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Chapter 9: The IS-LM Model: Macroeconomic Equilibrium with Fixed Prices Components of Expenditure E = C + I + G + NX E = [C 0 + Cy(1-t)Y] + [Io – Ir x r] + G + [XfYf^ + XԐ Ԑ - IMyY] E = A + MPE x Y Expenditure = Autonomous expenditure + Marginal Propensity to expend x Y Autonomous Spending A = C 0 + [Io – Ir x r] + G + [XfYf^ + XԐ Ԑ) A higher interest rate reduces autonomous spending (A) by reducing investment (Ir x r) r↑ A = (Ir x r) ↓ A higher interest rate also reduces autonomous spending by reducing net exports r↑ A = (XԐ Ԑ) ↓ ** Ԑ =Ԑ 0 - Ԑr x (r-rf) r↑ Ԑ↓ reducing net exports Autonomous Spending and the real interest rate (r) A = [C 0 + Io + G + (XfYf^ + XԐԐ 0 x rf)] – (Ir + XԐԐr) x r A one-percentage-point increase in the real interest rate (r) reduces autonomous spending by (Ir
- XԐԐr) o Now we see that r plays an important role in the determination of the equilibrium. The Investment-Saving (IS) Curve The relationship between the level of the real interest rate and the equilibrium level of real GDP is the IS curve To find a point on the IS curve: o pick a value for the real interest rate and determine the level of autonomous spending at that interest rate –use the income-expenditure diagram to determine the equilibrium level of real GDP o Repeat this procedure to find other points on the IS curve A = [C 0 + Io + G + (XfYf^ + XԐԐ 0 x rf)] – (Ir + XԐԐr) x r A 0 = determinants of autonomous spending that do not depend on real interest rate o A = [C 0 + Io + G + (XfYf^ + XԐԐ 0 x rf)] o A = A 0 – (Ir + XԐԐr) x r Recall that real GDP is equal to autonomous spending (A) divided by (1-MPE) o Y = A 0 – (Ir + XԐԐr) x r 1 – MPE
o Subbing in we get Y = [C 0 + Io + G + (XfYf^ + XԐԐ 0 x rf)] - (Ir + XԐԐr) 1 – (Cy (1 – t) – IMy) The term on the left is the horizontal intercept of the IS curve o the value of equilibrium real GDP if the real interest rate was equal to zero The term on the right is measures the responsiveness of real GDP to changes in the interest rate In drawing the IS curve economists measure the real interest rate on vertical axis and real GDP on the horizontal axis. In order to see what is the slope of the IS curve in that setting, re-solve the equation of the IS curve, expressing r as a function of Y: r = [C 0 + Io + G + (XfYf^ + XԐԐ 0 x rf)] - (Ir + XԐԐr) 1 – (Cy (1 – t) – IMy) IS SLOPE =
− 1 − C y ( 1 – t ) – ℑ y
( I r + X Ԑ Ԑ r ) =
1 − MPE
( I r + X ԐԐ r )
The first term is the inverse of the multiplier (1/1-MPE) The second term shows how large a change in investment or exports is generated by a change in the real interest rate The position of the IS curve depends on the baseline level of autonomous spending times the multiplier Changes in any of these determinants will shift the position of the IS curve Changes in the Interest Rate To calculate how much a change in the interest rate will shift the equilibrium level of real GDP, you need to know four things: o the marginal propensity to spend (MPE) o the interest sensitivity of investment (Ir) o the interest sensitivity of the exchange rate ( Ԑr) –the exchange rate sensitivity of exports (XԐ) Moving along the IS curve If the economy is above the IS curve: o real GDP > planned expenditure •inventories rise •firms cut production •employment, real GDP, and national income fall
o Money is liquid, and it can be easily spent Two facts about business and household demand for real money o First, real money demand is increasing in total real income or real GDP Y. o As Y increases there are more real balances needed to finance more transactions Second, real money demand is inversely related to the nominal interest rate: o When nominal interest rates go up, bonds become more attractive and the demand for real money goes down. Demand for money can be expressed:
Md
P
= L = Lo − Li x ( r + π e )+ Ly x Y
L 0 is the autonomous part of demand for real money balances Ly measures the sensitivity of real money demand to changes in Y Li measures the sensitivity of real money demand to changes in i Equilibrium In a sticky-price model, the price level is predetermined o it cannot move instantly to make money supply equal to money demand The nominal interest rate must adjust to keep the money market in equilibrium Bond Prices and Nominal Interest Rates a government bond promises to pay a fixed number of dollars, called the coupon, each time period until maturity. The nominal interest rate is the value of the coupon as percentage of the value of the bond. Hence, if the market value of a government bond increases and the value of the coupon as percentage of the value of the bond falls, the nominal interest rate i falls. Adjustment to equilibrium in Money Market If at a given nominal interest rate real money supply is more than real money demand then households and firms will try to convert part of their excess real money holdings into government bonds. This raises the demand for bonds As a result the price of bonds rises and the nominal interest rate falls. If at a given nominal interest rate real money supply is smaller than real money demand then households and firms will try to convert part of their bond holdings into money. This reduces the demand for bonds As a result the price of bonds falls and the nominal interest rate rises.
