IS-LM model Giovanni Di Bartolomeo giovanni.dibartolomeo@uniroma.it Note: These leture notes are inomplete without having attended letures
IS Curve Giovanni Di Bartolomeo giovanni.dibartolomeo@uniroma.it Note: These leture notes are inomplete without having attended letures
The Big Piture Keynesian Cross Theory of Liquidity Preferene IS urve LM urve IS-LM model Explanation of short-run flutuations Agg. demand urve Agg. supply urve Model of Agg. Demand and Agg. Supply
The Big Piture: IS/LM Keynesian Cross Theory of Liquidity Preferene IS urve LM urve IS-LM model Explanation of short-run flutuations Agg. demand urve Agg. supply urve Model of Agg. Demand and Agg. Supply
Context Long run pries flexible output determined by fators of prodution & tehnology unemployment equals its natural rate Short run pries fixed output determined by aggregate demand unemployment negatively related to output
Reall: The Keynesian Cross A simple losed eonomy model in whih inome is determined by expenditure. (due to J.M. Keynes) Notation: I = planned investment AE = C + I + G = planned expenditure Y = real GDP = atual expenditure Differene between atual & planned expenditure = unplanned inventory investment
Elements of the Keynesian Cross Consumption funtion: Govt poliy variables: for now, planned Investment is exogenous: planned expenditure: C C ( Y T ) G G, T T I I AE C Y T I G equilibrium ondition: Y AE atual expenditure = planned expenditure
Graphing planned expenditure AE planned expenditure AE =C +I +G MPC inome, output, Y
Graphing the equilibrium ondition AE planned expenditure AE =Y 45º inome, output, Y
The equilibrium value of inome AE planned expenditure AE =Y AE =C +I +G Equilibrium inome inome, output, Y
At Y, there is now an unplanned drop in inventory An inrease in government purhases G AE AE =C +I +G 2 AE =C +I +G so firms inrease output, and inome rises toward a new equilibrium. AE = Y Y AE 2 = Y 2 Y
Solving for Y Y C I G Y C I G C G MPC Y G Collet terms with Y on the left side of the equals sign: ( MPC) Y G equilibrium ondition in hanges beause I exogenous beause C = MPC Y Solve for Y : Y G MPC
The government purhases multiplier Definition: the inrease in inome resulting from a $ inrease in G. In this model, the govt purhases multiplier equals Y G MPC Example: If MPC = 0.8, then Y G 0.8 5 An inrease in G auses inome to inrease 5 times as muh!
Why the multiplier is greater than Initially, the inrease in G auses an equal inrease in Y: Y = G. But Y C further Y further C further Y So the final impat on inome is muh bigger than the initial G.
An inrease in taxes Initially, the tax inrease redues onsumption, and therefore E: AE AE =C +I +G AE =C 2 +I +G C = MPC T so firms redue output, and inome falls toward a new equilibrium AE 2 = Y 2 Y At Y, there is now an unplanned inventory buildup AE = Y Y
Solving for Y Y C I G C eq m ondition in hanges I and G exogenous MPC Y T Solving for Y : ( MPC) Y MPC T Final result: MPC Y T MPC
The tax multiplier Def: the hange in inome resulting from a $ inrease in T : Y T MPC MPC If MPC = 0.8, then the tax multiplier equals Y 0.8 0.8 T 0.8 0.2 4
The tax multiplier is negative: A tax inrease redues C, whih redues inome. is greater than one (in absolute value): A hange in taxes has a multiplier effet on inome. is smaller than the govt spending multiplier: Consumers save the fration ( MPC) of a tax ut, so the initial boost in spending from a tax ut is smaller than from an equal inrease in G.
Eonomi Senario: Walkthrough Example I: In the Keynesian Cross, assume that the onsumption funtion is given by: C = 475 + 0.75(Y-T) Planned Investment, I = 50, G = 250, T = 00. a. Graph planned expenditure as a funtion of inome b. What is the equilibrium level of inome. If government purhases inrease by 25, what is the new equilibrium inome? d. What level of government purhases is needed to ahieve an inome of 2600?
A Balaned Budget Approah Problem: Suppose that we fae our anonial problem where C=475 + 0.75(Y-T), T = 00 I = 50, G = 250 Suppose that the government wishes to inrease its spending by 00, but uses a balaned budget approah, thereby raising taxes by the same amount to finane its expenditures. Question: Is there any impat on GDP? Does it hange? If so, by how muh?
Balaned Budget Change in G Solution: Suppose G = T=00. Hene i.e. so our balaned budget multiplier = Why? 0 T G I Y AE In equilibrium,y 00 G G G T G Y G Y G T T T T G G G G G Y...... 4 3 2 4 3 2
The IS urve Definition: a graph of all ombinations of r and Y that result in goods market equilibrium i.e. atual expenditure (output) = planned expenditure The equation for the IS urve is: Y C ( Y T ) I ( r ) G
Deriving the IS urve AE AE =Y AE =C +I (r 2 )+G r I AE =C +I (r )+G AE I Y r Y Y 2 Y r r 2 Y Y 2 IS Y
Why the IS urve is negatively sloped A fall in the interest rate motivates firms to inrease investment spending, whih drives up total planned spending (AE ). To restore equilibrium in the goods market, output (a.k.a. atual expenditure, Y ) must inrease.
