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Chapter 13: Problem 11

Suppose there are only three goods \(\left(x_{1}, x_{2}, x_{3}\right)\) in an economy and that the excess demand functions for \(x_{2}\) and \(x_{3}\) are given by \\[ \begin{array}{l} E D_{2}=-\frac{3 p_{2}}{p_{1}}+\frac{2 p_{3}}{p_{1}}-1 \\ E D_{3}=\frac{4 p_{2}}{p_{1}}-\frac{2 p_{3}}{p_{1}}-2 \end{array} \\] a. Show that these functions are homogeneous of degree 0 in \(p_{1}, p_{2},\) and \(p_{3}\) b. Use Walras' law to show that, if \(E D_{2}=E D_{3}=0,\) then \(E D_{1}\) must also be 0. Can you also usc Walras' law to calculate \(E D_{1}\) ? 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
The excess demand functions are homogeneous of degree 0. If \(ED_2 = ED_3 = 0\), then \(ED_1 = 0\) by Walras' law. Equilibrium requires \(\frac{p_2}{p_1} = 3\), \(\frac{p_3}{p_1} = 5\), and \(\frac{p_3}{p_2} = \frac{5}{3}\).

Step by step solution

01

Verify Homogeneity Degree 0

To show homogeneity, multiply the prices by a scalar \( t \) and check if the excess demand \( ED_2 \) and \( ED_3 \) remain unchanged.For \( ED_2\):\[ ED_2(-\frac{3(tp_2)}{tp_1}+\frac{2(tp_3)}{tp_1}-1) = -\frac{3p_2}{p_1}+\frac{2p_3}{p_1}-1 = ED_2 \]For \( ED_3\):\[ ED_3(\frac{4(tp_2)}{tp_1}-\frac{2(tp_3)}{tp_1}-2) = \frac{4p_2}{p_1}-\frac{2p_3}{p_1}-2 = ED_3 \]Both functions are homogeneous of degree 0 as they are invariant to changes in scale.
02

Apply Walras' Law

Walras’ law states that the sum of weighted excess demands equals zero. If \( ED_2 = 0 \) and \( ED_3 = 0 \), we examine if \( ED_1 = 0 \) really is implied:\[ p_1 \times ED_1 + p_2 \times ED_2 + p_3 \times ED_3 = 0 \] - Given \( ED_2 = 0 \) and \( ED_3 = 0 \), the equation reduces to \( p_1 \times ED_1 = 0 \).Hence \( ED_1 = 0 \) since \( p_1 eq 0 \).
03

Calculate Excess Demand for Good 1 Using Walras' Law

If \( p_1 \times ED_1 + p_2 \times ED_2 + p_3 \times ED_3 = 0 \) and both \( ED_2 = 0 \) and \( ED_3 = 0 \) are satisfied, directly substituting provides no additional data as the entries are already zero: Therefore, if \( ED_2 = 0 \) and \( ED_3 = 0 \) hold true, then \( ED_1 = 0 \) must follow. Thus:\[ ED_1 = -(ED_2 + ED_3) = 0 + 0 = 0 \] so no calculation is needed as equation effectively cancels out.
04

Solve for Relative Prices in Equilibrium

With \( ED_2 = 0 \) and \( ED_3 = 0 \), solve the system:From \( ED_2 = 0 \):\[ -\frac{3p_2}{p_1} + \frac{2p_3}{p_1} - 1 = 0 \]This simplifies to:\[ 2p_3 = 3p_2 + p_1 \] From \( ED_3 = 0 \):\[ \frac{4p_2}{p_1} - \frac{2p_3}{p_1} - 2 = 0 \]This simplifies to:\[ 4p_2 = 2p_3 + 2p_1 \]Solving these two equations simultaneously:1. Substitute the expression from the first equation into the second: \[ 4p_2 = 2\left(\frac{3p_2 + p_1}{2}\right) + 2p_1 \] 2. Simplify and solve for \( p_2 / p_1\): \[ 4p_2 = 3p_2 + p_1 + 2p_1 \] \[ p_2 = 3p_1 \]3. Substitute \( p_2 = 3p_1 \) into the equation \( 2p_3 = 3p_2 + p_1 \): \[ 2p_3 = 3(3p_1) + p_1 \] \[ 2p_3 = 9p_1 + p_1 = 10p_1 \] \[ p_3 = 5p_1 \]Thus, the equilibrium relative prices are \( \frac{p_2}{p_1} = 3 \) and \( \frac{p_3}{p_1} = 5 \).
05

