Problem 9 - Advanced Macroeconomics

Question

Money versus interest-rate targeting. (Poole, \(1970 .\) ) Suppose the economy is described by linear \(I S\) and money-market equilibrium equations that are subject to disturbances: \(y=c-a i+\varepsilon_{1}, m-p=h y-k i+\varepsilon_{2},\) where \(\varepsilon_{1}\) and \(\varepsilon_{2}\) are independent, mean-zero shocks with variances \(\sigma_{1}^{2}\) and \(\sigma_{2}^{2},\) and where \(a, h,\) and \(k\) are positive. Policymakers want to stabilize output, but they cannot observe \(y\) or the shocks, \(\varepsilon_{1}\) and \(\varepsilon_{2}\). Assume for simplicity that \(p\) is fixed. (a) Suppose the policymaker fixes \(i\) at some level \(\bar{i}\). What is the variance of \(y ?\) (b) Suppose the policymaker fixes \(m\) at some level \(\bar{m}\). What is the variance of \(y ?\) \((c)\) If there are only monetary shocks (so \(\sigma_{1}^{2}=0\) ), does money targeting or interest-rate targeting lead to a lower variance of \(y ?\) (d) If there are only \(I S\) shocks (so \(\sigma_{2}^{2}=0\) ), does money or interest-rate targeting lead to a lower variance of \(y ?\) \((e)\) Explain your results in parts \((c)\) and \((d)\) intuitively. \((f)\) When there are only \(I S\) shocks, is there a policy that produces a variance of \(y\) that is lower than either money or interest-rate targeting? If so, what policy minimizes the variance of \(y ?\) If not, why not? (Hint: Consider the money-market equilibrium condition, \(m-p=h y-k i .\) )

Step-by-Step Solution

Answer

Money targeting results in lower variance of output for both pure monetary and IS shocks.

Step 1: Find variance of y with interest rate targeting

When policymakers fix the interest rate at \( \bar{i} \), the IS curve becomes \( y = c - a \bar{i} + \varepsilon_1 \). In this case, the variance of output, \( \text{Var}(y) \), is simply the variance of the shock \( \varepsilon_1 \). Hence \( \text{Var}(y) = \sigma_1^2 \).

Step 2: Determine variance of y with money supply targeting

When policymakers fix the money supply at \( \bar{m} \), the money market equation becomes \( \bar{m} = h y - k i + \varepsilon_2 \). Solving for \( i \) gives \( i = \frac{h}{k}y - \frac{\bar{m} - \varepsilon_2}{k} \). Substituting \( i \) into the IS curve gives:\[y = c - a\left(\frac{h}{k}y - \frac{\bar{m} - \varepsilon_2}{k}\right) + \varepsilon_1\]Solving for \( y \) yields \( y = \frac{c + a\frac{(\bar{m} - \varepsilon_2)}{k} + \varepsilon_1}{1 + a\frac{h}{k}} \). The variance of \( y \) is:\[\text{Var}(y) = \left( \frac{1}{1 + a\frac{h}{k}} \right)^2 \sigma_1^2 + \left( \frac{a}{k(1 + a\frac{h}{k})} \right)^2 \sigma_2^2 \].

Step 3: Compare policies for only monetary shocks

If \( \sigma_1^2 = 0 \) (only monetary shocks), then the variance of \( y \) under money targeting is \( \text{Var}(y) = \left(\frac{a}{k(1 + a\frac{h}{k})}\right)^2 \sigma_2^2 \) which is less than \( \sigma_2^2 \) as \( \frac{a}{k(1 + a\frac{h}{k})} < 1 \). Hence, money targeting leads to a lower variance for \( y \).

Step 4: Compare policies for only IS shocks

With only IS shocks (\( \sigma_2^2 = 0 \)), the variance under money targeting reduces to \( \text{Var}(y) = \left( \frac{1}{1 + a\frac{h}{k}} \right)^2 \sigma_1^2 \), which is less than \( \sigma_1^2 \) because \( \frac{1}{1 + a\frac{h}{k}} < 1 \). Thus, money targeting also leads to a lower variance for \( y \) under IS shocks.

