Lecture 14: The IS Curve

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Fall Semester ’06-’07 Akila Weerapana

I. OVERVIEW

• In the last lecture, we derived the most basic model of economic fluctuations using the Keyne- sian Cross Model. Starting with this class, we put together a more advanced macroeconomic model known as the IS-LM model.

• We can use this model to examine the impact of fiscal and monetary policy, as well as to study the impact of changes in investment, consumption and net exports in the economy. An important feature of the IS-LM model to keep in mind is that it is a short-run model of the economy; as you will see the model works with an exogenously given price level.

• Once we have completed our development of the IS-LM model, we will then extend it to do the Aggregate Demand model, which allows us to do intermediate term analysis, developing the ability to think intelligently about the impact of policy changes on inflation.

II. THE IS (Investment = Savings) CURVE

• The IS curve derives a relationship between interest rates and income in the short run. It modifies the Keynesian Cross by disposing of the unrealistic assumption that investment is exogeneously determined and instead specifies a simple negative relationship between investment and the interest rate.

• The linear investment function is of the form I = ¯I − βr where r is the real interest rate.

We can interpret ¯I as what Keynes termed animal spirits: changes in investment that are unrelated to the interest rate but come about because of investor confidence, for example.

• The parameter β measures how much investment responds to the real interest rate.

• The real interest rate = nominal interest rate - rate of inflation, i.e. r = i − π We use r to refer to an average real interest rate in the economy rather than the rate of return on any particular type of asset.

• An increase in the interest rate reduces investment by making it more expensive for firms to borrow money to make investment purchases and also by increasing the opportunity cost for those who plan to finance investment projects using their own funds.

• By reducing investment, high real interest rates will correspondingly reduce GDP. Having made these modifications we can now summarize the goods market in the economy as follows:

Y = C + I + G + N X C = C + b(1 − t)Y¯

I = I − βr¯ G = G¯ N X = N X¯

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• We combine these equations to derive the IS curve: all the combinations of Y and r that cause the market for goods and services to clear.

Y = [ ¯C + b(1 − t)Y ] + [ ¯I − βr] + ¯G + ¯N X

• This can be simplified as

Y = 1

1 − b(1 − t)[ ¯C + ¯I + ¯G + ¯N X] − 1

1 − b(1 − t)[βr]

• Since we showed that the multiplier is µ = _1−b(1−t)¹ , this can be re-written as Y = µC + ¯¯ I + ¯G + ¯N X − µ [βr]

• This is the equation for the IS curve, the relationship between Y and r that ensure that the goods market is in equilibrium: spending=production.

• Note that when interest rates are high, investment falls and therefore Y must fall as well; the IS curve should show a negative relationship between r and Y . This can be shown graphically as follows.

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@ r

Y IS Curve

III. THE SLOPE AND INTERCEPT OF THE IS CURVE

• One of the quirks of economists is our habit of writing equations of the form M = f (N ) and drawing graphs of the inverse N = g(M ). Examples include demand and supply curves, which we write as Q = f (P ) but then graph with P on the vertical axis and now the IS curve which we write as Y = f (r) but then draw with r on the vertical axis.

• The IS Curve, rewritten in a graph friendly (but not intuition friendly) manner is r = 1

β

C + ¯¯ I + ¯G + ¯N X − 1 βµ[Y ]

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The Slope

• The slope of the IS Curve is

∂r

∂Y = − 1 βµ

• The IS curve is steep when the absolute value of the slope is large. This occurs when 1. The parameter β is small.

2. The multiplier µ is small.

• What is the intuition for these results? Well, a steep IS curve means that Y does not respond very much to r. In other words, a change in the real interest rate does not have a significant impact on GDP.

• According to the above math, this happens when β is small, i.e. when investment does not respond very much to the real interest rate. It could also happen when the multiplier, µ, is small, because a given change in investment has fewer feedback effects when the multiplier is small.

The Intercepts

• The intercept of the IS curve can be found by setting Y = 0 as ¹_β[ ¯C + ¯I + ¯G + ¯N X].

• This can be thought of as the highest possible real interest rate in the economy but for the most part, we will have limited use for it in our analysis.

• Of slightly more interest is the horizontal intercept of the IS curve, which can be found by setting r = 0 to be µ[ ¯C + ¯G + ¯I + ¯N X] or _1−b(1−t)¹ [ ¯C + ¯I + ¯G + ¯N X] . This, you may recall is the Keynesian Cross solution of the problem, where we ignored real interest rates.

IV. SHIFTS, SLOPE CHANGES AND MOVEMENTS ALONG THE IS CURVE

Movements Along the Curve

• Since we are graphing the relationship between r and Y , the key thing to note is that changes in Y caused by changes in r are reflected as movements along the IS curve.

• When interest rates decrease, spending rises and as a result output increases as well. This is reflected in a movement to a lower point on the IS curve where interest rates are lower and output is higher.