Market For Loanable Funds Closed Eonomy Define S p Y-T-C(Y-T) and S g T-G (+) S S p + S g = Y C(Y-T) G = S(Y; G, T) (+) (-) (+) Capital Markets Equilibrium: S(Y;G,T) = I(r) (Loanable Funds) Or Equivalently: Y = Y d C(Y-T) + I(r) + G (-)
The IS urve and the loanable funds model (a) The L.F. model (b) The IS urve r S 2 S r r 2 r 2 r I (r ) r IS S, I Y 2 Y Y
Algebra of the IS Curve Suppose C = 0 + (Y-T) and I = I 0 br (Note: Blanhard also onsiders the effet of sales on Investment by inorporating Y, i.e. I=b 0 +b Y-b 2 r) Then Y= C + I + G = 0 + I 0 + G + (Y-T) br If we ollet like terms: Y 0 I0 G T b r
Slope of the IS Curve Hold everything exept Y and r fixed: Thus IS is relatively flat if either: (i) b is very large; or (ii) lose to unity. r b T G I Y 0 0 0 b Y r r b Y
Walkthrough Example II: Eonomi Senario: Consider the following IS-LM model: Goods Market: C = 200+0.5(Y-T); I = 200 000r; G = 250; T = 200 Question: Derive the IS relation (i.e. an equation with Y on one side and everything else on the other) Solution
Fisal Poliy and the IS urve We an use the IS-LM model to see how fisal poliy (G and T ) affets aggregate demand and output. Let s start by using the Keynesian ross to see how fisal poliy shifts the IS urve
Shifting the IS urve: G At any value of r, G AE Y so the IS urve shifts to the right. AE AE =Y AE =C +I (r )+G 2 AE =C +I (r )+G The horizontal distane of the IS shift equals r r Y Y 2 Y Y G MPC Y Y Y 2 IS IS 2 Y
LM Curve Giovanni Di Bartolomeo giovanni.dibartolomeo@uniroma.it Note: These leture notes are inomplete without having attended letures
The Big Piture Keynesian Cross Theory of Liquidity Preferene IS urve LM urve IS-LM model Explanation of short-run flutuations Agg. demand urve Agg. supply urve Model of Agg. Demand and Agg. Supply
The Theory of Liquidity Preferene Due to John Maynard Keynes. A simple theory in whih the interest rate is determined by money supply and money demand. Money supply is exogenous determined by Fed! People hold wealth in the form of either: Money Bonds Demand for money and demand for bonds!
Money supply The supply of real money balanes is fixed: r interest rate M P s s M P M P M P M/P real money balanes
Money demand People either hold: Money Bonds r interest rate M P s Demand for real money balanes: d M P L( r ) M P L (r ) M/P real money balanes
Equilibrium The interest rate adjusts to equate the supply and demand for money: r interest rate r M P s M P L( r ) L (r ) M P M/P real money balanes
How the Fed raises the interest rate To inrease r, Fed redues M r interest rate r 2 r M 2 P M P L (r ) M/P real money balanes
Monetary Tightening & Interest Rates Case study Late 970s: > 0% Ot 979: Fed Chairman Paul Volker announes that monetary poliy would aim to redue inflation Aug 979-April 980: Fed redues M/P 8.0% Jan 983: = 3.7% How do you think this poliy hange would affet nominal interest rates?
The effets of a monetary tightening on nominal interest rates model pries predition short run Liquidity preferene (Keynesian) stiky i > 0 long run Quantity theory, Fisher effet (Classial) flexible i < 0 atual outome 8/979: i = 0.4% 4/980: i = 5.8% 8/979: i = 0.4% /983: i = 8.2%
Now let s put Y bak into the money demand funtion: The LM Curve d M P L( r, Y ) Note: In Blanhard: M d YLi dy d 2 P i The LM urve is a graph of all ombinations of r and Y that equate the supply and demand for real money balanes. The equation for the LM urve is: M P L( r, Y )
Nominal or Real Rates in Money Demand? Money Market Equilibrium: M P What is real return to saving $? This is known as the Fisher Equation. So: M P L r r, Y i M P d L( i, Y) Treat M s as exogenous; for present set = 0. r (-)(+) i
Deriving the LM urve r (a) The market for real money balanes r (b) The LM urve LM r 2 L (r, Y 2 ) r 2 r r L (r, Y ) M P M/P Y Y 2 Y
Why the LM urve is upward sloping An inrease in inome raises money demand. Sine the supply of real balanes is fixed, there is now exess demand in the money market at the initial interest rate. The interest rate must rise to restore equilibrium in the money market.
