 to the AD-AS diagram we used earlier.

                We will start from the answer to Problem 3, part C.  At that point, we had an equilibrium Y of $6000, with I = $460..

                Now, if I rises from $460 to $560, we can re-calculate AE, and solve for equilibrium.  But, we already know, using the alternative approach outlined in
                  Problem 3, Part D, that a $100 increase in I will lead to a $1000 increase in equilibrium Y through the multiplier effect.

                                           DEquilibrium Y=k DA = k($100)=10($100)= $1000 .  Therefore, Equilibrium Y rises from $6000 to $7000.

We can see the results of this in the AE diagram, below.

As is shown in the graph, the equilibrium rises from Y = $6000 at point E, to Y = $7000 at point F.

The lines shift in AE(Y) to AE(Y)' is caused by the increase in I, as shown by the increase in autonomous spending from
A to A'.

 As AE shifts in the above diagram, AD shifts to the right in the AD-AS diagram, as shown below:


Point F in the AD-AS diagram above correspond to point F in the AE diagram.  Notice that the new equilibrium value in the AE diagram
is not a short-run equilibrium in the AD-AS diagram; the AE=Y equilibrium shows a point of aggregate demand, not full equilibrium in AD-AS space.

A.  The quantity of real GDP demanded rises to $7000.
B.  In the short run, real GDP does not rise to $7000, but to the level at point G in the above diagram.  Price increases begin, and that cuts the quantity of real GDP demanded somewhat over what it would have been, $7000, if the price level had been constant at P = 100.
C. In the long-run, prices will rise unitl the economy reaches long-run equilibrium at point H, with a price level higher than P = 100, but the same real GDP as we
    had originally, $6000.
D.  Price level rises in the short-run to the level at Point G.
E.  Price level rises in the long-run to the level at Point H.  At point H, the real wage is the same as it was originally, at point E.

     NOTE:  Though you were not asked this, you should realize that the horizontal axis on both of the above diagrams in the same, real GDP.
Therefore, to reach short-run equilibrium at point G in the lower diagram, we must have the AE(Y)' in the upper diagram shift back somewhat as the price level rises.
The higher price level causes the quantity demanded to drop a bit, from F to G.

Chapter 12:

Problem 1:

    Data for the problem:  mpc = 0.9
       I = $50
       G = $40
       Tx = $40

   We know the reduced form equation for finding equilibrium real GDP is:

                 Equilibrium Y = kA  and that DY = kDA

                 If the mpc=0.9, then k = 1/(1-0.9) = 10

     A.  If G drops from $40 to $30 billion, the A drops by $10 billion, since

             A = a+I+G-mpcTx for this problem; the drop in G is a drop in A.

            Therefore, the AE line shifts down by $10 billion in this case.

     B.  The multiplier is still k = 10.  Therefore, the decline in A of $10 billion would lead to a drop in
            equilibrium Y by $100 billion, since DY = kDA.   [$100 = 10[$10 billion]]

     C.  In this case, if G is maintained at $40 billion, but Tx is cut to $30 billion, we have:

                DA = -mpcDTx = -0.9[-$10 billion] = $9 billion

                The drop in taxes of $10 billion leads to an increase in autonomous spending of $9 Billion, as
               consumers increase their consumption spending.  The AE line shifts up by $9 billion.

      D.  Using the usual formula,  DY = kDA, we have DY = 10[$9 billion] = $90 billion.

             The book asks for the lump-sum tax multiplier.  That can be found this way:

                  DY = kDA = k[-mpcDTx] = [-mpc/(1-mpc)]DTx

                       The lump-sum tax multiplier is, then, -mpc/(1-mpc) = -0.9/(1-0.9) = 9

                      There is really no need to calculate this term, since using DY = kDA we can
                         get the result.  We just need to remember how A changes when Tx changes.

          E.  If the government cuts Tx by $10 billion and G by $10 billion at the same time, we get a
                balanced budget multiplier effect.

                Note, if G and Tx both change, then the change in A will be:

                      DA= DG-mpcDTx     but, DG = DTx = -$10 billion in this case.

               So,  DA = -(1-mpc)$10 billion = -$1 billion.   Therefore, the net effect on aggregate spending is
                  drop of $1 billion when both Tx and G are cut by $10 billion.

                     Notice what this implies [substituting for DA and then k]:

                      DY = kDA = k[(1-mpc)(-$10 billion)] = [1/(1-mpc)][1-mpc][-$10 billion] = - $10 billion.

                       This is the balanced budget multiplier effect:  With lump-sum taxes only, if we change G and Tx
                         by the same amount, then Y changes by that same amount.  The multiplier effect is reduced to
                         1 = (1-mpc)/(1-mpc).

Problem 2:

     A. and B.  As we saw above, DY = kDA, so if A falls by $10 billion, Y will fall by $100 billion.  Therefore, the AE line would shift down by  $10 billion.  This means that  the AD curve will shift left by $100 billion, via the multiplier effect.

