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.

We start by differentiating the expenditure function \(P_x X\) with respect to income \(I\). This gives us the following expression: $$ \frac{\partial P_x X}{\partial I} = P_x \frac{\partial X}{\partial I} $$ Now, we can substitute this expression back into the expenditure elasticity equation: $$ e_{P_x \cdot x, 1} = P_x \frac{\partial X}{\partial I} \cdot \frac{I}{P_x X} $$

To prove the first identity, let's first find the elasticity of expenditure with respect to quantity, \(x_j\). We start with the definition of elasticity: $$ e_{P_x \cdot x, x_j} = \frac{\partial P_x X}{\partial x_j} \cdot \frac{x_j}{P_x X} $$ Now, differentiate the expenditure function (\(P_x X = P_x x_j\)) with respect to \(x_j\): $$ \frac{\partial P_x x_j}{\partial x_j} = P_x $$ Substitute this back into the elasticity equation: $$ e_{P_x \cdot x, x_j} = P_x \cdot \frac{x_j}{P_x X} $$ We can now show the identity \(e_{P_x} \cdot x_j = e_{xj}\), by substituting the value of \(e_{P_x \cdot x, 1}\) we found in Step 1: $$ e_{P_x} \cdot x_j = P_x \frac{\partial X}{\partial I} \cdot \frac{I}{P_x X} \cdot x_j = P_x \cdot \frac{x_j}{P_x X} = e_{xj} $$

To prove the second identity, let's find the elasticity of expenditure with respect to price, \(P_x\). We start with the definition of elasticity: $$ e_{P_x \cdot x, P_x} = \frac{\partial P_x X}{\partial P_x} \cdot \frac{P_x}{P_x X} $$ Now, differentiate the expenditure function (\(P_x X = P_x x_j\)) with respect to \(P_x\): $$ \frac{\partial P_x x_j}{\partial P_x} = x_j $$ Substitute this back into the elasticity equation: $$ e_{P_x \cdot x, P_x} = x_j \cdot \frac{P_x}{P_x X} $$ The income elasticity of demand, \(e_{x, p_x}\), is given by: $$ e_{x, p_x} = \frac{\partial X}{\partial p_x} \cdot \frac{p_x}{X} $$ Now, differentiate the demand function (\(X = \frac{I}{P_x}\)) with respect to \(p_x\): $$ \frac{\partial \frac{I}{P_x}}{\partial p_x} = -\frac{I}{P_x^2} $$ Substitute this back into the elasticity equation: $$ e_{x, p_x} = -\frac{I}{P_x^2} \cdot \frac{p_x}{X} $$ Now we can show the identity \(e_{P_x \cdot x, P_x} = 1 + e_{x, p_x}\). First, let's find the value of \(1 + e_{x, p_x}\): $$ 1 + e_{x, p_x} = 1 - \frac{I}{P_x^2} \cdot \frac{p_x}{X} $$ Substitute the value of \(X = \frac{I}{P_x}\): $$ 1 + e_{x, p_x} = 1 - \frac{I}{P_x^2} \cdot \frac{p_x P_x}{I} = 1 - \frac{P_x}{P_x X} $$ Comparing this with the value we found for \(e_{P_x \cdot x, P_x}\): $$ e_{P_x \cdot x, P_x} = 1 + e_{x, p_x} $$ Thus, we have proven both identities and shown how income and price elasticity can be derived from expenditure elasticities.