The IS Curve: Derivation and Aggregation

CCBS/HKMA May 2004

The IS Curve - Some history

• Developed by Keynes (1936)• Made famous by Hicks’ IS-LM framework• Tool for macroeconomists during 1950s to

1970s• With rational expectations revolution,

approach seemed to fall out of favour• Recent literature (McCallum, Rotemberg,

Woodford, etc) has resurrected it

The IS curve - a reminder!

• Based on principle investment = saving

• An equilibrium condition since throughout the curve investment = saving

• In its simplest form it results in a relationship between output and the rate of interest

The IS curve - reminder!

• Constructed with expressions for consumption (C), investment (I), government expenditure (G) and exports (X) and imports (Z)

• C and Z depend on income (+): key parameters are marginal propensities

• I depends on the interest rate (-) and income (+): coefficient on rate is slope of IS curve

• G is exogenous

The IS-LM (&AS) framework

• Derivation of the liquidity = money curve gives us equilibrium in the money market

• The LM and IS curves lead to the aggregate demand schedule

• Together with an expression for aggregate supply based on the labour market, we have a description of the economy

• Easy, convenient and flexible framework, useful for policy-makers

IS-LM: Out of favour

• McCallum and Nelson (1997) give 6 ‘failures’:– 1. IS-LM analysis presumes a fixed, rigid price level– 2. No distinction between real and nominal rates– 3. Only 2 assets; money and bonds– 4. Only short-run, no steady states– 5. Capital stock fixed– 6. Not derived using microfoundations => Lucas

Critique• Rational Expectations revolution: Move away

from IS-LM (although equivalent expressions were still used!)

IS-(LM): Back in favour!

• From the mid 1990s some authors have claimed that an IS-looking schedule, together with an interest rate rule and an aggregate supply schedule can explain an economy’s dynamics

• LM schedule not needed as interest rate rule pins money market equilibrium down

• ‘Monetary policy without money’• Plus this can be useful to policy-makers

IS derivation

• Strongly micro-founded• Take a Rotemberg-Woodford (1997) or

McCallum-Nelson (1999) framework:1. Agents are consumers as well as producers2. Imperfect competition: agents cannot affect

prices; downward sloping demand curves3. Agents maximise lifetime utility subject to

constraint that all lifetime expenditure = lifetime resources

IS derivation

• Problem:

subject to

where

0

;;maxj

jtijtjt

ijt

jt yCuE

1,

11

0

1

dzzcC i

tit

it

jjt

ijtjtjttt

j

ijtjttt ATyipESE

0,

0, )(

1

11

0

1, dzzpPCPS ttitt

it

IS derivation

• FOCs

• Define

• Then we can write

2222 ; tittt

itct PECuE

ijt

jjtt

it ,

11,

tttt ER

ittt

it RE 1

IS derivation

• Assumption: all agents have identical initial wealth, and can insure against income risk (ie no precautionary saving!) thus ignore i superscripts

• log-linear approximation and solving forward gives expression for interest rates

10

ˆˆˆ

jtjtj

tltt REr

IS derivation

• A log-linear approximation of FOC for C:

• where σ is elasticity of substitution and C is the certainty equivalent level of consumption that guarantees a constant level of marginal utility

• Assumption: aggregate demand given by Y=C+G

lttttt rECCE 222 ˆˆ~

IS derivation

• Log-linearise aggregate demand:

• Use FOC for C to get IS curve

Y

CsGCsY CttCt ,

~ˆˆ

ttCtt

Ctlttt

CEsGG

sGrEY

2

21

~ˆ

~ˆˆˆ

Interpretation

• (Negative) slope of IS curve given by: elasticity of substitution and share of consumption in aggregate demand

• Expectations important

• Long-rate enters the IS, not the short rate

• ‘Aggregate block’: IS, long-rate expression and monetary policy rule (say a Taylor rule)

Useful or useless?

• Difficult to derive yet beautiful. But does it serve any purpose? Why do we use it?

