Olivier Coibion The IS-LM Model and Aggregate Demand These notes will construct our model of aggregate demand. This will be the standard “IS-LM” approach to aggregate demand, based on Keynes and Hicks. Essentially, we will describe two important macroeconomic markets: the market for savings and the market for money holdings. We will consider each market independently, taking the level of output as exogenous in each market but solving for the equilibrium interest rate in each market. We will then combine the two markets, using the IS-LM approach, to determine the equilibrium level of output and interest rates that ensure that both markets clear. Part 1: The Savings (Loanable Funds) MarketThis is the market for savings and investment. We will examine what the supply and demand for savings are, as well as to characterize how the price of savings (the real interest rate) is determined. The supply of savings Total saving in the economy can be divided into two components: public and private saving. Private savingSpis the saving done by consumers and is equal to after-tax income minus consumption: Sp=Y-T-C. Public savingSgis the difference between tax revenues and government expenditures Sg= T-G. Total savingSis the sum of public and private saving, and so S=Sp+Sg=(Y-T-C)+(T-G)=Y-C-G. Note that it appears that total savings does not directly depend on taxes. However, to the extent that changes in taxes can affect consumption or output, total saving will in general change with exogenous changes in taxes. Recall from our work on consumption that aggregate consumption could be described by a function of current after-tax income, consumer sentiment (as a proxy for permanent income), and the interest rate: C=C(Y-T,CS,r). Hence, plugging this into our expression for total saving, we get S=Y-C(Y-T,CS,r)-G.

Recall that consumption is increasing in after-tax income Y-Tand consumer sentiment CSbut decreasing in the real interest rate r. Hence, aggregate savings must be increasing in rbut decreasing in CS. How does an increase in Yaffect savings? An increase in Yhas a direct positive effect on saving (since it is on the RHS of the expression above) but it also increases consumption which tends to decrease savings. We will refer to the increase in consumption from a 1$ change in after-tax income the marginal propensity to consume(mpc). Suppose the mpcis 0.8, then if income goes up by a dollar, consumers will tend to raise their consumption by $0.80. From the permanent income hypothesis, we know that transitory changes in income have small (less than one-for-one) effects on consumption, which tells us that as long as the change in Y is not permanent, then it must be that 0<mpc<1. Consumption rises when income rises but by less than one-for one. Therefore, when Y goes up by a dollar, consumption goes up by less than a dollar, which means that total savings must rise. The amount by which savings rises is known as the marginal propensity to save(mps). Since income is either consumed or saved, it must be that mps=1-mpc.