Problem 1. Acme Manufacturing
Acme has decided to (I omit the last three zeros):
Hence:
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
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:
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):
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:
Problem 6 -- Multiplier Analysis.
Consider the following economy:
Problem 7 -- Keynesian equilbrium
Basic equations for the economy:
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:
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.