Increase in the Money Supply Suppose the central bank increases the money supply through an open market operation o that is, it buys bonds from households and firms in the bond market in exchange for money. This action by the central bank will increase the demand for bonds in the bonds market, thereby raising bond price and lowering the nominal interest Decrease in Real Money Demand The demand for real money balances can fall due to certain financial innovations: o credit cards are more widely used o Introduction of automated banking machines (ATMs), telephone banking and internet banking. To reduce their excess money balances, households and firms will buy bonds o Demand for bonds increases. o Bond prices increase and i falls Real Money Demand and Interest Rates Because the real money demand depends on the level of real GDP, if the money stock is constant, the equilibrium nominal interest rate will vary whenever real GDP varies When Y falls the liquidity preference will shift to the left and the interest rates fall. When Y increases the liquidity preference will shift to the right and the interest rates increase. The LM Curve The LM curve shows the relationship between the level of real GDP and the equilibrium nominal interest rate that clears the money market
The Multipliers Again We derived the government expenditure multiplier, which gave us the increase in output corresponding to one unit increase in government expenditures with fixed prices and fixed real interest rates. In the present chapter we can derive similar multipliers in a more complete model where the real interest rate is not fixed, but is determined as part of the equilibrating process. The equilibrium output in the IS-LM model: Y = YA + YG x G + YM x M/P YG tells us how much the equilibrium output is in the IS-LM model will increase if G increases by one unit o Government expenditures multiplier YG = ΔY = 1 1 ΔG (1-MPE) + Ir x (^) r + XԐԐr) x Ly/Li YM tells us by how much the equilibrium output in the model will increase if the real money supply increases by one unit o The money supply multiplier: YM = ΔY = 1/ Li 1 ΔM (1-MPE) + Ir x (^) r + XԐԐr) x LyLi Interest Rate Targets When the central bank targets the interest rate the effective LM curve is horizontal at the target interest rate o Since the central bank is willing to supply as much money as is demanded at the target interest rate, the money market is in equilibrium at that interest rate, regardless of the level of output Changes that effect the LM Curve Any change in the nominal money stock, in the price level, or in the demand for real money balances Any change in the trend velocity of money will shift the LM curve Any change in the interest sensitivity of real money demand will change the slope of the LM curve The IS-LM diagram is drawn with the long-term, risky, real interest rate on the vertical axis The LM curve is a relationship between the short-run nominal interest rate and the level of real GDP at a fixed level of the money supply
As long as the spread between the short-term, safe, nominal interest rate and the long-term, risky, real interest rate is constant, there are no complications in drawing the LM curve onto the same diagram as the IS curve If the expected rate of inflation, the risk premium, or the term premium between short- and long-term interest rates changes, the LM curve will shift o changes in financial market expectations of future central bank policy, future interest rates, or changes in the risk tolerance of bond traders will shift the LM curve Changes that affect the IS Curve Shifts in the IS curve are more frequent than shifts in the LM curve Any change in the interest sensitivity of investment, the sensitivity of exports to the exchange rate, or the sensitivity of the exchange rate to the domestic interest rate will change the slope of the IS curve Anything that affects MPE will change the slope and the position of the IS curve o this includes changes in the MPC, tax rates, and the propensity to import Any change in autonomous spending will shift the IS curve The IS-LM Framework and the exchange rate ( Ԑ) In the sticky-price model, the real exchange rate is equal to: Ԑ =Ԑ 0 - Ԑr (r -rf) As long as the domestic real interest rate does not change, domestic conditions will have no impact on the exchange rate Changes in the IS and LM curves that change the domestic real interest rate will alter the real exchange rate by an amount equal to: (- Ԑr x Δr) o A rightward shift in the IS curve or a leftward shift in the LM curve will lower the real exchange rate o A leftward shift in the IS curve or a rightward shift in the LM curve will raise the real exchange rate Changes in the domestic interest rate affect the real exchange rate which affects gross exports Changes in total income affects imports The effect on net exports is the difference between these two effects ΔNX = ΔGX – ΔIM
