Macro Notes: IS Curve

Alan G. Isaac

American University

Bathtub Model of Natural Rate

The labor force is the employed plus the unemployed:

Et + Ut = L (1)

Unemployment is changed by separations and by job finding:

∆Ut+1 = sEt − f Ut (2)

Steady state = natural rate: unemployment stops changing:

0 = sEt − f Ut

= s(L− Ut) − f Ut

= s L− (s + f )Ut

(3)

Solve for the steady state unemployment rate (u = U/L)

u∗ =s

s + f(4)

Apply Bathtub Model

Recall:

u∗ =s

s + f(5)

Implication: the only way to change the natural rate is tochange the separation rate or the job-finding rate!Crude monthly averages: s = 0.013 and f = 0.25.

u∗ =0.013

0.013 + 0.25≈ 0.049 (6)

SR Unemployment Forecasts

We don’t just want to predict the natural rate. We also wantto know the SR effects of shocks and policy changes.

Source: Jones (2011)

We will now develop a description of equilibrium in the goodsmarket known as the IS curve. The IS curve will show theincrease in equilibrium income that is associated with adecrease in the interest rate. Why does equilibrium incomeincrease?

Here is one reason. When the interest rate is lower it costs lessto borrow, making investment projects look more attractive.An increase in desired investment is an increase in aggregatedemand. At the old level of income there is now excessdemand, which can be removed by an increase in supply. Thisimplies an increase in the level of income compatible withgoods market equilibrium.

So this is our basic story: an increase in the interest ratecauses a fall in investment which causes a decline in output.

IS Curve

An important question arises at this point: which interest rateare we talking about? When we think about the cost ofborrowing, we should think about the cost in purchasingpower. That is, we should think about the real interest rate.The ex ante real interest rate should be one of thedeterminants of desired investment.

The ex post real interest rate:

Rt = it − πt (7)

The ex ante real interest rate

Ret = it − πe

t (8)

This poses a problem for macroeconomists, because it is theex ante real interest rate that is relevant for saving andinvestment behavior. How are we going to deal withexpectations?Static expectations: expectations are exogenously fixed; theydo not change.Adaptive expectations: adjust to past experience. (“I’ll believeit when I see it.”) For example, if inflation was higher thanexpected last period, then you will increase your inflationexpectation.“Rational” expectations: by this we just meanmodel-consistent expectations. We assume that expectationsare the same as the forecast of our economic model.Learning models of expectations.For now we will not try to model expectations. Instead we willtreat them as given (i.e., static).

Our Keynesian IS curve is the combinations of R and Y suchthat the goods market is in equilibrium.Along this equilibrium locus, R and Y are negativelycorrelated.

Model of SR equilibrium:

Y = C + I + G + EX− IM (9)

C = acY (10)

G = ag Y (11)

EX = aex Y (12)

IM = aimY (13)

I

Y= ai − b(R − r) (14)

PILCH Model

Source: Jones (2011)

Model of SR equilibrium:

C/Y = ac (15)

G/Y = ag (16)

EX/Y = aex (17)

IM/Y = aim (18)

I

Y= ai − b(R − r) (19)

Y

Y=

C

Y+

I

Y+

G

Y+

EX− IM

Y(20)

Solution of model of SR equilibrium:

Y

Y=

C

Y+

I

Y+

G

Y+

EX− IM

Y= ac + ai − b(R − r) + ag + aex − aim

(21)

So

Y =Y

Y− 1

= (ac + ai + ag + aex − aim − 1)︸ ︷︷ ︸a

−b(R − r)

= a − b(R − r)

(22)

Comment: in the LR, Y = Y or equivalently Y = 0, and alsoR = r , so we can say a = 0. However, shocks to the economypush a away from 0 in the SR. We will call a the aggregatedemand shock.

IS Curve

Source: Jones (2011)

IS Curve: Increase in Real Interest Rate

Source: Jones (2011)

IS Curve: Aggregate Demand Shock

Source: Jones (2011)

IS Curve: Interest Rate Cut

Source: Jones (2011)

IS Curve: Consumer Confidence Falls

Source: Jones (2011)

IS Curve: IT Improvements

Source: Jones (2011)