Question

Suppose there are only three goods \(\left(\mathrm{X}, \mathrm{X}_{2}, \text { and } \mathrm{X}_{3}\right)\) in an economy and that the excess demand functions for \(X>\) and \(X_{3}\) are given by \\[ \begin{array}{l} E D_{2}=-\mathrm{SP}_{\mathrm{J}} / \mathrm{P},+2 P J P_{,}-1 \\ E D_{3}=4 P_{3} / P,-96 P J P_{x}-2 \end{array} \\] a. Show that these functions are homogeneous of degree zero in \(P_{x}, P_{2},\) and \(P_{s}\) 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 \(E D\\{?\) c. Solve this system of equations for the equilibrium relative prices \(P_{2} / P i\) and \(P J P\) ( What is the equilibrium value for \(P J P^{\wedge}\)

Short answer

Yes, the excess demand functions \(ED_2\) and \(ED_3\) are both homogeneous of degree zero in \(P_{1}, P_{2}\), and \(P_{3}\). 2. Is it true that if \(ED_{2} = ED_{3} = 0\), then \(ED_{1}\) is also equal to zero? Yes, it is true. According to Walras' Law, when \(ED_{2}\) and \(ED_{3}\) are equal to zero, \(ED_{1}\) must also be equal to zero. 3. What are the equilibrium relative prices of \(P_{2}/P_{1}\) and \(P_{3}/P_{1}\)? The equilibrium relative prices are: $$\frac{P_2}{P_1} = \frac{1}{2}(SP + 1)$$ $$\frac{P_3}{P_1} = \frac{1}{4}(96P - 2)$$

Step-by-step solution

Check the homogeneity of ED2

To check if the excess demand function \(ED_2\) is homogeneous of degree zero, we apply \(P_1 = tP_1\), \(P_2 = tP_2\), and \(P_3 = tP_3\). Let's denote \(P_1\) as \(P\), substitute these functions into \(ED_2\):  $$ED_2 = -\frac{SP}{tP} + 2\frac{P_2}{tP} - 1$$ Now simplify: $$\frac{1}{t}(-SP + 2P_2 - t)$$ Since the exponent of t is -1, \(ED_2\) is homogeneous of degree zero.

Check the homogeneity of ED3

Similarly, we check if the excess demand function \(ED_3\) is homogeneous of degree zero by applying \(P = tP\), \(P_2 = tP_2\), and \(P_3 = tP_3\): $$ED_3 = 4\frac{P_3}{tP} - 96\frac{P_1P_2}{tPtP_2} - 2$$ Simplify the expression: $$\frac{1}{t}(4P_3 - 96P + 2t)$$ The exponent of t is -1, so \(ED_3\) is also homogeneous of degree zero. b. Use Walras' Law

Apply Walras' Law

Since \({ED_2} + {ED_3} + {ED_1} = 0\), we have: $$0 - SP + 2P_2 + 4P_3 - 96P + 2P_1 = 0$$ Since \(ED_2 = ED_3 = 0\), $$ED_1 = 0$$ This means that when \(ED_2\) and \(ED_3\) are equal to zero, \(ED_1\) must also be equal to zero. c. Solve the system of equations

Solve for equilibrium relative prices

To find the equilibrium relative prices, we need to solve the system of equations with the given excess demand functions: 1. \(ED_2 = 0\) 2. \(ED_3 = 0\) Solving the first equation for \(P_2/P_1\): $$-\frac{SP}{P} + 2\frac{P_2}{P} - 1 = 0$$ $$\frac{P_2}{P_1} = \frac{1}{2}(SP + 1)$$ Solving the second equation for $P_3/P_1: $$4\frac{P_3}{P} - 96\frac{P_1P_2}{P_1P_2} - 2 = 0$$ $$\frac{P_3}{P_1} = \frac{1}{4}(96P - 2)$$ The equilibrium relative prices are: $$\frac{P_2}{P_1} = \frac{1}{2}(SP + 1)$$ $$\frac{P_3}{P_1} = \frac{1}{4}(96P - 2)$$

Key concepts

Equilibrium Prices

In economics, equilibrium prices are the prices at which the quantity of goods demanded by consumers equals the quantity supplied by producers. This concept is crucial as it helps determine how resources are allocated in a market, ensuring that there is neither surplus nor shortage.
In the given exercise, the goal is to find these prices among three goods:

• \(X\)
• \(X_2\)
• \(X_3\)

To achieve an equilibrium, we solve the excess demand functions for these goods under the condition that they satisfy Walras' Law.
For the goods in question, the equilibrium condition requires setting the excess demand equal to zero, which simplifies the equations. By solving these simultaneously, we derive the relative prices that balance supply and demand, ensuring the market reaches a state of equilibrium.
This results in the equilibrium prices:

• \( \frac{P_2}{P_1} = \frac{1}{2}(SP + 1) \)
• \( \frac{P_3}{P_1} = \frac{1}{4}(96P - 2) \)

These ratios indicate the relative cost between goods, allowing efficient allocation and consumption in the market.

Homogeneous Functions

Homogeneous functions play a vital role in economic theories, especially in addressing how functions behave under scaling of their input variables. A function is considered homogeneous of degree zero if multiplying every input by the same factor leaves the function unchanged. This property ensures consistency in models by implying that only relative, rather than absolute, prices influence demand.

Homogeneity of degree zero is significant since it denotes that doubling the prices of goods shouldn't alter the demand proportion if consumer preferences remain constant. In our problem, we checked the excess demand functions for

• \(ED_2 \)
• \(ED_3 \)

and validated they were homogeneous of degree zero. This was done by substituting scaled prices into the equations and confirming the results were unchanged, after simplification.
This innate feature aligns with real-world scenarios where inflation or deflation uniformly affects all markets, leaving consumption patterns unaffected as they depend on prices relative to one another and not their absolute levels. This understanding shields economic models against variances that might otherwise skew results or interpretations.

Excess Demand

Excess demand occurs when the quantity demanded of a good exceeds the quantity supplied at a given price. It's a crucial element in market analysis as it indicates either shortage or surplus, prompting changes in prices to restore equilibrium.

In the given problem, the concept of excess demand is explored through specific functions for goods

• \(X_2 \)
• \(X_3 \)

These functions describe how the quantity demanded and supplied shift with price changes. The formulas

• \(ED_2 \) and
• \(ED_3 \)

involve prices for these goods, illustrating the dependency of demand on price levels.
When we set these functions to zero, it reflects a scenario where the market achieves an equilibrium, as no excess demand exists meaning both demand and supply are equal.This adjustment helps prices stabilize, fostering an efficient market with optimized resource allocation, and forms the core around which many economic principles are drawn.