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Chapter 16: Problem 7

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

Expert verified
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

01

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.
02

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
03

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
04

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

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Most popular questions from this chapter

The purpose of this problem is to examine the relationship among returns to scale, factor intensity, and the shape of the production possibility frontier. Suppose there are fixed supplies of capital and labor to be allocated between the production of good \(X\) and good \(Y\). The production function for \(X\) is given by \\[ X=K^{\alpha} L^{\beta} \\] and for \(Y\) by \\[ Y=K^{\gamma} L^{s} \\] where the parameters \(\alpha, \beta, \gamma, \delta\) will take on different values throughout this problem. Using either intuition, a computer, or a formal mathematical approach, derive the production possibility frontier for \(X\) and \(Y\) in the following cases: a. \(\quad \alpha=\beta=\gamma=\delta=\frac{1}{2}\) b. \(\quad \alpha=\beta=\frac{1}{2}, \gamma=\frac{1}{3}, \delta=\frac{2}{3}\) \(c_{*} \quad \alpha=\beta=\frac{1}{2}, \gamma=\delta=\frac{2}{3}\) d. \(\alpha=\beta=\gamma=\delta=\frac{2}{3}\) e. \(\quad \alpha=\beta=.6, \gamma=.2, \delta=1.0\) f. \(\quad \alpha=\beta=.7, \gamma=.6, \delta=.8\) Do increasing returns to scale always lead to a convex production possibility frontier? Explain.

Suppose silver is used as the medium of exchange in the economy described in Example \(16.4 .\) Does this economy exhibit the classical dichotomy?

Suppose an economy produces only two goods, \(X\) and \(Y\). Production of good \(X\) is given by where \(K_{x}\) and \(L_{x}\) are the inputs of capital and labor devoted to \(X\) production. The production function for good Fis given by \\[ =\sin ^{3} y^{-4} \lg ^{2} x^{2} \\] where \(K_{\text {; }}\) a.nd \(\mathrm{L}_{\text {, are the inputs of capital and labor devoted to } \mathrm{F} \text { production. The supply of }}\) 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} K_{x}+K_{Y}=K_{T}=100 \\ L_{x}+L_{Y}=L_{T}=200 \end{array} \\] Using this information, complete the following questions. a Show how the capital-labor ratio in \(X\) production \(\left(K / L_{x}=k_{x}\right)\) must be related to the capital-labor ratio in \(\mathrm{F}\) production \(\left(K_{y} / L_{Y}=k_{y}\right)\) if production is to be efficient. b. Show that the capital-labor ratios for the two goods are constrained by \\[ a_{x} k_{x}+\left(l-a_{x}\right) k_{y}=\underline{K}_{T}-\underline{100}_{-} \\] where \(a_{x}\) is the share of total labor devoted to \(X\) production [that is, \(a_{x}=L, / L_{r}=L_{s} /\) \((L x+L y) J\) c. Use the information from parts (a) and (b) to compute the efficient capital-labor ratio for good Xfor any value of \(a_{x}\) 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.

The country of Podunk produces only wheat and cloth, using as inputs land and labor. Both are produced by constant retums-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 (I'). Smith's utility function is given by \\[ U_{r}=X_{-}^{3} Y \\] whereas Jones' is given by \\[ U_{j}=X^{s} Y^{s} \\] The individuals do not care whether they produce \(X\) or \(Y\), and the production function for cach 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 Xand Fwill Smith and Jones demand? (Hint: Set the wage equal to 1 here. c. How should labor be allocated between \(X\) and \(Y\) to satisfy the demand calculated in part (b)?
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