Question

Show that for a two-good world, $$s_{x} e_{x, r_{x}}+s_{y} e_{y, r_{x}}=-s_{x}$$ If the own-price elasticity of demand for \(X\) is known, what do we know about the cross-price elasticity for \(Y\) ? (Hint: Bcgin by taking the total differential of the budget constraint and set\(\operatorname{ting} d I=0=d P_{y}\)

Short answer

#Answer# In a two-good world, as the demand for good X becomes more sensitive to its price changes, the demand for good Y will also become more sensitive in response to the price changes of good X. This means that an increase in own-price elasticity for good X leads to an increase in cross-price elasticity for good Y when holding the expenditure shares constant.

Step-by-step solution

Write down the budget constraint

The budget constraint for a two-good world is given by: $$M = P_x X + P_y Y$$ Where \(M\) is the total amount of income spent, \(P_x\) and \(P_y\) are the prices of good X and Y respectively, and \(X\) and \(Y\) are the quantities of goods X and Y consumed.

Take the total differential of the budget constraint

Differentiating the budget constraint with respect to income \(M\), price \(P_x\), and quantity \(X\) we get: $$dM = P_x dX + X dP_x + P_y dY + Y dP_y$$

Set \(dI=0\) and \(dP_y=0\)

Since the income and the price of good Y are constant (not changing), we can set \(dM=0\) and \(dP_y=0\): $$0 = P_x dX + X dP_x + P_y dY$$

Calculate shares and elasticities

In this step, we'll calculate the share of expenditure on each good and elasticities of demand for both goods. The share of expenditure on good X is \(s_x = \frac{P_x X}{M}\). The own-price elasticity of demand for good X is \(e_{x,r_x} = \frac{dX}{dP_x} \cdot \frac{P_x}{X}\). The cross-price elasticity of demand for good Y with respect to the price of good X is \(e_{y,r_x} = \frac{dY}{dP_x} \cdot \frac{P_x}{Y}\).

Rewrite the differential equation using shares and elasticities

Now, we'll rewrite the differential equation from Step 3 using the calculated shares and elasticities: $$0 = s_x M e_{x,r_x}\frac{dP_x}{P_x} + s_y M e_{y,r_x}\frac{dP_x}{P_x}$$

Simplify the equation

We can cancel the \(M\) and \(\frac{dP_x}{P_x}\) terms for both sides of the equation and simplify to: $$0 = s_x e_{x,r_x} + s_y e_{y,r_x}$$ Now, rearrange the equation to get: $$s_x e_{x, r_{x}} + s_y e_{y, r_{x}} = -s_x$$ Thus, the equation is proven true for a two-good world.

Discuss the relationship between own-price elasticity and cross-price elasticity

Given the proven equation \(s_x e_{x, r_{x}} + s_y e_{y, r_{x}} = -s_x\), if we know the own-price elasticity of demand for good X (\(e_{x, r_{x}}\)), we can deduce the cross-price elasticity for good Y (\(e_{y, r_{x}}\)). As the equation shows, an increase in own-price elasticity leads to an increase in cross-price elasticity when holding the expenditure shares constant. This means that as the demand for good X becomes more sensitive to its price changes, the demand for good Y will also become more sensitive in response to the price changes of good X.

Key concepts

Own-Price Elasticity of Demand

When we speak of the own-price elasticity of demand, we're referring to how the quantity demanded of a good responds to a change in its own price. It is a measure of the sensitivity or responsiveness of consumers to price changes of a particular product. In mathematical terms, this elasticity is denoted by the formula:

\( e_{x,r_x} = \dfrac{dX}{dP_x} \cdot \dfrac{P_x}{X} \)

Where \(dX\) represents the change in quantity demanded, \(dP_x\) is the change in price, and \(P_x\) and \(X\) are the initial price and quantity respectively. If the value of this elasticity is greater than 1, the demand is considered elastic, implying consumers are highly responsive to price changes. Conversely, if it is less than 1, demand is described as inelastic, indicating that consumers are relatively unresponsive to price changes. A key point to understand is that an increase in own-price elasticity implies that consumers will significantly adjust their purchasing behavior if the price of the product changes.

Budget Constraint

The budget constraint represents the combinations of goods and services that a consumer can purchase given their income and current prices. In the context of a two-good world, the constraint is depicted by the equation:

\( M = P_x X + P_y Y \)

With \(M\) being the income available for spending, \(P_x\) and \(P_y\) the prices of goods X and Y, and \(X\) and \(Y\) the quantities of these goods. Understanding the budget constraint is crucial because it encapsulates the idea of trade-offs; a consumer can only buy more of one good by giving up some of another, as long as they are bound by their total budget. In the exercise provided, setting \(dM=0\) and \(dP_y=0\) signifies that the consumer's income and the price of good Y remain constant. This aspect is specifically important when calculating how a change in the price of one good affects the quantity demanded, keeping the consumer’s budget in mind.

Total Differential

The total differential in economics provides a way to approximate how changes in different variables affect the overall outcome. When taking the total differential of the budget constraint, we investigate the impact on the consumer's expenditure resulting from changes in prices and quantities of goods. The differential of the budget constraint is expressed as:

\( dM = P_x dX + X dP_x + P_y dY + Y dP_y \)

This expression helps us link infinitesimal changes in the quantities and prices of goods to the total money spent. In the scenario where the consumer's income and the price of one good remain constant, signified by \(dM=0\) and \(dP_y=0\), it simplifies the calculation as it indicates that there are no external changes affecting the budget besides the price of good X. This simplification bridges the concept of the budget constraint and elasticity, allowing for deeper insights into consumer behavior and market dynamics. Overall, the total differential is a valuable tool in economic analysis, facilitating the understanding of complex relationships in a simplified manner.