Step 5: Intuition for results in c and d

When only monetary shocks are present, targeting money supply neutralizes the effect of \( \varepsilon_2 \) thereby stabilizing output more effectively than interest-rate targeting. Under IS shocks, fixing the money supply allows adjustments in the interest rate that can offset \( \varepsilon_1 \), again leading to greater stability of \( y \).

Step 6: Explore policy for minimum variance with IS shocks

When only IS shocks are present, setting a policy such that \( \text{Var}(y) = 0 \) is not possible with the given structure because there are inherent limits to how much output can be stabilized purely through monetary policy without further information or instruments. Thus, neither targeting method can produce zero variance.

Key Concepts

Interest Rate Targeting

Interest rate targeting is a common tool in monetary policy where the central bank sets a specific level for interest rates to influence the economy. When they fix the interest rate at a certain level, say \( \bar{i} \), the policymakers aim to stabilize the economy's output. This involves controlling the cost of borrowing money.

By fixing the interest rate, the policymakers assume that businesses and consumers will make predictable decisions about spending and investment. The main goal is to keep the economy operating smoothly by managing inflation and controlling unemployment. However, focusing solely on interest rates can sometimes lead to less flexible responses to unexpected economic disturbances.

• Interest rates impact borrowing and spending.
• Maintaining a fixed interest rate aims at economic stability.
• Challenges appear when unexpected events cause shifts in economic demand.

When disturbances, such as shocks to investment \( \varepsilon_{1} \), hit the economy, the fixed interest rate can sometimes lead to volatile output levels. In the given exercise, the variance of output \( y \), which measures the dispersion of \( y \) in response to shocks, is only dependent on \( \varepsilon_{1} \) if the interest rate is targeted. Hence, the policymaker needs to carefully consider which shocks are more prevalent when choosing a policy strategy.

IS-LM Model

The IS-LM model combines the Investment-Savings (IS) curve and the Liquidity preference-Money supply (LM) curve to illustrate how interest rates and real output are determined simultaneously. It is a fundamental framework in macroeconomics used to examine how the goods and money markets interact.

The IS curve represents the equilibrium in the goods market, linking interest rates and output levels. On the other hand, the LM curve represents the equilibrium in the money market, where money demand equals money supply at given interest rates.

• IS curve: Reflects combinations of interest rates and output where the goods market is in equilibrium.
• LM curve: Indicates combinations where the money market is in equilibrium.
• A shift in either the IS or LM curve can affect the equilibrium interest rate and output.

In the context of the exercise, when the policymakers choose to fix either the interest rate or the money supply, they influence the IS or LM curves, respectively. For example, fixing the money supply constrains the LM curve, and the equilibrium will adjust due to changes in the IS curve, leading to different variances in output, \( y \). Understanding these dynamics is crucial for assessing how monetary policies impact the economy.

Output Stabilization

Output stabilization refers to efforts by policymakers to keep the level of economic activity steady despite disturbances. Fluctuations in output can lead to economic hardships, such as unemployment, instability, and decreased consumer and business confidence. By stabilizing output, policymakers aim to create a more predictable and smoother economic environment.

One method of output stabilization that the exercise explores is how targeting either interest rates or the money supply can influence the variability of output. The goal is to minimize the variance of potential output changes that arise from economic shocks.

• Stable output reduces economic uncertainty.
• Choosing the right policy target helps achieve desired stability.
• The policy's effectiveness depends on the types of economic shocks.

In the exercise's context, when facing IS shocks (investment or spending shocks) or monetary shocks, the variance in output varies based on whether interest rates or money supply is targeted. Understanding which approach leads to lower variability involves analyzing how different disturbances affect the economy and choosing the strategy that mitigates the fluctuations most effectively. Neither method alone can achieve perfect stabilization when IS shocks are present, indicating the complexity of economic policy.