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Chapter 7: Problem 8

Show that for a two-good world, \({"} X \text { & } X_{1}, P_{x}\) If the own-price elasticity of demand for Xis known, what do we know about the cross-price elasticity for \(Y ?\) (Hint: Begin by taking the total differential of the budget constraint and set\(\left.\operatorname{ting} d l=0=d P_{Y} .\right)\)

Short Answer

Expert verified
Answer: We can find the cross-price elasticity of demand for good Y using the derived formula: \(\varepsilon_{Y,X}=-\frac{Py}{Px}\left(\varepsilon_X+1\right)\).

Step by step solution

01

Write the Budget Constraint for a Two-Good World

In a two-good world, we have the following budget constraint: I = Px * X + Py * Y. Here, I represents the income, Px and Py are the prices of good X and good Y respectively, and X and Y are the quantities of good X and good Y respectively.
02

Take the Total Differential of the Budget Constraint

To find the total differential of the budget constraint, we differentiate each term with respect to Px, X, Py, and Y, and set dI = 0 and dPy = 0 as given in the hint. Doing this, we get: 0 = X * dPx + Px * dX + Y * dPy + Py * dY.
03

Substitute dPy = 0 in the Total Differential Equation

Given that dPy = 0, the derived equation in Step 2 becomes: 0 = X * dPx + Px * dX + Py * dY.
04

Own-Price and Cross-Price Elasticity Definitions

Own-price elasticity of demand for good X is defined as: \(\varepsilon_{X}=\frac{d X}{d P_{X}}\left(\frac{P_{X}}{X}\right)\). Cross-price elasticity of demand for good Y with respect to the price of good X is defined as: \(\varepsilon_{Y, X}=\frac{d Y}{d P_{X}}\left(\frac{P_{X}}{Y}\right)\).
05

Find the Relationship between Own-Price Elasticity and Cross-Price Elasticity

We can rewrite the equation from Step 3 as: \(X * dPx + Px * dX = - Py * dY\). Now, divide both sides by \(Px * X * Py * Y\). This gives us: \(\frac{dX}{X} + \frac{dPx}{Px} = -\frac{dY}{Y} \cdot \frac{Px}{Py}\). We can now plug in the definitions of own-price elasticity and cross-price elasticity: \(\varepsilon_X + 1 = -\varepsilon_{Y,X} \cdot \frac{Px}{Py}\). Rearranging this to find an expression for the cross-price elasticity, we get: \(\varepsilon_{Y,X}=-\frac{Py}{Px}\left(\varepsilon_X+1\right)\).
06

Conclusion

Given the own-price elasticity of demand for good X, we can find the cross-price elasticity of demand for good Y using the derived formula: \(\varepsilon_{Y,X}=-\frac{Py}{Px}\left(\varepsilon_X+1\right)\).

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Most popular questions from this chapter

Suppose there are \(n\) individuals, each with a linear demand curve for \(Q\) of the form $$Q i=\mathrm{a},+b_{i} P+c j+d i P^{\prime} \quad i=1, n$$ where the parameters \(a, b_{h} c_{h}\) and \(d,\) differ among individuals. Show that at any point, the price elasticity of the market demand curve is independent of \(P^{\prime}\) and the distribution of income. Would this be true if each individual's demand for \(Q\) were instead linear in logarithms? Explain.

The "expenditure elasticity" for a good is defined as the proportional change in total expenditures on the good in response to a 1 percent change in income. That is, $$\mathbf{T}^{*}-\mathbf{M} \quad \overline{dl} \quad \overline{p_{x} x^{2}}$$ Prove that \(e_{R \cdot} \quad x=e_{X} .\) Show also that \(e_{P \cdot} \quad x z_{x}=1+e_{x z} .\) Both of these results are useful for empirical work in cases where quantity measures are not available, because income and price elasticities can be derived from expenditure elasticities.

A luxury is defined as a good for which the income elasticity of demand is greater than 1 Show that for a two-good economy, both goods cannot be luxuries. (Hint: What happens if both goods are luxuries and income is increased by 10 percent?)

In Example 7.2 we showed that with two goods the price elasticity of demand of a compensated demand curve is given by where \(s_{x}\) is the share of income spent on good \(X\) and \(c r\) is the substitution elasticity. Use this result together with the elasticity interpretation of the Slutsky equation to show that: a. if \(\mathrm{cr}=1\) (the Cobb-Doublas case), $$e_{x}, P_{x}+e_{Yy, Py}=-2$$ b. if \(a>1\) implies \(e_{x} p_{x}+e_{y} p_{Y}<-2\) and \(c r<1\) implies \(e_{x z_{x}}+e_{x x_{y}}>\sim 2 .\) These results can easily be generalized to cases of more than two goods.

Suppose that ham and cheese are pure complements- -they will always be used in the ratio of one slice of ham to one slice of cheese to make a sandwich. Suppose also that ham and cheese sandwiches are the only goods that a consumer can buy and that bread is free. Show that if the price of a slice of ham equals the price of a slice of cheese, a. The own-price elasticity of demand for ham is \(-| ;\) and b. The cross-price elasticity of a change in the price of cheese on ham consumption is also - - c. How would your answers to (a) and (b) change if a slice of ham cost twice as much as a slice of cheese? \(\\{\text {Hint}\) : Use the Slutsky equation - what is the substitution elasticity here?)
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