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Unformatted text preview: >> Chapter 11 Appendix: Deriving the Multiplier Algebraically This appendix shows how to derive the multiplier algebraically. First recall that in this chapter planned aggregate spending, AEPlanned is the sum of consumer spending, C, which is determined by the consumption function, and planned investment spending, IPlanned. Rewriting equation 11-9 to express all its terms fully, we have: (11A-1) AEPlanned = A + MPC × YD + IPlanned Because there are no taxes or government transfers in this model, disposable income is equal to GDP, so equation 11A-1 becomes (11A-2) AEPlanned = A + MPC × GDP + IPlanned The income–expenditure equilibrium GDP, Y*, is equal to planned aggregate spending: (11A-3) Y* = AEPlanned = A + MPC × Y* + IPlanned in income–expenditure equilibrium Just two more steps. Subtract MPC × Y* from both sides of Equation 11-12: (11A-4) Y* − MPC × Y* = Y* × (1 − MPC) = A + IPlanned Finally, divide both sides by (1 − MPC): (11A-5) Y* = A + IPlanned 1 − MPC Equation 11A-5 tells us that a $1 autonomous change in planned aggregate spending— a change in either A or IPlanned—causes a $1/(1 − MPC) change in income–expenditure equilibrium GDP, Y*. The multiplier in our simple model is therefore: (11A-6) Multiplier = 1/(1 − MPC) PROBLEMS 1. In an economy without government purchases, government transfers, or taxes, aggregate autonomous consumer spending is $500 billion, planned investment spending is $250 billion, and the marginal propensity to consume is 0.5. 2. Compare the following table by calculating the value of the multiplier and identifiying the change in Y* due to the change in autonomous spending. How does the value of the multiplier change with the marginal propensity to consume? Value of multiplier Change in spending Change in Y* a. Write the expression for planned aggregate spending as in Equation 11A-1. MPC b. Solve for Y* algebraically. c. What is the value of the multiplier? d. How will Y* change if autonomous consumer spending falls to $450 billion? 0.5 0.6 0.75 0.8 0.9 C = + $50 million I = − $10 million C = − $25 million I = + $20 million C = − $2.5 million 291 ...
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This note was uploaded on 03/17/2009 for the course ECON 102 taught by Professor Erus during the Spring '09 term at Boğaziçi University.

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