Aggregate Demand and Output -- Keynesian Macroeconomics (Chapter 13)

Problem 1. Acme Manufacturing
Acme has decided to (I omit the last three zeros):

  1. Produce $ 4,000 worth of goods -- we assume they in fact do so.
  2. Invest $ 1,500 in plant and equipment -- since contracts for the purchase of this must be signed in advance, we are safe in assuming that they will in fact do so (the text statement of the problem does not make this clear).
  3. Does NOT plan to add or subtract anything from its inventory of $ 500. Whether they will or will not actually invest in inventory depends on whether or not they sell the entire $ 4,000 worth of goods they produce.

Note that Change in inventories = Production - Sales

Hence:

  1. If Acme sells only $ 3,850 of the $ 4,000 produced, their inventories rise by $ 150 . We can expect that they will reduce production next year to allow the inventories to run down -- and if every firm does likewise, the economy will contract
  2. If Acme sells all that is produced, there is no change in inventories With no change in inventory, there is no reason to change the production target. If Acme is a typical firm, next year GDP will remain unchanged
  3. If Acme sells $ 4,200 but has only produced $ 4,000, they will reduce inventories by $ 200. In order to continue meeting orders, they must replenish their inventories -- which means they must expand production. If Acme is a typical firm, the economy will expand .

Problem 2. Simpsons and the MPC
Income Taxes Disposable Income Consumption dC/dYD
25,000 3,000 22,000 20,000 xxxxx
27,000 3,500 23,500 21,350 0.90
28,000 3,700 24,300 22,070 0.90
30,000 4,000 26,000 23,600 0.90


The calculation which is a bit hard to see is the MPC = dC / dYD
Change in consumption divided by the change in disposable income
For example, in the second row, disposable income increases by 1,500 and consumption increases by 1,350 (above their values in the first row).
The Marginal Propensity to Consume is therefore
MPC = 1,350 / 1,500 = 0.90
The same calculation can be repeated for the next row:
Change in disposable income = 24,300 - 23,500 = 800.
Change in consumption = 22,070 - 21,350 = 720.
Marginal propensity to consume = 720 / 800 = 0.90

Problem 2.II -- The Simpsons, Part II

  1. Given that the MPC is 0.90, we know that the Simpson's consumption function is of the general form:
    C = Co + 0.9 YD

    This equation must hold for any specific point in the table. For example:
    If disposable income is 22,000, it must predict a consumption of 20,000
    This yields 20,000 = Co + .9 (22,000) = Co + 19,800
    For this to be the case, we must have Co = 200 , and the consumption function of the Simpsons is therefore:

    C = 200 + 0.9 YD

  2. If the Simpson's disposable income is $ 27,000, we use the consumption function to find:

    C = 200 + 0.9 (27,000) = 24,300

  3. Given the lottery win, their consumption function shifts up by $1,000 In terms of the algebra, their consumption function becomes:

    C = 1,200 + 0.9 YD

    Which means, for example, that at disposable income of $ 27,000, they will consume $ 25,300.

    Problem 3 -- Keynesian equilbrium: algebraic analysis
    We are given the following information (I use a star rather than the text bar over a variable to denote the exogenously given variables):

    Note that we are given potential rather than actual output. To find actual output, we use the
    GDP identity: Y = C + I + G + NX
    substituting in the consumption function but keeping other values for the moment in starred form:
    Y = 1,800 + 0.6 Y - 0.6 T + I + G + NX

    or 0.4 Y = 1,800 - 0.6 T + I + G + NX

    And to get the multiplier , we multiply through by 2.5 (or divide through by 0.4):
    Y = 4,500 - 1.5 T + 2.5 I + 2.5 G + 2.5 NX

    Once we have this equation, we can easily solve for equilibrium output and for the impact of any change.

    Problem 4 -- Keynesian equilibrium output

    To solve for equilibrium, simply substitute the given values into the equation above:


    Y = 4,500 - 1.5 T + 2.5 I + 2.5 G + 2.5 NX


    Y = 4,500 - 1.5 (1,500) + 2.5 (900) + 2.5 (1,500) + 2.5 (100)


    Y = 4,500 - 2,250 + 2,250 + 3,750 + 250 = 8,500

    Note that Keynesian equilibrium output is below potential

    To be precise, the output gap is 9000 - 8500 = 500, or about 5 percent of overall output.
    Okun's law tells us that an output gap of 5 percent should correspond to an unemployment rate 2.5 percent above the natural rate of unemployment.
    Given the textbook statement that the NAIRU is 4 percent, we should therefore expect an unemployment rate of 6.5 percent in this economy.

