Policy and Shocks in the IS-LM World

Abstract

We put together the behaviors of the IS and LM curves laid out in Chap. 14 to see what predictions the IS-LM model makes about different policy actions and external shocks and how they affect output and interest rates. The goal of policy is defined as keeping actual output close to potential output. Depending on the relative slopes of the two curves, we may be able to say that fiscal policy is particularly effective or, with other slopes, that monetary policy is particularly effective.

Appendices

Appendix I: The LM Curve and the Real Interest Rate

Chapter 14 introduced the algebraic representation of the LM curve in terms of GDP and the nominal interest rate:

$$\displaystyle{ Y = \frac{1} {m_{1}}M^{S} -\frac{m_{0}} {m_{1}} + \frac{m_{2}} {m_{1}}i. }$$

(15.1)

When you lend or borrow money, the actual real interest rate that you will receive or pay is unknown, because you don’t know what inflation will turn out to be over the duration of the loan. All that you know is the expected real interest rate, which is the (known) nominal rate i and the expected inflation rate π ^e:

$$\displaystyle{r\cong i -\pi ^{e}.}$$

We can use this relationship to take the nominal interest rate out of Eq. 15.1 above describing the LM curve:

$$\displaystyle{Y = \frac{1} {m_{1}}M^{S} -\frac{m_{0}} {m_{1}} + \frac{m_{2}} {m_{1}}(r +\pi ^{e}).}$$

Now if we regroup the inflation expectation, we get an equation for the LM curve in Y and r, showing the position of the curve as a function of both the money supply and the expected level of inflation:

$$\displaystyle{ Y = \left ( \frac{1} {m_{1}}M^{S} -\frac{m_{0}} {m_{1}} + \frac{m_{2}} {m_{1}}\pi ^{e}\right ) + \frac{m_{2}} {m_{1}}r }$$

(15.2)

Appendix II: Algebraic Solution of IS-LM

We now have equations for the IS and LM curves, both of them expressing Y as a function of the real interest rate, and other stuff.

The LM curve is above, at the end of Appendix I, and the IS curve (derived in Chap. 14) is:

$$\displaystyle{Y = \frac{C_{0} + I_{0} + G + G\!X_{Y }Y ^{f} + G\!X_{\varepsilon }\varepsilon _{0} + G\!X_{\varepsilon }\varepsilon _{r}r^{f}} {1 -\text{MPE}} -\frac{G\!X_{\varepsilon }\varepsilon _{r} + I_{r}} {1 -\text{MPE}} \cdot r.}$$

Equilibrium for the economy as a whole is where there is equilibrium in both the money market and the goods market—the pair of values of Y and r that are on both the IS and the LM curve simultaneously. In other words, equilibrium for the economy as a whole is when IS = LM:

$$\displaystyle\begin{array}{rcl} \frac{C_{0} + I_{0} + G + G\!X_{Y }Y ^{f} + G\!X_{\varepsilon }\varepsilon _{0} + G\!X_{\varepsilon }\varepsilon _{r}r^{f}} {1 -\text{MPE}} & -& \frac{G\!X_{\varepsilon }\varepsilon _{r} + I_{r}} {1 -\text{MPE}} \cdot r {}\\ & =& \left ( \frac{1} {m_{1}}M^{S} -\frac{m_{0}} {m_{1}} + \frac{m_{2}} {m_{1}}\pi ^{e}\right ) + \frac{m_{2}} {m_{1}}r {}\\ \end{array}$$

This is now a single linear equation in one unknown (that is, r). If you bring the r terms over to one side, then divide by the items that are multiplied by r, you can solve for the equilibrium level of r.

Once you know the equilibrium value of r, you can plug it back into either the IS or the LM curve to find the corresponding equilibrium level of Y.

Problems

Problem 15.1

Consider an economy with the following parameters:

Table 1

Assume potential output is 1,117.

• (a) Determine the equation for the IS curve.

• (b) Determine the equation for the LM curve.

• (c) Use your IS and LM curves to determine the economy’s short-run equilibrium output y and real interest rate r.

• (d) In percentage terms, what is the size of the output gap?

Problem 15.2

Except where specified, use the same parameters as in Problem 15.1.

• (a) If government expenditure falls from $200 to $194, how much of a change in Y do you expect, based simply on the expenditure multiplier of Chap. 11?

• (b) Which curve (IS or LM) is affected by the change in G?

• (c) Rewrite the curve you identified in (b), to reflect the change in G.

• (d) What are the new equilibrium values of Y and r? How much did Y actually change?

• (e) What is the new size of the output gap?

Problem 15.3

Go back to the parameters provided in Problem 15.1, with G = $200.

• (a) If M ^s is increased to $270, which curve is affected?

• (b) Rewrite the curve you identified in (a), to reflect the new value of M ^s.

• (c) Solve for the new equilibrium values of Y and r.

• (d) Qualitatively, how has this outcome been similar to and different from the outcome in Problem 15.2?

Problem 15.4

Once again, go back to the parameters of Problem 15.1. This time, consider expected inflation π ^e falling from 3% to 2.9%.

• (a) Which curve is affected?

• (b) Rewrite the curve you identified in (a), to reflect the new value of π ^e.

• (c) Solve for the new equilibrium values of Y and r.

• (d) Explain conceptually why the fall in expected inflation had the effects on output and real interest that you identified in (c).

Problem 15.5

Revisit Problem 14.6.

• (a) How do your answers there change your view of the efficacy of fiscal policy?

• (b) How do your answers to Problem 14.6 change your view of the efficacy of monetary policy?