Question

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$\mathrm{ting}dl=0=d{P}_Y.)$

Short answer

Answer: We can find the cross-price elasticity of demand for good Y using the derived formula: ${\epsilon }_{Y,X}=-\frac{Py}{Px}({\epsilon }_X+1)$.

Step-by-step solution

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.

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.

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.

Own-Price and Cross-Price Elasticity Definitions

Own-price elasticity of demand for good X is defined as: ${\epsilon }_X=\frac{dX}{d{P}_X}\left(\frac{P_X}{X}\right)$. Cross-price elasticity of demand for good Y with respect to the price of good X is defined as: ${\epsilon }_{Y,X}=\frac{dY}{d{P}_X}\left(\frac{P_X}{Y}\right)$.

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

We can rewrite the equation from Step 3 as: $X\ast dPx+Px\ast dX=-Py\ast dY$. Now, divide both sides by $Px\ast X\ast Py\ast Y$. 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: ${\epsilon }_X+1=-{\epsilon }_{Y,X}\cdot \frac{Px}{Py}$. Rearranging this to find an expression for the cross-price elasticity, we get: ${\epsilon }_{Y,X}=-\frac{Py}{Px}({\epsilon }_X+1)$.

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: ${\epsilon }_{Y,X}=-\frac{Py}{Px}({\epsilon }_X+1)$.

Key concepts

Own-Price Elasticity

In economics, own-price elasticity of demand measures how much the quantity of a good responds to changes in its own price. It tells us about the sensitivity of consumers' demand when the price for the same good changes. The own-price elasticity (${\epsilon }_X$) is calculated using the formula: $${\epsilon }_X=\frac{\partial Q_X}{\partial P_X}\cdot \frac{P_X}{Q_X}$$where $Q_X$ is the quantity demanded of good X, and $P_X$ is the price of the good.

• If ${\epsilon }_X<-1$, the demand is elastic, meaning consumers respond significantly to price changes.
• If ${\epsilon }_X>-1$ and ${\epsilon }_X<0$, the demand is inelastic, with slight consumer response to price changes.
• If ${\epsilon }_X=-1$, demand is unitary elastic, meaning the percentage change in quantity demanded is exactly the same as the percentage change in price.

Understanding this elasticity helps businesses and policymakers predict changes in demand and adjust accordingly. Consider, for example, the prices of beverages: if the price of a popular soda increases slightly and consumption drops considerably, the demand for that soda is considered elastic.

Cross-Price Elasticity

Cross-price elasticity measures how the quantity demanded of one good (let's call it Y) reacts to a change in the price of another good (say X). This concept is crucial in assessing the relationships between different goods in the market. The formula for cross-price elasticity (${\epsilon }_{Y,X}$) is given by: $${\epsilon }_{Y,X}=\frac{\partial Q_Y}{\partial P_X}\cdot \frac{P_X}{Q_Y}$$

• If ${\epsilon }_{Y,X}>0$, then goods X and Y are substitutes, meaning an increase in the price of X leads to more demand for Y.
• If ${\epsilon }_{Y,X}<0$, X and Y are complements, meaning an increase in the price of X leads to less demand for Y.
• If ${\epsilon }_{Y,X}=0$, X and Y are unrelated products, so price changes in X don't affect Y's demand.

It's helpful to think of goods like coffee and tea. If the price of coffee rises, and more people start buying tea instead, they are seen as substitutes. On the other hand, if the price of cars goes up and the demand for tires falls, they are regarded as complements.

Budget Constraint

A budget constraint represents the combinations of goods and services that a consumer can purchase given their income and the prices of goods. In a two-good world, it is represented as: $I=P_X\cdot X+P_Y\cdot Y$, where $I$ is the income, $P_X$ and $P_Y$ are the prices of goods X and Y, and $X$ and $Y$ are the quantities of these goods.

• The budget constraint line illustrates trade-offs, showing how much of each good a consumer can afford given their budget.
• If prices change, the slope of this line will change, demonstrating how the consumer's ability to purchase varies with price changes.
• The budget constraint also highlights opportunity cost, as choosing more of one good typically results in less of another.

An example in daily life could be a student with $100tospendonbooksandschoolsupplies.Ifatextbookcosts$50 and a supply bag costs $10, the student can buy two textbooks (and no supplies) or one textbook and five supply bags. Understanding this helps us better plan our spending and make informed choices about what's affordable.