     C.  With an upward sloping short-run aggregate supply curve, real GDP will increase by less than $100 billion at the new short run equilibrium.

     D.  In the long run, real GDP will return to its original level, as resource owners bid-up nominal wages to restore the  real wage.

     E.  In the short run, the price level rises.

     F.  In the long run, the price level rises.

Problem 3:

      A.  At point a, P =150, real GDP = 600

      B.  Immediate effect is a move to point b, P = 160, real GDP = about 650.

      C.  System moves to point d, a lower price than at b, a higher real GDP than at b.

       D.  The continued shifting of the SAS curves leads the system to point e, a lower price level than at
            point a, and a higher real GDP than at point a.     However, this assumes a large supply side effect from
            the tax cut.

3.  3.  We are given the following data on consumption spending in a simple economy.

Disposable Income      Consumption Spending        Saving
        0                                   $ 1.0 million        -$1.0 million
$ 1 million                             $ 1.75 million       -$0.75 million
$ 2 million                                2.5                      -0.50
$ 3 million                                3.25                    -0.25
$ 4 million                                4.0                       0.0
$ 5 million                                4.75                      0.25
$ 6 million                                5.50                      0.50
$ 7 million                                6.25                      0.75
$ 8 million                                7.00                      1.0
$ 9 million                                7.75                      1.25
$10 million                               8.50                      1.50

A.  Define MPC.  Assuming it is constant, calculate it and use it to fill in the rest of the table.  Define the multiplier.  What is its value for this example?

    Marginal Propensity to Consumer: The change in consumption spending brought about by a change in disposable income.

         mpc = 0.75

B.   There are no taxes or transfer payments or depreciation in this model, for now.  But, we have the following additional data:

Investment Spending = $ 2 million

Government Spending = $ 0.5 million  {Only Federal, no State and
Exports = $ 3 million

Imports = $ 1 million

Given this information, what is the equilibrium level of GDP?

     Autonomous spending = A = a+I+G+X-IM

          a = autonomous consumption spending = $1 million, from the table. Therefore, using the other data
                given in the problem:

            A = 1+2+0.5+3-1 = 5.5

             k = 1/(1-mpc) = 1/(1-0.75) = 4

          Equilibrium Y = kA = 4[5.5] = $22 million.

C.  Economists working for the Department of Commerce predict that next year, imports will rise by $ 2 million, and exports will rise by only $0.5 million.  Assuming that investment and government spending remain the same, and that the consumption function remains the same:

   a. What will be the new equilibrium level of GDP next year if these economists have forecasted correctly?

          DA = DX - DIM = 0.5 -2.0 = -$1.5 million.

             Since DY = kDA = 4[-$1.5 million] = -$6 million.  Therefore, the new level of real GDP demanded is

                  $22 million -$6 million = $16 million.

   b.  What will be the level of consumption spending at this new equilibrium GDP?  What will be the new level of savings? Is it true that savings equal investment?  Explain.

       C = 1+0.75Y = 1+0.75($16 million) = $13 million.

       S = -1+0.25Y = -1+0.25($16 million) = $3 million.

       No,  I = S+(Tx-G)-(X-IM)  as before.  But, Tx = 0 in this problem.

                 I = $3 - 0.5 -0.5 = $2, as we see above.

             S+(Tx-G)-(X-IM) equals the resources available for use by investors; ie. total savings in the open
                                          economy.  In equilibrium, investment equals this total level of savings.

D.  [To answer this question, we will begin with the data and the results in (B), and ignore what went on in (C).]

     Most members of Congress agree that the Federal government has run a deficit for long enough, and that steps must be taken to eliminate it.  Two plans have been discussed.  One says to cut government spending from its current level of $0.5 million to zero.  The other says taxes should be raised until the budget is balanced;  assuming only autonomous taxes (sometimes called lump-sum taxes), this would mean that taxes must be raised from their current level of zero to $0.5 million.

     a.  Test each policy by finding the new equilibrium level of GDP each would bring about.  Why do the equilibrium levels of GDP differ?

      If G is cut from $0.5 to zero, then DA = $0.5 million, and DY = kDA = 4[0.5] = $2 million.
           So, the new equilibrium would be $20 million.

     If Tx is raised from 0 to $0.5 million, then DA = -mpcDTx = -0.75[$0.5 million] = -$0.375 million.

       Therefore,    DY = kDA = 4[-$0.375 million] = - $1.5 million. So, the new equilibrium would be $20.5 million.

     The equilibria differ because taxes act through the mpc on consumption.  An increase in taxes of $0.5 million leads to a reduction in autonomous spending of only $0.375 million, or 75% of the increase in taxes.

      b. Discuss the implications of each policy for the composition of equilibrium GDP, i.e., how does the proportion of various goods produced and sold differ under each scheme, and who is buying them?.

      In the first case, when G falls to zero,  we have :  C = 1+0.75Y = 1+0.75[$20 million] = $16 million.

       In the second case, when Tx is raised to $0.5 million, C = 1+0.75[Y-Tx} = 1+0.75[$20.5- $0.5] = $16 million.