Equilibrium in the Bond Market? There are two assets (money and bonds), but only one equilibrium ondition. Do we need to worry about bond market equilibrium as well? Answer: No! M s Bs A P with Areal wealth. M P d Bd So : M s P M P d Bs Bd This is an example of Walras Law.
Algebra of the LM Curve Write: M P d m ky hr 0 With M and P fixed: 0 = ky - h r Slope of LM Curve : LM urve relatively flat if either: (i) (ii) k small; or r Y k h h large ( Liquidity Trap ) 0
How M shifts the LM urve r (a) The market for real money balanes r (b) The LM urve LM 2 r 2 r 2 LM r L (r, Y ) r M 2 P M P M/P Y Y
M ky Y P M Pk What happens if k 0? Hold Y fixed: Shifts in LM urve (r fixed) M hr P r M Ph So vertial shift is independent of k
Walkthrough Example III: Eonomi Senario: Suppose that the money demand funtion is: (M/P) d = 000 00r where r is the interest rate (in perent). The money supply M is 000, and the prie level is 2. a. Graph the supply and demand for real money balanes. b. What is the equilibrium interest rate?. Assume that the prie level is fixed. What happens to the equilibrium interest rate if the supply of money is raised from 000 to 200? d. If the Fed wishes to raise the interest rate to 7 perent, what money supply should it set?
IS/LM Equilbrium Giovanni Di Bartolomeo giovanni.dibartolomeo@uniroma.it Note: These leture notes are inomplete without having attended letures
The Big Piture Keynesian Cross Theory of Liquidity Preferene IS urve LM urve IS-LM model Explanation of short-run flutuations Agg. demand urve Agg. supply urve Model of Agg. Demand and Agg. Supply
The short-run equilibrium The short-run equilibrium is the ombination of r and Y that simultaneously satisfies the equilibrium onditions in the goods & money markets: r LM Y C ( Y T ) I ( r ) G M P L( r, Y ) Equilibrium interest rate IS Equilibrium level of inome Y
Equilibrium With Fixed Pries IS Curve 0 0 ) ( ), ; ( br T G I Y r I G T Y S or LM Curve (+)(-)(+) ) 0 P M (or ), ( hr ky m Y r L P M (-)(+) Solve for Y and r in terms of G,T, M and P.
Equilibrium in the IS-LM Model r LM Equilibrium interest rate IS Inome and Output Equilibrium level of inome Y
Equilibrium Adjustment Is there any reason to expet it to onverge to this equilibrium from arbitrary r and Y? If there is an exess demand for money (exess supply of bonds) this should drive the return on bonds up, and vie versa. If savings exeeds planned investment, then onsumers must be spending less and produers will be aumulating unwanted inventories. So they will ut bak prodution, and vie versa. Hene the system should onverge.
What is this??? h bk hp bm h bm T G I h P M ky m b T G I Y / / / 0 0 0 ) / 0 ( 0 0 h 0 or 0 as b and M Y 0 0 as h and Hene: 0 ) / ( 0 / h bk hp b h bk G Y
Walkthrough Example IV: Eonomi Senario: Consider the following IS-LM model: Goods Market: C = 200+0.5(Y-T); I = 200 000r; G = 250; T=200 Money Market: M = 3000; P = 2; L(r,Y)= 2Y 0000r a. Derive the IS relation (i.e. an equation with Y on one side and everything else on the other) b. Derive the LM relation. Solve for equilibrium real GDP (Y*) and interest rate (r*) d. Solve for the equilibrium values of C, I and G (- verify you get Y by adding up C+I+G) e. Suppose that the Fed inreases money supply by 280, i.e. M=280. Solve for Y*, r*, C and I. Desribe what happens. f. Set M bak to its initial value. Now suppose the government inreases spending by 70, i.e. G = 70. Solve for Y*, r*, C and I. Desribe what happens.
Fisal Expansion r G h bk / LM r 2 B r A G IS 2 IS Y Y 2 Y Inome and Output
Monetary Expansion r LM LM 2 r A r 2 B IS Y Y 2 Y Inome and Output
Summary. Keynesian ross basi model of inome determination takes fisal poliy & investment as exogenous fisal poliy has a multiplier effet on inome. 2. IS urve omes from Keynesian ross when planned investment depends negatively on interest rate shows all ombinations of r and Y that equate planned expenditure with atual expenditure on goods & servies
Summary 3. Theory of Liquidity Preferene basi model of interest rate determination takes money supply & prie level as exogenous an inrease in the money supply lowers the interest rate 4. LM urve omes from liquidity preferene theory when money demand depends positively on inome shows all ombinations of r and Y that equate demand for real money balanes with supply
Summary 5. IS-LM model Intersetion of IS and LM urves shows the unique point (Y, r ) that satisfies equilibrium in both the goods and money markets.