• Conversely, when interest rates increase, spending falls and as a result output decreases as well. This is reflected in a movement to a higher point on the IS curve where interest rates are higher and output is lower.

Parallel Shifts of the IS Curve

• On the other hand changes in Y that are brought about by factors other than interest rates will cause Y to change, regardless of the level of interest rates in the economy.

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• The IS Curve, rewritten in a graph friendly manner is r = 1

β

C + ¯¯ I + ¯G + ¯N X − 1 βµ[Y ]

• We can see that changes in exogenous spending (consumer confidence, investor confidence etc.) will not change the slope but will change the intercept: in other words, they will cause the IS curve to shift. What direction will the IS curve shift in response to changes in these variables? We can summarize the changes as follows.

1. Increases in consumer confidence ( ¯C), investor confidence ( ¯I), government purchases or net exports ( ¯N X) will raise expenditure, and therefore production of goods and services, shifting the IS curve out. Reductions in consumer or investor confidence or government purchases or net exports will reduce expenditure and shift the IS curve in.

2. A reduction in lump-sum taxes ( ¯T ) [if they happen to be in the model, they are not in the model above] will increase expenditure on goods and services and therefore increase output as well. This will cause the IS curve will shift out. Conversely, an increase in lump-sum taxes ( ¯T ) [if they happen to be in the model] will shift the IS curve in.

EXAMPLE 1

• An increase in G will raise spending, and therefore production, of domestic goods and services. This will cause the IS curve to shift outwards. From the equation for the IS curve, Y = µ[ ¯C + ¯G + ¯I +N X] − µβr, we can see that¯ ^dY_{d ¯}_G = µ ≡ _1−b(1−t)¹ . In other words, an increase in government purchases of ∆ ¯G will cause the IS curve to shift out by _1−b(1−t)¹ [∆ ¯G].

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@ r

Y New IS Curve Old IS Curve

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∆ ¯G 1−b(1−t)

EXAMPLE 2

• A fall in investor confidence will reduce investment spending on goods and services and therefore cause the IS curve to shift in. From the equation for the IS curve, Y = µ[ ¯C + ¯G + I + ¯¯ N X] − µβr, we can see that ^dY_{d ¯}_I = µ ≡ _1−b(1−t)¹ , i.e. a decrease in exogenous investment of ∆ ¯I will cause the IS curve to shift in by _1−b(1−t)¹ [∆ ¯I].

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@ r

Y Old IS Curve New IS Curve

∆ ¯I 1−b(1−t)

Slope Shifts of the IS Curve

• Remember that the IS Curve, rewritten in a graph friendly manner is r = 1

β

C + ¯¯ I + ¯G + ¯N X − 1 βµ[Y ]

• Changes in the multiplier, µ, will not change the intercept but will change the slope of the IS curve. As we discussed before, if the multiplier becomes larger, the IS curve becomes flatter and if it becomes smaller the IS curve becomes steeper.

• Given the expression for the spending multiplier of µ = _1−b(1−t)¹ we need to think about how changes in b and changes in t affect the multiplier.

• The intuition is fairly simple, the multiplier increases (the slope becomes flatter) if tax rates fall or if the marginal propensity to consume rises.

1. Increases in the marginal propensity to consume (b) will increase the multiplier and make the IS curve flatter. Conversely, decreases in b will make the IS curve steeper.

2. Decreases in income tax rates (t) will increase the multiplier and make the IS curve flatter. Conversely, increases in t will make the IS curve steeper.

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EXAMPLE 1

• An increase in the marginal propensity to consume, b,(which raises the multiplier µ =

1

1−b(1−t)) and will cause the IS curve to have a flatter slope.

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@ Q

Q Q

Q Q r

Y Old IS Curve

New IS Curve

EXAMPLE 2

• A higher income tax rate will lower the multiplier µ = _1−b(1−t)¹ and will cause the IS curve to have a steeper slope.

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@ Q

Q Q

Q Q r

Y New IS Curve

Old IS Curve

V. DO I REALLY HAVE TO MEMORIZE ALL OF THIS?

• No, you do not have to memorize each of these changes. The goal is to have an intuitive understanding of the IS curve.

• Keep in mind that the IS curve is summarizing the inverse relationship between interest rates and GDP in the economy. If you are in doubt as to how a particular change should be reflected in the IS curve, you need to only ask two questions.

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• First, is the change in Y or in r? If so, then it should be simply a matter of moving to a different spot on the IS curve.

• Second, does the change increase or decrease expenditure on domestic goods and services? If it is an increase then it should cause Y to rise, i.e. be a rightward movement of the IS curve, if the answer is a decrease then it should cause Y to fall, it should be a leftward movement of the IS curve.

• That’s 95% of the battle. All that remains is whether the rightward (or leftward) movement is a parallel shift or a slope change. If the driving force is a change in exogenous spending there will be a parallel shift. If the driving force is a change in the multiplier there will be a slope change.