Fiscal Policy Appendices

Fiscal Policy Appendices Appendices to Chapter 10 of Essentials of Economics in Context APPENDIX A: AN ALGEBRAIC APPROACH TO THE MULTIPLIER, WITH A LUMP-SUM TAX A lump-sum tax is a tax that is simply levied on an economy as a flat amount. This amount does not change with the level of income. Suppose that a lump-sum tax is levied in an economy with a government (but no foreign sector). Consumption in this economy is: C = C + mpc Yd (the consumption function from Chapter 9, but using after-tax or disposable income in the formula). Since disposable income is: Yd = Y – T + TR we can write the consumption function as: C = C + mpc (Y – T + TR) Thus aggregate expenditure in this economy can be expressed as: AE = C + I + G = C + mpc (Y – T + TR) + I + G = ( C – mpc T + mpc TR + I + G) + mpc Y By substituting this into the equation for the equilibrium condition, Y = AE, we can derive an expression for equilibrium income in terms of all the other variables in the model: Y= ( C – mpc T + mpc TR + I + G) + mpc Y Y - mpc Y = ( C – mpc T + mpc TR + I + G) (1-mpc) Y = ( C – mpc T + mpc TR + I + G) 1 𝑌= ( C – 𝑚𝑝𝑐 T + 𝑚𝑝𝑐 𝑇𝑅 + 𝐼 + 𝐺) (1 − 𝑚𝑝𝑐) If autonomous consumption, investment, or government spending change, these each increase equilibrium income by mult = 1/(1 – mpc) times the amount of the original change. If the level of lump-sum taxes or transfers changes, these change Y by either negative or positive (mult)(mpc) times the amount of the original change. To see this explicitly, consider the changes that would come about in Y if there were a change in the level of the lump sum tax from T0 to a new level, T1, if everything else stays the same. We can solve for the change in Y by subtracting the old equation from the new one: [AUTHOR NAME] 1 1 ( C + 𝐼 + 𝐺 – 𝑚𝑝𝑐 𝑇̅1 + 𝑚𝑝𝑐 𝑇𝑅) (1 − 𝑚𝑝𝑐) 1 𝑌0 = ( C + 𝐼 + 𝐺 – 𝑚𝑝𝑐 𝑇̅0 + 𝑚𝑝𝑐 𝑇𝑅) (1 − 𝑚𝑝𝑐) 𝑌1 = 𝑌1 − 𝑌0 = 1 ( C − 𝐶̅ + 𝐼 − 𝐼 + 𝐺 − 𝐺 – 𝑚𝑝𝑐 𝑇̅1 + 𝑚𝑝𝑐 𝑇̅0 + 𝑚𝑝𝑐 𝑇𝑅 − 𝑚𝑝𝑐 𝑇𝑅) (1 − 𝑚𝑝𝑐) But C , I, G, TR (and the mpc) are all unchanged, so most of the subtractions in parentheses come out to be 0. We are left with (taking the negative sign out in front): 1 𝑌1 − 𝑌0 = − 𝑚𝑝𝑐 (𝑇̅1 − 𝑇̅0 ) (1 − 𝑚𝑝𝑐) or ∆Y = – (mult)(mpc)∆ T As explained in the text, the multiplier for a change in taxes is smaller than the multiplier for a change in government spending, because taxation affects aggregate expenditure only to the extent that people spend their tax cut or pay their increased taxes by reducing consumption. Because people may also save part of their tax cut or pay part of their increased taxes out of their savings, not all the changes in taxes will carry over to changes in aggregate expenditure. The tax multiplier has a negative sign, since a decrease in taxes increases consumption, aggregate expenditure, and income, while a tax increase decreases them. APPENDIX B: AN ALGEBRAIC APPROACH TO THE MULTIPLIER, WITH A PROPORTIONAL TAX With a proportional tax, total tax revenues are not set at a fixed level of revenues, as was the case with a lump sum tax but, rather, are a fixed proportion of total income. That is, T = tY where t is the tax rate. The equation for AE becomes AE = C + mpc (Y – tY + TR) I + G = C + mpc TR + I + G) + mpc (Y – tY) = ( C + mpc TR + I + G) + mpc (1 – t) Y Substituting in the equilibrium condition, Y = AE, and solving yields: Y = ( C + mpc TR + I + G) + mpc (1 – t) Y Y – mpc (1 – t) Y = C + mpc TR + I + G [AUTHOR NAME] 2 (1 – mpc (1 – t)) Y = C + mpc TR + I + G 1 𝑌=[ ] (𝐶̅ + 𝑚𝑝𝑐 𝑇𝑅 + 𝐼 + 𝐺) 1 − 𝑚𝑝𝑐(1 − 𝑡) The term in brackets is a new multiplier, for the case of a proportional tax. It is smaller than the basic (no proportional taxation) multiplier, reflecting the fact that now any change in spending has smaller feedback effects through consumption. (Some of the change in income “leaks” into taxes.) For example, if mpc = 0.8 and t = 0.2, then the new multiplier is 1/(1 – 0.64), or approximately 2.8, compared to the simple model multiplier 1/(1 – 0.8), which is 5. Changes in autonomous consumption or investment (or government spending or transfers) now have less of an effect on equilibrium income—the “automatic stabilizer” effect mentioned in the text. Is there a multiplier for the tax rate, t? That is, could we derive from the model a formula for how much equilibrium income should change with a change in the rate (rather than level) of taxes? For example, if the tax rate were to decrease from 0.2 to 0.15, could we calculate the size of the change from Y0 to Y1 illustrated in Figure 10.7? Yes, but deriving a general formula for a multiplier relating the change in Y to the change in the tax rate requires the use of calculus, which we will not pursue here. (If you are familiar with calculus, you can use the last formula above to calculate the change in Y resulting from a change in t). [AUTHOR NAME] 3