Question

Suppose there are only three goods \(\left(X_{1}, X_{2}, \text { and } X_{3}\right)\) in an economy and that the excess demand functions for \(X_{2}\) and \(X_{3}\) are given by \\[ \begin{array}{l} \boldsymbol{E D}_{2}=-3 \boldsymbol{P}_{2} / \boldsymbol{P}_{1}+2 \boldsymbol{P}_{3} / \boldsymbol{P}_{1}-1 \\ \boldsymbol{E} \boldsymbol{D}_{3}=4 P_{2} / \boldsymbol{P}_{1}-2 \boldsymbol{P}_{3} / \boldsymbol{P}_{1}-2 \end{array} \\] a. Show that these functions are homogeneous of degree zero in \(P_{1}, P_{2},\) and \(P_{3}\) b. Use Walras' law to show that if \(E D_{2}=E D_{3}=0, E D_{1}\) also must be \(0 .\) Can you also use Walras' law to calculate \(F D_{1} ?\) c. Solve this system of equations for the equilibrium relative prices \(P_{2} / P_{1}\) and \(P_{3} / P_{1}\). What is the equilibrium value for \(P_{3} / P_{2} ?\)

Short answer

Based on the given excess demand functions for goods \(X_2\) and \(X_3\), we have shown that these functions are homogeneous of degree zero in \(P_1, P_2,\) and \(P_3\). Using Walras' law, we determined that if the excess demand functions for goods \(X_2\) and \(X_3\) are zero, the excess demand function for good \(X_1\) must also be zero, but we cannot directly calculate \(E D_{1}\) without knowing its functional form. Finally, we solved the system of equations for the equilibrium relative prices, finding that \(P_2 / P_1 = \frac{1}{3}\), \(P_3 / P_1 = 1\), and the equilibrium value for \(P_3 / P_2\) is 3.

Step-by-step solution

Multiply the prices by a constant factor

Let \(k\) be a positive constant, and let \(\widetilde{P_i} = k \cdot P_i\), where \(i = 1, 2, 3\). Now replace \(P_i\) with \(\widetilde{P_i}\) in the excess demand functions: $\widetilde{E D}_{2}=-3 \widetilde{P}_{2} / \widetilde{P}_{1}+2 \widetilde{P}_{3} / \widetilde{P}_{1}-1$ $\widetilde{E D}_{3}=4 \widetilde{P}_{2} / \widetilde{P}_{1}-2 \widetilde{P}_{3} / \widetilde{P}_{1}-2$

Check the change in the excess demand functions

Now replace \(\widetilde{P_i}\) with \(k \cdot P_i\): $\widetilde{E D}_{2}=-3 (k \cdot P_{2}) / (k \cdot P_{1})+2 (k \cdot P_{3}) / (k \cdot P_{1})-1$ $\widetilde{E D}_{3}=4 (k \cdot P_{2}) / (k \cdot P_{1})-2 (k \cdot P_{3}) / (k \cdot P_{1})-2$ We can cancel out the constant \(k\): $\widetilde{E D}_{2}=-3 P_{2} / P_{1}+2 P_{3} / P_{1}-1 = E D_{2}$ $\widetilde{E D}_{3}=4 P_{2} / P_{1}-2 P_{3} / P_{1}-2 = E D_{3}$ Since the excess demand functions do not change when multiplying all the prices by a constant factor, both functions are homogeneous of degree zero in \(P_1, P_2\), and \(P_3\). b. Use Walras' law to show \(E D_{1} = 0\) and check its direct calculation

Apply Walras' law to the given excess demand functions

According to Walras' law: \(E D_{1} + E D_{2} + E D_{3} = 0\) We have the excess demand functions for goods \(X_2\) and \(X_3\). If \(E D_{2} = E D_{3} = 0\), we can use Walras' law to show that \(E D_{1}\) also must be 0. However, we cannot use Walras' law directly to calculate \(E D_{1}\) without knowing its functional form. c. Solve for the equilibrium relative prices and find the equilibrium value for \(P_3 / P_2\)

