Question

corresponding share elasticities. a. Show that the elasticity of a good's budget share with respect to income \(\left(e_{s_{x}, I}=\partial s_{x} / \partial I \cdot I / s_{x}\right)\) is equal to \(e_{x, I}-1 .\) 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 \(\left[e_{x \cdot p_{a}, p_{a}}=\partial\left(p_{x} \cdot x\right) / \partial p_{x} \cdot 1 / x\right]\) 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 \(\left(e_{s_{x}, p_{y}}=\partial s_{x} / \partial p_{y} \cdot p_{y} / s_{x}\right)\) is equal to \(e_{x}, p_{y}\) c. 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 \(5.56 ; e_{x^{\prime}, p_{x}}=-\left(1-s_{x}\right) \sigma\) Hint: This problem can be simplified by assuming \(p_{x}=p_{y}\) in which case \(s_{x}=0.5\)

Short answer

Short Answer: The elasticity of a good's budget share with respect to income (represented as \(e_{s_{x}, I}\)) is equal to the income elasticity of the good minus 1 (\(e_{x, I}-1\)). This relationship can be demonstrated by calculating the income elasticity of the good, defining its budget share, and differentiating it with respect to income. After a series of substitutions, it is proven that \(e_{s_{x}, I} = e_{x, I} -1\). This conclusion helps understand how the budget share of a good changes with respect to income levels and can be further analyzed with numerical examples.

Step-by-step solution

To show that \(e_{s_{x}, I}=e_{x, I}-1\), let's first write the definition of income elasticity: $$ e_{x, I} = \frac{\partial x}{\partial I} \cdot \frac{I}{x} $$ Now, let's define \(s_x\), the budget share as: $$ s_x = \frac{p_x \cdot x}{I} $$ Next, rearrange to isolate \(x\): $$ x = \frac{s_x \cdot I}{p_x} $$ #tag_step2#Step 2: Differentiating \(x\) with respect to \(I\)

Now we will differentiate \(x\) with respect to \(I\): $$ \frac{\partial x}{\partial I} = \frac{s_x}{p_x} $$ #tag_step3#Step 3: Substituting the expression for \(\partial x / \partial I\) into the income elasticity formula

We will now substitute the expression for \(\frac{\partial x}{\partial I}\) that we found in step 2 into the income elasticity formula: $$ e_{x, I} = \frac{s_x}{p_x} \cdot \frac{I}{x} $$ #tag_step4#Step 4: Substituting the expression for \(x\) into the income elasticity formula

Next, we will substitute the expression for \(x\) that we found in step 1 into the income elasticity formula: $$ e_{x, I} = \frac{s_x}{p_x} \cdot \frac{I}{\frac{s_x \cdot I}{p_x}} = \frac{s_x \cdot p_x}{s_x \cdot I} \cdot I = \frac{s_x \cdot p_x}{s_x \cdot I} \cdot I = 1 + \frac{I}{s_x} \cdot \left( \frac{\partial s_x}{\partial I} \right) $$ Now, solve for \(e_{s_{x}, I}\): $$ e_{s_{x}, I} = e_{x, I} - 1 $$

Key concepts

Income Elasticity of Demand

Income elasticity of demand is a measure that economists use to understand how a change in consumer income affects the quantity demanded of a good or service. It's computed with the following formula:
\[ e_{x, I} = \frac{\partial x}{\partial I} \cdot \frac{I}{x} \]
where \( \frac{\partial x}{\partial I} \) represents the change in the quantity demanded when income changes, \( I \) is the income and \( x \) is the quantity of the good.

A positive income elasticity indicates that the product is a normal good, meaning demand for it increases as income increases. Conversely, a negative income elasticity shows that the product is an inferior good, with demand decreasing as income increases. It's an essential concept for businesses and policymakers as they forecast changes in market dynamics in response to shifts in consumer income levels. Numerical examples can illustrate this: suppose a 10% increase in income leads to a 5% increase in demand for a product, the income elasticity is 0.5, indicating that the product is a normal good but not a luxury item (which typically has an elasticity greater than 1).

Expenditure Elasticity

Expenditure elasticity refers to how sensitive the amount of money spent on a good is to the change in the price of that good. It's another form of price elasticity and is given as:
\[ e_{x \cdot p_{x}, p_{x}} = \frac{\partial(p_{x} \cdot x)}{\partial p_{x}} \cdot \frac{1}{x} \]
This measures the percentage change in expenditure on good \( x \) (\( p_{x} \cdot x \) is the expenditure on good \( x \) where \( p_{x} \) is the price of good \( x \) and \( x \) is the quantity) relative to a percentage change in its own price (\( p_{x} \) ). A value of expenditure elasticity greater than 1 suggests that the good is a luxury, as consumers spend a higher proportion of their budget on it when the price increases. Conversely, a value of less than 1 indicates a necessity, as the increase in price doesn't lead to a proportionally significant increase in expenditure.

CES Utility Function

The Constant Elasticity of Substitution (CES) utility function is a popular type of utility function used in economic models to depict consumer preferences. It has the general form:
\[ U(x, y) = \left(a \cdot x^{\rho} + b \cdot y^{\rho}\right)^{\frac{1}{\rho}} \]
where \( a \) and \( b \) are constants, \( x \) and \( y \) represent quantities of different goods, and \( \rho \) determines the substitution elasticity, denoted by \( \sigma \) (where \( \rho = \frac{1}{\sigma - 1} \) ). This function assumes substitutability between goods; the higher the value of \( \sigma \) , the easier it is to substitute one good for another. The CES function can be tailored to specific contexts, making it highly versatile for analyzing consumer behavior and market mechanisms. In the context of the textbook exercise, understanding the CES utility function helps in examining how income shares devoted to certain goods change with prices, a concept crucial for comprehending budget share elasticities.

Elasticity of Substitution

Elasticity of substitution is a key concept in microeconomic theory that measures the ease with which consumers can substitute one good for another as relative prices change. It is specifically linked to the CES utility function. An elasticity greater than 1 indicates that the goods are easy to substitute, whereas a value less than 1 implies that the goods are not as easily replaced by one another. This notion is instrumental in understanding how consumers react to changes in market conditions and pricing strategies. For instance, if two products are close substitutes, a rise in the price of one could significantly increase the demand for the other. The textbook exercise involves the elasticity of substitution when it employs a CES utility function to determine the income share devoted to a good and how it varies with changes in price, which in turn affects the calculation of budget share elasticity.