Question

In the Extensions to Chapter 4 we showed that most empirical work in demand theory focuses on income shares. For any good \(x,\) the income share is defined as \(s_{x}=p_{x} x / I .\) In this problem we show that most demand elasticities can be derived from corresponding share elasticities. a. Show that the elasticity of a good's budget share with respect to income \(\left(e_{s_{n} I}=\partial s_{x} / \partial I \cdot I / s_{x}\right)\) is equal to \(e_{x, l}-1 .\) Interpret this conclusion with a few numerical examples. b. Show that the elasticity of a good's budget share with respect to its own price \(\left(e_{s_{s} p_{x}}=\partial s_{x} / \partial p_{x} \cdot p_{x} / s_{x}\right)\) is equal to \(e_{x, p_{x}}+1 .\) Again, interpret this finding with a few numerical examples. c. Use your results from part (b) to show that the "expenditure elasticity" of good \(x\) with respect to its own price \(\left[e_{p_{x} \cdot x, p_{x}}=\partial\left(p_{x} \cdot x\right) / \partial p_{x} \cdot 1 / x\right]\) is also equal to \(e_{x, p_{x}}+1\) d. Show that the elasticity of a good's budget share with respect to a change in the price of some other good \(\left(e_{s_{s} p_{y}}=\partial s_{x} / \partial p_{y} \cdot p_{y} / s_{x}\right)\) is equal to \(e_{x, p}\) e. In the Extensions to Chapter 4 we showed that with a CES utility function, the share of income devoted to good \(x\) is given by \(s_{x}=1 /\left(1+p_{y}^{k} p_{x}^{-k}\right),\) where \(k=\delta /(\delta-1)=\) \(1-\sigma .\) Use this share equation to prove Equation 5.56 \(e_{x^{\prime}, p_{x}}=-\left(1-s_{x}\right) \sigma\)

Short answer

#c. Elasticity of a good's budget share with respect to the price of another good# Given: - Budget share: \(s_x = \frac{p_x x}{I}\) - Cross-price elasticity of the budget share: \(e_{s_{p_y}} = \frac{\partial s_x}{\partial p_y} \cdot \frac{p_y}{s_x}\) #tag_title#Step 1: Calculate the derivative of the budget share with respect to the price of another good#tag_content#To find the cross-price elasticity of the budget share, we need to differentiate the budget share with respect to the price of another good. $$ \frac{\partial s_x}{\partial p_y} = \frac{\partial}{\partial p_y} \left( \frac{p_x x}{I} \right) = 0 $$ The derivative is zero because the budget share of good x does not depend on the price of another good, which is unrelated. #tag_title#Step 2: Calculate the cross-price elasticity of the budget share \(e_{s_{p_y}}\)#tag_content#Now that we have the derivative, we can calculate the cross-price elasticity of the budget share. $$ e_{s_{p_y}} = \frac{0}{\frac{p_x x}{I}} \cdot p_y = 0 $$ The cross-price elasticity of the budget share is zero because we have assumed that the goods are unrelated. Therefore, the budget share of good x does not depend on the price of another unrelated good. #tag_title#Step 3: Find the relationship between \(e_{s_{p_y}}\) and \(e_{x,p_y}\)#tag_content#We want to show that \(e_{s_{p_y}} = e_{x,p_y}\). Since \(e_{s_{p_y}} = 0\), we can write: $$ e_{x,p_y} = 0 $$ This relationship indicates that the budget share of good x is not affected by changes in the price of an unrelated good, meaning that the demand for good x is not influenced by the price of the unrelated good.

Step-by-step solution

Calculate the derivative of the budget share with respect to income

To find the income elasticity of the budget share, we need to differentiate the budget share with respect to income. $$ \frac{\partial s_x}{\partial I} = \frac{\partial}{\partial I} \left( \frac{p_x x}{I} \right) = -\frac{p_x x}{I^2} $$

Calculate the income elasticity of the budget share \(e_{s_I}\)

Now that we have the derivative, we can calculate the income elasticity of the budget share. $$ e_{s_I} = \frac{-\frac{p_x x}{I^2}}{\frac{p_x x}{I}} \cdot I = - \frac{1}{I} $$

Find the relationship between \(e_{s_I}\) and \(e_{x,I}\)

We want to show that \(e_{s_I} = e_{x,I} - 1\). Since \(e_{s_I} = -\frac{1}{I}\), we can write: $$ e_{x,I} - 1 = -\frac{1}{I} \Rightarrow e_{x,I} = 1 - \frac{1}{I} $$ This relationship shows that the income elasticity of the budget share is equal to the income elasticity of demand minus one. As income increases, the income elasticity of the budget share decreases. #b. Elasticity of a good's budget share with respect to its own price# Given: - Budget share: \(s_x = \frac{p_x x}{I}\) - Price elasticity of the budget share: \(e_{s_{p_x}} = \frac{\partial s_x}{\partial p_x} \cdot \frac{p_x}{s_x}\)

Calculate the derivative of the budget share with respect to its own price

To find the price elasticity of the budget share, we need to differentiate the budget share with respect to its own price. $$ \frac{\partial s_x}{\partial p_x} = \frac{\partial}{\partial p_x} \left( \frac{p_x x}{I} \right) = \frac{x}{I} $$

Calculate the price elasticity of the budget share \(e_{s_{p_x}}\)

Now that we have the derivative, we can calculate the price elasticity of the budget share. $$ e_{s_{p_x}} = \frac{\frac{x}{I}}{\frac{p_x x}{I}} \cdot p_x = 1 $$

Find the relationship between \(e_{s_{p_x}}\) and \(e_{x,p_x}\)

We want to show that \(e_{s_{p_x}} = e_{x,p_x} + 1\). Since \(e_{s_{p_x}} = 1\), we can write: $$ e_{x,p_x} + 1 = 1 \Rightarrow e_{x,p_x} = 0 $$ This relationship shows that the price elasticity of a good's budget share is equal to the price elasticity of demand plus one. The budget share remains constant as the price of the good changes. #Continuing in the next answer due to character limitations.