Set the excess demand functions to zero and solve for the relative prices

To find the equilibrium relative prices, we set the excess demand functions to zero: $0 = -3 P_{2} / P_{1}+2 P_{3} / P_{1}-1$ $0 = 4 P_{2} / P_{1}-2 P_{3} / P_{1}-2$ Solve these equations for \(P_2 / P_1\) and \(P_3 / P_1\): \(P_2/P_1 = \frac{1}{3}\) \(P_3/P_1 = 1\)

Find the equilibrium value for \(P_{3} / P_{2}\)

Use the results obtained in Step 1 to find the value of \(P_3 / P_2\) in equilibrium: \(P_{3} / P_{2} = (P_{3} / P_{1}) / (P_{2} / P_{1}) = \frac{1}{\frac{1}{3}} = 3\) In equilibrium, the relative price \(P_3 / P_2 = 3\).

Key concepts

Homogeneity of Demand

The concept of *homogeneity of demand* in economics refers to the characteristic of demand functions to remain unchanged when all prices are multiplied by a common factor. This implies that the consumer's choice doesn't change if all prices and income are scaled by the same amount. In our exercise, this principle is illustrated by showing that the excess demand functions for goods have a degree of zero.

To understand this, consider the given excess demand functions:

• \( ext{ED}_2 = -3 \frac{P_2}{P_1} + 2 \frac{P_3}{P_1} - 1 \)
• \( ext{ED}_3 = 4 \frac{P_2}{P_1} - 2 \frac{P_3}{P_1} - 2 \)

By multiplying each price by a factor \(k\), the function does not change. After substituting \(P_i\) with \(kP_i\), and canceling \(k\), the equations remain unchanged. This shows how demand is homogeneous of degree zero. It means if all prices increase by the same factor, the excess demand functions remain unaffected. This is an essential property in economic theory, ensuring consistent consumer preferences irrespective of overall price levels.

Walras' Law

*Walras' Law* is a fundamental principle ensuring that the combined excess demand across all markets must sum to zero when markets are in equilibrium. In simpler terms, if some markets have excess supply, others must have excess demand of the same magnitude, or vice versa.

In the context of our exercise, when \( ext{ED}_2 = 0 \) and \( ext{ED}_3 = 0 \), Walras' Law dictates that \( ext{ED}_1 \) must also be zero. This means the market for good \(X_1\) must also clear when \(X_2\) and \(X_3\) do. Although we cannot directly compute \( ext{ED}_1 \) without its equation, knowing \( ext{ED}_1 + ext{ED}_2 + ext{ED}_3 = 0 \) provides assurance of this balance.

These relationships underscore how interconnected economic markets are. When one good's demand shifts, others respond to maintain economic consistency. It demonstrates the holistic view of markets in economic theory, where price changes have reciprocal effects, maintaining balance.

Equilibrium Relative Prices

In economic theory, finding the *equilibrium relative prices* means determining the price ratio at which the supply and demand across all markets in an economy are balanced. In our situation, we achieved this by setting the excess demand functions to zero, finding the relative prices that equalize demand and supply of goods.

From the equations:

• \( 0 = -3 \frac{P_2}{P_1} + 2 \frac{P_3}{P_1} - 1 \)
• \( 0 = 4 \frac{P_2}{P_1} - 2 \frac{P_3}{P_1} - 2 \)

We solved to find \( \frac{P_2}{P_1} = \frac{1}{3} \) and \( \frac{P_3}{P_1} = 1 \). This means in equilibrium, \(P_2\) is one-third of \(P_1\), while \(P_3\) equals \(P_1\). Moreover, these price ratios allow us to determine \( \frac{P_3}{P_2} \), which turns out to be 3.

These *relative prices* represent the economic balance within the market, ensuring that every good is optimally distributed according to demand and supply forces. Understanding equilibrium helps mechanics of resource allocation in economies and the effect of price mechanisms on market dynamics.