• A next step is to calibrate the model and check whether it matches aggregate data

• To do this need to identify exogenous shocks, estimate structural parameters (or impose numbers!)

• If one gets a close approximation to the data it is possible to do policy experiments to examine impact of monetary policy

Useful or Useless?

• So mechanics not too straight-forward but nonetheless rich and may serve as a good benchmark

• Model based on many assumptions and some of the structural parameters have to imposed

• Model may match time series data, but will the model hold at all time periods? (eg are the assumptions made correct?)

• => Lucas’ Critique revisited?• Furher (1997) argues so

Some of the assumptions made

• Agents can insure themselves against income risk (no precautionary saving)

• All agents have identical initial wealth (no considerations about the wealth distribution)

• Agents live forever, no life-cycle considerations (implications for an ageing economy?)

• Capital markets are perfect ( liquidity constraints)• Functional form of aggregate demand• Interest rate rule can be ad hoc

Precautionary saving

• If there is income risk, the level of wealth (and therefore the capital stock) should be higher

• The consumption rule will be concave leading to aggregation problems. Using a ‘representative’ agent may not lead to ‘representative’ conclusions

• Evidence on the latter point mixed: Carroll (2001) argues this is important, Gourinchas (2000) argues not that quantitatively important

• Carroll points out that the distribution of wealth important

Life-cycle considerations

• Theory: young dis-save, mid-age save, old dis-save

• Response to shocks likely to be different

• Implications for capital stock?

• Ignores age composition of the economy

Perfect capital markets

• Not everyone can borrow at a constant interest rate• Credit supply normally upward sloping• But it has been noticed that when companies’

balance sheets healthy, these can borrow at more favourable terms: financial accelerator effect

• This effect can also be found for households• Effect of financial liberalisation• Interactions with liquidity constraints?

Aggregate demand

• No role for an external sector

• Is this a good assumption for your economy?

• What is the role of the government sector?

• This is likely to complicate the model

• If the model becomes more complex, will it be able to match time series data?

External sector: Svensson

• Svensson includes an external sector; other approaches see Clarida, Gali and Gertler or McCallum and Nelson

• See Batini Haldane (1999) for a UK model that includes the external sector

• What needs to be modified in the previous set up?• We will need to modify all the expressions (IS,

long-rate expression, policy rule, AS)• Why?

External sector: Svensson

• Consumption now made up of domestic and foreign goods (modify utility and price and expenditure aggregators)

• Consumption will depend on the exchange rate (modified Euler equation)

• Inflation now made of domestic and foreign prices• Resource constraint now made of domestic and foreign

consumption of domestically produced goods• Real long rate now depends on inflation that is determined

by domestic and foreign prices

External sector: Svensson

• Thus IS depends on (modified) real rates, government/shocks and the exchange rate

• Production must serve domestic and foreign markets thus depends on the exchange rate

• Wages ‘home determined’• Probability of changing prices exogenous, not determined

by foreign competition• Problem set as before; yields an expression for inflation as

a function of expectations, the exchange rate, the gap and future inflation

• Two more equations needed: exchange rate plus an expression for inflation (and foreign equations!)

Other issues

• Data measurement:– How do you measure output gap?– What interest rate do you use?– What price index to use? GDP deflator, CPI?

• Is there a role for money? • What expression for the exchange rate condition? • What exchange rate data?• Homogeneity in equations• Econometric techniques: IV (endogeneity), GMM

(forward looking variables and RE)

Other issues

• Functional forms:– Role for habit formation?

– Deviations from Cobb-Douglas production functions?

• Specific role for labour market and sticky wages?• Role for other transmission mechanism?• Log-linearisation of the model?• Around what steady state?

Conclusions

• Model examined is interesting and can give us plenty of insights

• Role of expectations and shocks very important for policy makers

• But before going away to develop a simple model one needs to think about one’s economy

• Use variables and equations that will give you the results you want!