          Therefore, consumers have the same level of spending in each case.  But in the second case, the government
        purchases $0.5 million in output, paid for by taxes.  In the second case, the government has no role in the economy.

4.  We are given the following data for a simple economy.

 [In millions of dollars]
Consumption Spending = C = a(r)+0.8*DI
a(r)= 100-5r    where r - interest rate in percents
                [a(r), autonomous consumption spending, is the component of consumption spending   that does not depend on DI, but depends on the rate of interest.]

Investment = I(r) = 200 - 10r
Autonomous Government Spending = G = 100
Autonomous Taxes = Tx = 50
Autonomous Exports= X = 200
Autonomous Imports = IM = 175
r = 10 percent

A.  For every one percentage point increase in the interest rate, how much does consumption spending change?  In what direction?  Give two examples of this type of consumption spending.

    Every time r rises by 1, a(r) falls by $5 million.

      Automobiles, Household Appliances.

B.  Use the information given above to write total autonomous spending as a function of the interest rate.
Give its value when r = 10.

    A(r) = a(r)+I(r)+G+X-IM-mpcTx = 100-5r + 200-10r +100+200-175-0.8(50)
            = 385-15r

    A(r=10) = 385-150 = 235
C.  Calculate the equilibrium level of GDP.

    Y=kA(r) = kA(r=10) = 5[235] =     $1175

D.  If interest rates rise from 10 percent to 12 percent, find the new level of autonomous consumption, the new level  of investment, and the new value of autonomous spending?  What is the new equilibrium level of GDP?

    a(r=12) = 100-5(12) = 40
    I(r-12) = 200-10(12) = 80
   A(r=12) = 385-15(12) = 180 = 205

    Y = kA(r=12) = 5[205] = $1025

E.  If interest rates fall to 8 percent, find the new level of autonomous consumption, the new level of investment, and  the new level of autonomous spending.  What is the new equilibrium level of GDP?

    a(r=8) = 100 -5(8) = 60
    I(r=8) = 200 - 10(8) = 120
   A(r=8) = 385-15(8) = 265

    Y = kA(r=8) = 5[265] = $1325

5.   We are given the following information about an economy:
 [In millions of dollars]

Consumption Spending = C = a(r)+0.8*DI
a(r)= 200-10r    where r - interest rate in percents
   [a(r), autonomous consumption spending, is the component of consumption spending that does not depend on DI, but depends on the rate of interest.]

Investment Spending = I(r) = 300 - 15r
Autonomous Government Spending = G = 500
Autonomous Taxes = T0 = 50
 Autonomous Exports= X = 200
Autonomous Imports = IM = 175
r = 10 percent
DI = Y - Taxes
Y = Real GDP

A.  Find equilibrium real GDP for the data given above.

     A(r) = a(r)+I(r)+G+X-IM-mpcTx = 200-10r+300-15r+500+200-175-0.8(50)

              = 985-25r

     A(r=10) = 985-250 = 735

       Y = kA(r=10) = 5[735] =  $3675

B.  What is the state of the government budget?  Calculate the surplus or deficit.

       Tx-G = 50-500 = -450.  There is a deficit of $450 million.

C.  The Congress decides to institute an income tax in this economy to balance the government's budget.
Government spending will not change, but Instead of having only autonomous taxes, the tax level will be
determined by the following equation:

    Taxes = T0 + t*Y                        Where T0 = 50, as above, and the tax rate, t = 0.25

      a.    Write down the equation for the multiplier with an income tax.

            With an income tax, the expenditure multiplier becomes k' = 1/[1-mpc(1-t)]

      b.  For this particular problem, what is the value of the expenditure multiplier once the income tax is implemented?

               k' = 1/[1-mpc(1-t)] = 1/[1-0.8+0.2]=2.5   In this problem, the expenditure multiplier is cut in half by the income tax.

      c.   Find the equilibrium real GDP after the imposition of the income tax.

             Y =k'A(r) = 2.5[735] = $1837.5 million.  There is no change in any autonomous variables, but the slope of the AE
                                                                          line is reduced.

      d.  How did the imposition of the income tax change the government's budget balance;  did the income tax meet            the policy goal set by the Congress? Explain.

          Under the income tax,  Y = $1837.5  so Tx = 50 + 0.25($1837.5) = 50+459.375 = $509.375.

              There is no a slight surplus, so the goal has been nearly met.

e.   After the imposition of the income tax, there is a surge in Autonomous Exports, which rise from 200 to 500.
         Calculate the change in equilibrium Y this will bring about.

        If exports surge, then DA(r) = 300.  The new equilibrium is found by:  DY=k'DA = 2.5[300] = $750

            So, the new equilibrium Y = $2587.5 million

   f.  After the surge in exports and the change in equilibrium Y you calculated above, calculate the new level of
           tax revenue, and the new state of the government's budget balance.

           Tx = 50+0.25[2587.5] = 696.875
            G = 500

           Tx-G = 697-500 = 197  in surplus.