Question

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, $${\mathbf{T}}^{\ast }-\mathbf{M}\overline{dl}\overline{p_x{x}^2}$$ Prove that $e_{R\cdot }x=e_X.$ Show also that $e_{P\cdot }x{z}_x=1+e_{xz}.$ 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.

Short answer

Answer: The two identities related to expenditure elasticity for a good proven in the solution are: 1) $e_{P_x}\cdot x_j=e_{xj}$, and 2) $e_{P_x\cdot x,P_x}=1+e_{x,p_x}$.

Step-by-step solution

Differentiate the expenditure function

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: $$e_{P_x\cdot x,1}=P_x\frac{\partial X}{\partial I}\cdot \frac{I}{P_{x}X}$$

Prove the first identity

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: $$e_{P_x}\cdot x_j=P_x\frac{\partial X}{\partial I}\cdot \frac{I}{P_{x}X}\cdot x_j=P_x\cdot \frac{x_j}{P_{x}X}=e_{xj}$$

Prove the second identity

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.

Key concepts

Income Elasticity

Income elasticity of demand helps us understand how the quantity demanded of a good changes with consumer income. It's calculated as the percentage change in quantity demanded divided by the percentage change in income. If we denote income as $I$ and quantity demanded as $Q$, income elasticity $E_I$ can be expressed as:$$E_I=\frac{\mathrm{\%}\text{change in }Q}{\mathrm{\%}\text{change in }I}$$Income elasticities are typically categorized as follows:

• **Normal Goods**: These have a positive income elasticity (greater than 0), meaning consumption increases as income rises.
• **Inferior Goods**: These have a negative income elasticity (less than 0), meaning consumption decreases as income increases.
• **Luxury Goods**: These exhibit a high positive income elasticity (greater than 1), and demand grows even faster than income increases.
• **Necessities**: These have an income elasticity between 0 and 1, indicating that demand grows but not as rapidly as income increases.

To apply this concept, think about what happens when someone gets a raise in salary. They'll likely buy more luxury goods, while they might reduce their consumption of inferior goods, like lower-cost store-brand products.

Price Elasticity

Price elasticity of demand measures how sensitive the quantity demanded of a good is to a change in its price. It is calculated by the percentage change in quantity demanded divided by the percentage change in the good's price. The formula is:$$E_P=\frac{\mathrm{\%}\text{change in }Q}{\mathrm{\%}\text{change in Price}}$$There are several types of price elasticity:

• **Elastic Demand**: When elasticity is greater than 1, meaning a small price change results in a larger change in quantity demanded.
• **Inelastic Demand**: When elasticity is less than 1, indicating consumers aren't very responsive to price changes.
• **Unitary Elastic Demand**: When elasticity is exactly 1, so price changes result in proportional changes in quantity demanded.
• **Perfectly Elastic Demand**: When elasticity approaches infinity, consumers will only purchase at one price, such as a currency exchange rate.
• **Perfectly Inelastic Demand**: When elasticity is 0, indicating quantity demanded is not affected by price changes; often seen in necessary goods, like insulin for diabetics.

Understanding these concepts is critical for businesses when setting prices, as it helps determine how a change in price might affect total revenue.

Elasticity Concepts

Elasticity is a fundamental concept in economics. It describes how one economic variable responds to a change in another. This concept includes income elasticity, price elasticity, and more. In practice, elasticity is used to:

• **Forecast Consumer Reaction**: Understanding elasticity can help businesses anticipate changes in consumer behavior as a result of varying prices or incomes.
• **Government Policy Making**: Policymakers use elasticity to predict the impacts of fiscal policies, such as tax changes, on consumption and revenue.
• **Market Strategy**: Companies can utilize elasticity to set optimal pricing strategies. For instance, if demand for a product is highly elastic, lowering the price could lead to higher overall sales volume.

The broader elasticity measure covers several contexts, such as cross-price elasticity, which examines how the quantity demanded of one good responds to price changes of another good, or supply elasticity, which looks at how quickly producers adjust quantity in response to price changes. Elasticity analysis provides crucial insights across various sectors, helping stakeholders make informed decisions by understanding the delicate interplay between price, income, and consumption.