    Problem 5 -- Keynesian equilbrium: multiplier analysis
    Given the equation from the last problem


    Y = 4,500 - 1.5 T + 2.5 I + 2.5 G + 2.5 NX

    we can easily identify the multipliers:

    1. Tax multiplier: - 1.5
      Every dollar increase in taxes will reduce output by $ 1.50. Hence a $ 100 decrease in taxes will increase output by $ 150.
    2. Investment multiplier: 2.5
      Every dollar increase in investment spending will increase output by $ 2.50. Hence a $ 100 decrease in investment would reduce output by $ 250.
    3. Government spending multiplier: 2.5 Every dollar increase in government spending will have the same short-term impact on the economy as a dollar increase in investment. Hence a $ 100 increase in government purchases would increase output by $ 250 -- and would exactly offset the impact of the decline in investment spending in the other part of the problem.

    Problem 6 -- Multiplier Analysis.
    Consider the following economy:

    C = .75 (Y - T)

    Y = C + I + G (no net exports)

    Y = .75Y - .75 T + I + G

    .25 Y = - .75 T + I + G

    Y = - 3 T + 4 I + 4 G


    1. If planned investment declines by 150, GDP will decline by 600.
      In the text language, there will be a recessionary gap of 600.
    2. A increase in government purchases of 150 would exactly offset the recessionary gap.
    3. A cut in taxes would have to be larger. Since the tax multiplier is -3, it would take a tax cut of 200 to make up the recessionary gap.
    4. Note that if taxes increased by 600 and government spending increased by 600, there would be a net expansionary effect: the tax increase would reduce output by 3 times the amount of the tax increase, and the new government spending uld increase output by 4 times the amount of the increase. Hence, the total increase in GDP would be
      Increase in GDP = 4(600) - 3(600) = 600

      This illustrates what is known as the balanced budget multiplier .

      Problem 7 -- Keynesian equilbrium
      Basic equations for the economy:

      C = 40 + 0.8 (Y - T)

      Y = Co + .8 Y - .8T + Ip + G + NX

      .2Y = Co - .8T + Ip + G + NX

      Y = 5 (Co + Ip + G + NX) - 4T

      Y = 5 PAE - 4T , where PAE = planned aggregate expenditure

      Ip = 70, G = 120, NX = 10, T = 150

      Y = 40 + .8(Y - 150) + 70 + 120 + 10

      Y = 120 + .8 Y

      .2Y = 120

      Y = 5 (120) = 600

      This is 20 above the equilbrium output of 580; hence, government purchases must be reduced or the taxes increased.
      Note from the formula in bold that the government multiplier is 5 and the tax multiplier is 4; hence you can either:

      • Reduce government spending by 4 , or
      • Increase taxes by 5

        If the full-employment output were 630, we would have a recessionary gap of 30; hence we would want to increase government spending by 6 or cut taxes by 7.5.

        Problem 8 -- Keynesian equilbrium
        Collect the information given into the Keynesian equilibrium equation:


        Y = 3000 + 0.5 Y - 0.5 T + 1500 + 2500 + 200


        Y = 7200 + 0.5 Y - 0.5 (2000) (substituting in taxes)


        Y = 6200 + 0.5 Y (substituting in taxes)


        0.5 Y = 6200


        Y = 12,400 (multiplying by 2)

        Note that Keynesian equilbrium output is above potential GDP by 400.

        In order to close the output gap, government spending could be reduced by 200 or taxes increased by 400.

        Problem 9 -- Keynesian equilbrium
        The economy in this problem is identical to the last except for the fact that
        NX = 0.
        Y = 3000 + 0.5 Y - 0.5 T + 1500 + 2500 + 0


        Y = 6000 + 0.5 Y (substituting in taxes)


        0.5 Y = 6000


        Y = 12,000 (multiplying by 2)

        When net exports go down by 200, Keynesian equilbrium GDP goes down by 400.
        Note in this case there is no particular problem -- Keynesian equilbrium has simply gone down to potential, and has not generated higher than normal unemployment.

        Problem 10 -- Income taxes and automatic stablizers
        A numerical example might be easier than the algebraic approach in the text:

        Assume the consumption function is C = 500 + 0.8 (Y - T).

        Then, if taxes = 0.25 Y, we have C = 500 + 0.8 (Y - 0.25 Y) = 500 + .8 (.75 Y)
        Hence, the consumption function is: C = 500 + 0.6 Y

        You now have a consumption function to substitute into the Keynesian equilibrium equation:

        Y = 500 + 0.6 Y + 1,500 + 2,000 + 0 (using the values in the text problem)
        Y - 0.6 Y = 4000

        0.4 Y = 4000

        Y = 2.5 (4000)

        Note that the multiplier is 2.5, not the 5 we met in the last problem -- despite the fact that the marginal propensity to consume is the same in both problems.