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## Intuition

The intuition comes from the fact that the marginal propensity to consume (MPC) is positive. MPC is the money people spend when they get an extra dollar of income. When MPC = 0.8, for example, when people gets an extra dollar of income, they spend 80 cents of it. So the Keynesian multiplier works as follow, assuming for simplicity, MPC = 0.8. Then when the government increases expenditure by 1 dollar on a good produced by agent A, this dollar becomes A's income. As MPC = 0.8, A will spend 80 cents of this extra income on something is wants to consume. Suppose A spends the 80 cents on a good produced by B, then B would have an extra income of 80 cents. B would then spend 0.8 of this 80 cents, ie, 64 cents, on something else. This 64 cents becomes someone else's income, and this someone will spend 0.8 of it. The process repeats itself. The GDP added to the economy is the sum of all the spending, 1 + 0.8 + 0.64 + 0.512 + ... which has a larger effect than the 1 dollar that the government originally spent. In other words, the government spending is "multiplied".

Mathematically, the sum 1 + 0.8 + 0.64 + ... is a geometric series. When you sum them up, it takes the form ${\frac {1}{1-MPC}}={\frac {1}{1-0.8}}=5$ . The effect of the government spending is multiplied 5 times, and thus the multiplier is ${\frac {1}{1-MPC}}$ ## Algebra representation

The key to solving these question is to isolate GDP (Y) on one side. That gives us the multiplier. We will see an example below:

Q Consider the following multiplier model:

C = 100 + 0.8Yd; I = 200; T = 0; G = 0; NX = 0 (Yd is disposable income)

1. What is the aggregate expenditure? Show the equation that describes it.
2. What is the equilibrium condition? Derive the equilibrium output.
3. Suppose that, for some reason, the level of GDP is 2000. What will the level of involuntary inventory accumulation be?
4. If Government expenditure (G) rises to 100, what will the effect be on equilibrium output? (Note: assume that taxes do not rise to cover the increase G)
5. What is the value of the government spending multiplier? Interpret its value.
6. Suppose now that investment changes to: I = 50 + 0.05Y. Find now the value of the government multiplier. Is it larger or lower than in part 5? Why?

A The steps are:

• $AE=C+I+G+NX=100+0.8Y_{d}+200+0+0=300+0.8Y_{d}$ • The equilibrium condition is $Y=AE$ , output equals aggregate expenditure. Therefore:

{\begin{aligned}Y^{*}&=300+0.8Y^{*}\\0.2Y^{*}&=300\\Y^{*}&=1500\end{aligned}} • If Y = 2000, then aggregate expenditure is:

{\begin{aligned}AE&=300+0.8Y=300+1600=1900\\Y&>AE\end{aligned}} In this case firms will increase their inventories by 100 and unplanned (inventory) investment is 100.

• Now, {\begin{aligned}Y=AE=100+0.8(Y-T)+200+100+0=400+0.8Y\\Y^{*}={\frac {1}{1-0.8}}\times 400=2000\end{aligned}} • The government spending multiplier is 1/(1-0.8) = 5, which means that for every $1 increase in government spending, the equilibrium level of output increases by$5.
• Again using the equilibrium condition, we have:
{\begin{aligned}Y^{*}&=C+I+G+NX=100+0.8(Y^{*}-T)+50+0.05Y^{*}+G\\(1-0.85)Y^{*}&=150+G\\Y^{*}&={\frac {1}{0.15}}\times 150+{\frac {1}{0.15}}\times G\\Y^{*}&=1000+{\frac {1}{1-0.8}}\times G\end{aligned}} .


Thus the government multiplier is now 1/0.15 > 4. The multiplier is larger because as G increases we have all the previous effects plus now investment also increases, adding to the overall effect of increased government spending. Investment is not exogenous anymore.

## Notable cases for Keynesian multiplier

Transfer/lump-sum tax T

1. Disposable income with lump-sum tax = Y - T
2. Consumption with lump-sum tax: C = a + b(Y - T)

When tax depends on income at a rate t

1. Disposable income with tax rate t = Y - tY or (1-t)Y
2. Consumption with tax rate t: C = a + bY(1-t)

Note that in the case with both the lump-sum tax and the tax at rate t, the GDP becomes:

$Y={\frac {a}{1-b+bt}}-{\frac {b}{1-b+bt}}T+{\frac {1}{1-b+bt}}(I+G)$ Balance budget condition: set G = T