Log out Go to App
Go to App

Learning Materials

Features

Discover

Chapter 16: Problem 7

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

Expert verified
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

01

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: $\widetilde{E D}_{2}=-3 \widetilde{P}_{2} / \widetilde{P}_{1}+2 \widetilde{P}_{3} / \widetilde{P}_{1}-1$ $\widetilde{E D}_{3}=4 \widetilde{P}_{2} / \widetilde{P}_{1}-2 \widetilde{P}_{3} / \widetilde{P}_{1}-2$
02

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
03

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 \(P_3 / P_2\)
04

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 \(P_3 / P_1\): \(P_2/P_1 = \frac{1}{3}\) \(P_3/P_1 = 1\)
05

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 \(P_3 / P_2 = 3\).

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Suppose the production possibility frontier for cheeseburgers ( \(C\) ) and milkshakes (M) is given by \\[ C+2 M=600 \\] a. Graph this function. b. Asuming that people prefer to eat two cheeseburgers with every milkshake, how much of each product will be produced? Indicate this point on your graph. c. Given that this fast-food economy is operating efficiently, what price ratio \(\left(P_{c} / P_{u}\right)\) must prevail?

The country of Podunk produces only wheat and cloth, using as inputs land and labor. Both are produced by constant returns-to-scale production functions. Wheat is the relatively landintensive commodity. a. Explain, in words or with diagrams, how the price of wheat relative to cloth \((p)\) deter mines the land-labor ratio in each of the two industries. b. Suppose that \(p\) is given by external forces (this would be the case if Podunk were a "small" country trading freely with a "large" world). Show, using the Edgeworth box, that if the supply of labor increases in Podunk, the output of cloth will rise and the output of wheat will fall.

Suppose two individuals (Smith and Jones) each have 10 hours of labor to devote to producing either ice cream \((X)\) or chicken soup \((Y) .\) Smith's utility function is given by \\[ \boldsymbol{U}_{s}=\boldsymbol{X}^{3} \boldsymbol{Y}^{\boldsymbol{\eta}} \\] whereas Jones" is given by \\[ U_{J}=X^{.5} Y^{3} \\] The individuals do not care whether they produce \(X\) or \(Y\), and the production function for each good is given by \\[ \begin{array}{l} X=2 L \\ Y=3 L \end{array} \\] where \(L\) is the total labor devoted to production of each good. Using this information, a. What must the price ratio, \(P_{x} / P_{Y},\) be? b. Given this price ratio, how much \(X\) and \(Y\) will Smith and Jones demand? (Hint: Set the wage equal to \(1 \text { here. })\) c. How should labor be allocated between \(X\) and \(Y\) to satisfy the demand calculated in part (b)?

Suppose the production possibility frontier for guns \((X)\) and butter \((Y)\) is given by \\[ X^{2}+2 Y^{2}=900 \\] a. Graph this frontier. b. If individuals always prefer consumption bundles in which \(Y=2 X\), how much \(X\) and \(Y\) will be produced? c. At the point described in part (b), what will be the \(R P T\) and hence what price ratio will cause production to take place at that point? (This slope should be approximated by considering small changes in \(X\) and \(Y\) around the optimal point.) d. Show your solution on the figure from part (a).

Suppose an economy produces only two goods, \(X\) and \(Y\). Production of good \(X\) is given by \\[ X=K_{x}^{1 / 2} L_{x}^{1 / 2} \\] where \(K_{X}\) and \(L_{X}\) are the inputs of capital and labor devoted to \(X\) production. The production function for good \(Y\) is given by \\[ Y=K_{r}^{1 / 3} \mathbf{L}_{v}^{2 / 3} \\] where \(K_{Y}\) and \(L_{\gamma}\) are the inputs of capital and labor devoted to \(Y\) production. The supply of capital is fixed at 100 units and the supply of labor is fixed at 200 units. Hence, if both units are fully employed, \\[ \begin{array}{l} \boldsymbol{K}_{\mathbf{x}}+\boldsymbol{K}_{\boldsymbol{r}}=\boldsymbol{K}_{\boldsymbol{T}}=100 \\\ \boldsymbol{L}_{\mathbf{x}}+\boldsymbol{L}_{\boldsymbol{\gamma}}=\boldsymbol{L}_{\boldsymbol{T}}=200 \end{array} \\] Using this information, complete the following questions. a. Show how the capital-labor ratio in \(X\) production \(\left(K_{x} / L_{x}=k_{x}\right)\) must be related to the capital-labor ratio in \(Y\) production \(\left(K_{Y} / L_{\gamma}=k_{Y}\right)\) if production is to be efficient. b. Show that the capital-labor ratios for the two goods are constrained by \\[ \alpha_{x} k_{x}+\left(1-\alpha_{x}\right) k_{r}=\frac{K_{T}}{L_{T}}=\frac{100}{200}=\frac{1}{2} \\] where \(\alpha_{x}\) is the share of total labor devoted to \(X\) production [that is, \(\alpha_{x}=L_{x} / L_{r}=L_{x} /\) \(\left.\left(L_{x}+L_{y}\right)\right]\) c. Use the information from parts (a) and (b) to compute the efficient capital-labor ratio for good \(X\) for any value of \(\alpha_{x}\) between 0 and 1 d. Graph the Edgeworth production box for this economy and use the information from part (c) to develop a rough sketch of the production contract curve. e. Which good, \(X\) or \(Y\), is capital intensive in this economy? Explain why the production possibility curve for the economy is concave. f. Calculate the mathematical form of the production possibility frontier for this economy (this calculation may be rather tedious!). Show that, as expected, this is a concave function.
See all solutions

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.

Sign-up for free