Calculate Equilibrium Value for Price Ratio \(p_3 / p_2\)

With the equilibrium prices \( \frac{p_2}{p_1} = 3 \) and \( \frac{p_3}{p_1} = 5 \):Calculate the price ratio:\[ \frac{p_3}{p_2} = \frac{\frac{p_3}{p_1}}{\frac{p_2}{p_1}} = \frac{5}{3} \]Thus, the equilibrium value for \( \frac{p_3}{p_2} \) is \( \frac{5}{3} \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Homogeneity of Degree 0
Understanding homogeneity of degree 0 is crucial in analyzing excess demand functions. A function is homogeneous of degree 0 if, when we multiply all input variables by the same factor, the output remains unchanged. This implies that the function is scale-independent.

Consider the excess demand functions given in the problem:
  • For good 2: \(ED_2 = -\frac{3 p_2}{p_1} + \frac{2 p_3}{p_1} - 1 \)
  • For good 3: \(ED_3 = \frac{4 p_2}{p_1} - \frac{2 p_3}{p_1} - 2\)
To check for homogeneity, observe what happens when each price \(p_1, p_2,\) and \(p_3\) is multiplied by a common scalar \(t\):
  • \(ED_2(-\frac{3(tp_2)}{tp_1} + \frac{2(tp_3)}{tp_1} - 1) = -\frac{3p_2}{p_1} + \frac{2p_3}{p_1} - 1 = ED_2 \)
  • \(ED_3(\frac{4(tp_2)}{tp_1} - \frac{2(tp_3)}{tp_1} - 2) = \frac{4p_2}{p_1} - \frac{2p_3}{p_1} - 2 = ED_3 \)
Both functions retain their forms, confirming homogeneity of degree 0.
Walras' Law
Walras' Law is a foundational concept in general equilibrium theory, stating that the total value of excess demands across all markets in an economy sums to zero when evaluated at market prices. This implies that if some markets have excess supply, others must have excess demand.

Given that \( ED_2 = 0 \) and \( ED_3 = 0 \), Walras' Law helps us examine what happens to \(ED_1\).

According to the law,:
  • \(p_1 \times ED_1 + p_2 \times ED_2 + p_3 \times ED_3 = 0\)
  • Substituting the given conditions \((ED_2 = 0, ED_3 = 0)\), we get \(p_1 \times ED_1 = 0\)
Since prices are positive, \(ED_1\) must be zero. This demonstrates how zero excess demand in some goods influences the overall market conditions, ensuring that the sum of all excess demands equates to zero.
Equilibrium Relative Prices
Equilibrium relative prices are the price ratios that balance the economy's supply and demand without any excess. They enable us to analyze the price structure within the economy.

From the zero excess demand conditions \((ED_2 = 0, ED_3 = 0)\), we derive two key equations:
  • From \(ED_2 = 0\): \( 2p_3 = 3p_2 + p_1 \)
  • From \(ED_3 = 0\): \( 4p_2 = 2p_3 + 2p_1 \)
By solving these equations simultaneously:
  • We substitute and rearrange them to find \( p_2 = 3p_1\)
  • Substitute \(p_2\) in the first equation, leading to \(p_3 = 5p_1\)
Therefore, the equilibrium relative prices are\( \frac{p_2}{p_1} = 3 \) and \( \frac{p_3}{p_1} = 5 \). These ratios represent a stable price structure where no changes in supply or demand disturb market balance.
Price Ratios
Price ratios provide insight into the relative valuation of goods within an economy. They are essential for understanding both consumer choices and producer allocations. After determining equilibrium relative prices, we can deduce the price ratio between any two goods.

For the exercise, we've solved for:
  • \( \frac{p_2}{p_1} = 3 \)
  • \( \frac{p_3}{p_1} = 5 \)
Now, we find \( \frac{p_3}{p_2} \):
  • Express \( \frac{p_3}{p_2} \) as \( \frac{\frac{p_3}{p_1}}{\frac{p_2}{p_1}}\)
  • Substitute the known values: \(\frac{p_3}{p_2} = \frac{5}{3} \)
This ratio \(\frac{5}{3}\) is a reflection of the equilibrium where good 3 is valued higher than good 2, indicating their relative scarcity or utility in the market. Price ratios such as this one are pivotal for analyzing shifts in market dynamics.

