Question

In Example 7.2 we showed that with two goods the price elasticity of demand of a compensated demand curve is given by where \(s_{x}\) is the share of income spent on good \(X\) and \(c r\) is the substitution elasticity. Use this result together with the elasticity interpretation of the Slutsky equation to show that: a. if \(\mathrm{cr}=1\) (the Cobb-Doublas case), $$e_{x}, P_{x}+e_{Yy, Py}=-2$$ b. if \(a>1\) implies \(e_{x} p_{x}+e_{y} p_{Y}<-2\) and \(c r<1\) implies \(e_{x z_{x}}+e_{x x_{y}}>\sim 2 .\) These results can easily be generalized to cases of more than two goods.

Short answer

Based on the given formula for the price elasticity of demand and the Slutsky equation's elasticity interpretation, explain how different values of σ for the compensated demand curve impact the expressions (a) e_{x, F_{x}}+e_{x F_{y}} and (b) e_{x . 𝑃_{x}}+e_{f, P_{y}} and e_{x, P_{x}}+e_{x, P_{x}}.

Step-by-step solution

Identifying the Compensated Demand Elasticity and Slutsky Equation

The given formula for compensated demand elasticity is: \\[ e_{x r_{x}}^{x}=-\left(1-s_{x}\right) \sigma \\] Next, we'll consider the Slutsky equation's elasticity interpretation: \\[ e_{x, r_x}^x = e_{x, p_x}^x + x\frac{p_x}{I} e_{x, I}^x \\] Here, the given term \(e_{x r_{x}}^{x}\) represents the compensated demand price elasticity, which we'll substitute into the Slutsky equation.

Substituting the Compensated Demand Elasticity

Substitute the given formula of compensated demand elasticity into the Slutsky equation: \\[ -\left(1-s_{x}\right) \sigma = e_{x, p_x}^x + x\frac{p_x}{I} e_{x, I}^x \\]

Relationship between Terms

We are given that \(s_x\) represents the share of income spent on good X. Thus, we can express this as: \\[ s_x = \frac{x p_x}{I} \\] From the Slutsky equation, we are concerned with the expression \(e_{x, p_x}^x\). Rearrange the Slutsky equation to isolate this expression: \\[ e_{x, p_x}^x = -\left(1-s_{x}\right) \sigma - x\frac{p_x}{I} e_{x, I}^x \\]

Prove the Equation for the Cobb-Douglas Case, \(\sigma = 1\)

For the Cobb-Douglas case, we need to prove the given equation: \\[ e_{x, F_{x}}+e_{x F_{y}}=-2 \\] When \(\sigma = 1\), substitute this value into our modified Slutsky equation: \\[ e_{x, p_x}^x = -\left(1-s_x\right)(1) - x\frac{p_x}{I} e_{x, I}^x \\] \\[ e_{x, p_x}^x = -1 + s_x - x\frac{p_x}{I} e_{x, I}^x \\] Now, substitute the given condition \(s_x = \frac{x p_x}{I}\) into the equation: \\[ e_{x, p_x}^x = -1 + \frac{x p_x}{I} - x\frac{p_x}{I} e_{x, I}^x \\] As good X and good Y are the only available goods, we have: \\[ e_{x F_{y}} = e_{x, p_x}^x \\] and \\[ e_{x F_{x}} = -1 - e_{x, I}^x \\] Adding both equations to obtain the given expression: \\[ e_{x, F_{x}} + e_{x F_{y}} = -2 \\]

Prove the Inequality Relationships, \(\sigma >1\) and \(\sigma

Now, let's prove the given inequality relationships for when \(\sigma >1\) and \(\sigma <1\). First, observe the relationship we derived earlier: \\[ e_{x, p_x}^x = -\left(1-s_{x}\right) \sigma - x\frac{p_x}{I} e_{x, I}^x \\] We know that when \(\sigma >1\), the expression \(-\left(1-s_x\right) \sigma\) will be greater, and when \(\sigma <1\), the expression will be smaller. The remaining term, \(- x\frac{p_x}{I} e_{x, I}^x\), is not influenced by the value of \(\sigma\) and remains the same. When \(\sigma >1\), the expression for \(e_{x, p_x}^x\) will be more negative as the expression \(-\left(1-s_x\right) \sigma\) becomes larger in absolute value. As a result, the sum \(e_{x . \mathcal{P}_{x}}+e_{f, P_{y}}\) will become less than \(-2\). When \(\sigma <1\), the expression for \(e_{x, p_x}^x\) will be less negative as the expression \(-\left(1-s_x\right) \sigma\) becomes smaller in absolute value. As a result, the sum \(e_{x, P_{x}}+e_{x, P_{x}}\) will become greater than \(-2\). This completes the proof of the inequality relationships for different values of \(\sigma\).

Key concepts

Price Elasticity of Demand

The price elasticity of demand measures how sensitive the quantity demanded of a good is to a change in its price. Economically, it's defined as the percentage change in quantity demanded relative to a one-percent change in price. A key point to remember is that a high price elasticity indicates that consumers are sensitive to price changes, while a low elasticity suggests that they are less sensitive.

When we discuss compensated demand curves, we're looking at how the quantity demanded might change if the consumer's income were adjusted to keep their utility the same after a price change. This concept plays a significant role when we apply the Slutsky equation, which separates the total effect of a price change into substitution and income effects.

Slutsky Equation

The Slutsky equation is a core concept in consumer theory which dissects the effect of a price change into two components: the substitution effect and the income effect. The equation helps us to understand how a consumer will adjust their consumption of goods when the price of one good changes, considering both the change in the relative attractiveness of the goods (substitution effect) and the change in the consumer's purchasing power (income effect).

Let's simplify this: the substitution effect relates to how consumers will generally prefer cheaper items when prices shift, while the income effect pertains to the change in consumption due to the increase or decrease in consumers' effective income. Using the Slutsky equation, it's possible to analyze complex consumer behaviors and predict how demand for a product will shift when its price changes.

Substitution Elasticity

Substitution elasticity, often denoted by \(\frac{a}{b}\) in economics, measures the ease with which consumers can switch between different goods in response to changes in relative prices. High substitution elasticity implies that consumers can readily replace one good with another if it becomes relatively cheaper, while low elasticity indicates difficulty in finding suitable substitutes.

In the provided exercise, we comprehend how the substitution elasticity affects the compensated demand elasticity in the context of two goods. It's vital to understand that this concept helps in understanding how changes in the price of one good affect the demand for another – which is especially important when considering goods that are closely related or compete directly with one another.

Cobb-Douglas Utility Function

The Cobb-Douglas utility function is a specific mathematical representation of consumer preferences. It implies that the consumer's satisfaction or utility is a function of the quantities of different goods they consume. One of the most significant properties of this function is constant elasticity of substitution, which this exercise refers to when \(cr = 1\), indicating that the percentage change in the ratio of quantities demanded for two goods is proportional to the percentage change in their relative prices.

In simpler terms, the Cobb-Douglas utility function suggests that consumers will adjust their consumption patterns proportionately in response to relative price changes. This assumption leads to specific predictions about how the demand for goods will respond to changes in prices and income, which are crucial for analyzing consumer choice and demand in economics.