# multiplier dirivation

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DERIVATION OF A SIMPLE INCOME MULTIPLIER
Let’s start with a very simple closed economy, where GDP or Y = C + I + G only and there are no exports (X) or imports (M).
(1) Substituting the Consumption Function for C:
Y = (co + c1 YD) + I + G = (co + c1 {Y - T}) + I + G = co + c1Y – c1T + I + G

(2) Collecting terms in Y on the left hand side, and factoring out Y:
Y – c1Y = Y (1 – c1) = co - c1T + I + G

(3) Placing (1 – c1) back on the right hand side, we have derived an expression for equilibrium GDP or Y:
Y = (1/{1 – c1}) (co – c1T + I + G)

(4) The expression in the first set of parentheses
Y = (1/{1 – c1}) (co – c1T + I + G) is the income multiplier.
(If c1 is the marginal propensity to consume, and \$1 is spent in the economy, the person who gets that dollar will spend c1 of it, the person who receives that c1 will spend c1(c1) or c12, and so forth. Without deriving it, it turns out that 1 + c1 + c12 + c13 + …….. + c1n converges to 1/{1 – c1}, or the multiplier.) Let’s say we estimate a consumption function from the available data, and the marginal propensity to consume is estimated to be c1 = .80 (meaning that, on average, people in the economy spend \$0.80 out of every \$1.00 they earn). Then the multiplier would be 1/{1 - .8} = 1/.2 = 5.

That’s high for a multiplier. But if that were the case, it would have significance in the formulation of fiscal policy. If government policymakers wanted GDP (Y) to rise by \$100 billion, they would only have to increase government spending G by \$20 billion -- (5 x 20) = 100.

If they wanted to achieve the same result with a tax cut, they would have to cut taxes by \$25 billion. Why would they have to cut taxes more than they would have to increase spending? The entire \$20 billion spending increase gets injected into the economy, but part (20%) of the tax cut is saved and not spent.

But the bottom line is that if any one of the four terms in the second set of parentheses of Equation (4) above

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