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Most popular questions from this chapter

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 land-intensive commodity. a. Explain, in words or with diagrams, how the price of wheat relative to cloth ( \(p\) ) determines the land-labor ratio in cach 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 then the output of cloth will rise and the output of wheat will fall. Note: This result was discovered by the Polish economist Tadeusz Rybczynski. It is a fundamental result in the theory of international trade.

Consider an economy with just one technique available for the production of each good. $$\begin{array}{lcc} \hline \text { Good } & \text { Food } & \text { Cloth } \\ \hline \text { Labor per unit output } & 1 & 1 \\ \text { Land per unit output } & 2 & 1 \\ \hline \end{array}$$ a. Suppose land is unlimited but labor cquals 100 . Write and sketch the production possibility fronticr. b. Suppose labor is unlimited but land equals \(150 .\) Write and sketch the production possibility frontier. c. Suppose labor equals 100 and land equals \(150 .\) Write and sketch the production possibility frontier. Hint: What are the intercepts of the production possibility fronticr? When is land fully employed? Labor? Both? d. Explain why the production possibility frontier of part (c) is concave. e. Sketch the relative price of food as a function of its output in part (c). f. If consumers insist on trading 4 units of food for 5 units of cloth, what is the relative price of food? Why? g. Explain why production is exactly the same at a price ratio of \(p_{F} / p_{C}=1.1\) as at \(p_{F} / p_{C}=1.9\) h. Suppose that capital is also required for producing food and clothing and that capital requirements per unit of food and per unit of clothing are 0.8 and \(0.9,\) respectively. There are 100 units of capital available. What is the production possibility curve in this case? Answer part (e) for this case.

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).

In the country of Ruritania there are two regions, \(A\) and \(B\). Two goods \((x \text { and } y)\) are produced in both regions. Production functions for region \(A\) are given by \\[ \begin{array}{l} x_{A}=\sqrt{l_{x}} \\ y_{A}=\sqrt{l_{y}} \end{array} \\] here \(l_{x}\) and \(l_{y}\) are the quantities of labor devoted to \(x\) and \(y\) production, respectively. Total labor available in region \(A\) is 100 units; that is, \\[ l_{x}+l_{y}=100 \\] Using a similar notation for region \(B\), production functions are given by \\[ \begin{array}{l} x_{B}=\frac{1}{2} \sqrt{l_{x}} \\ y_{B}=\frac{1}{2} \sqrt{l_{y}} \end{array} \\] There are also 100 units of labor available in region \(B:\) \\[ l_{x}+l_{y}=100 \\] a. Calculate the production possibility curves for regions \(A\) and \(B\). b. What condition must hold if production in Ruritania is to be allocated efficicntly between regions \(A\) and \(B\) (assuming labor cannot move from one region to the other)? c. Calculate the production possibility curve for Ruritania (again assuming labor is immobile between regions \() .\) How much total \(y\) can Ruritania produce if total \(x\) output is \(12 ?\) Hint: \(\mathrm{A}\) graphical analysis may be of some help here.

Suppose that Robinson Crusoe produces and consumes fish \((F)\) and coconuts \((C) .\) Assume that, during a certain period, he has decided to work 200 hours and is indifferent as to whether he spends this time fishing or gathering coconuts. Robinson's production for fish is given by \\[ F=\sqrt{l_{F}} \\] and for coconuts by \\[ C=\sqrt{l_{C}} \\] where \(l_{F}\) and \(l_{C}\) are the number of hours spent fishing or gathering coconuts. Consequently, \\[ l_{C}+l_{F}=200 \\] Robinson Crusoe's utility for fish and coconuts is given by \\[ \text { utility }=\sqrt{\boldsymbol{F} \cdot \boldsymbol{C}} \\] a. If Robinson cannot trade with the rest of the world, how will he choose to allocate his labor? What will the optimal levels of \(F\) and \(C\) be? What will his utility be? What will be the \(R P T\) (of fish for coconuts)? b. Suppose now that trade is opened and Robinson can trade fish and coconuts at a price ratio of \(p_{F} / p_{C}=2 / 1 .\) If Robinson continues to produce the quantities of \(F\) and \(C\) from part \((a)\) what will he choose to consume once given the opportunity to trade? What will his new level of utility be? c. How would your answer to part (b) change if Robinson adjusts his production to take advantage of the world prices? d. Graph your results for parts (a), (b), and (c).
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