§5 An Extended IS-LM Model

Nominal vs Real Interest Rates

• Nominal interest rate is the interest rate in terms of dollars, $i$.

• Real interest rate is the interest rate in terms of a basket of goods, $r$.

• Ex-ante, the difference is the expected inflation for the corresponding period.

• Why do we care about this distinction?

  • Because private sector decisions, such as (physical) investment, depend on the real, not the nominal interest rate.

• One-year real interest rate at time $t$, $r_t$:

  $$1 + r_t=(1 + i_t)\frac{P_t}{P^{e}_{t + 1}}$$

• Denote the expected inflation between $t$ and $t + 1$ by $\pi_{t+1}^{e}$:

  $$\pi_{t + 1}^{e}\equiv\frac{P_{t+1}^{e}-P_t}{P_t}$$

  to imply

  $$1 + r_t=\frac{1 + i_t}{1+\pi_{t+1}^{e}}$$

  or, approximately:

  $$r_t\approx i_t-\pi_{t+1}^{e}$$

Risk and Risk Premium

• Since some bonds are risky, bond holders require a risk premium to hold these bonds, $x_t$

  $$r_t^F = r_t + x_t$$

• The risk premium moves a lot during the business cycle; it tends to be higher during recessions.

• Why do we care about the risk premium?

  • Because private sector decisions, such as (physical) investment, depend on the risk-adjusted interest rate, $r_t^F$.

• The risk premium is determined by

  • The probability of default;
  • The degree of risk aversion of bond holders.

• Let‘s ignore the second channel (which is quite important and we will revisit much later in the course), and denote the probability of default by $p_t$, then:

  $$\begin{aligned} 1 + r_t&=(1 - p_t)\left(1 + r_t^F\right)+p_t*0\\ &=(1 - p_t)(1 + r_t + x_t) \end{aligned}$$

  To imply

  $$x_t=\frac{p_t}{1 - p_t}(1 + r_t)$$

• During severe recessions, $p_t$ can rise a lot (and a lot more than $r_t$ can decline, so $r_t^F$ goes up).

Extending the IS-LM Model

• Let‘s extend the basic IS-LM model to incorporate these new concepts.

• All the key changes are in the IS:

  $$Y = C(Y - T)+I(Y,i-\pi^{e}+x)$$

• LM remains the same (warning: difference with book)

  $$i=\bar{i}$$

— Apr 13, 2025

§5 An Extended IS-LM Model by Lu Meng is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License. Permissions beyond the scope of this license may be available at About.