Question

In general, uncompensated cross-price effects are not equal. That is, $$\frac{\partial x_{i}}{\partial p_{j}} \neq \frac{\partial x_{j}}{\partial p_{i}}$$ Use the Slutsky equation to show that these effects are equal if the individual spends a constant fraction of income on each good regardless of relative prices. (This is a generalization of Problem \(6.1 .)\)

Short answer

ds_i}{dp_j} = \frac{x_i \cdot \delta_{ij} + p_i \cdot \frac{dx_i}{dp_j}}{I} $$ Where: - \(\delta_{ij}\) is the Kronecker delta, which equals 1 if \(i = j\) and 0 if \(i \ne j\) #tag_title# Step 4: Plug in the generalized Slutsky equation #tag_content# Now, let's replace \(\frac{dx_i}{dp_j}\) with the generalized Slutsky equation: $$ \frac{ds_i}{dp_j} = \frac{x_i \cdot \delta_{ij} + p_i \cdot \left( h_{ij} \cdot \Delta p_j + \frac{I}{p_i} \cdot \Delta I \right)}{I} $$ Since we are only interested in uncompensated cross-price effects, we can ignore the change in income term: $$ \frac{ds_i}{dp_j} = \frac{x_i \cdot \delta_{ij} + p_i \cdot h_{ij} \cdot \Delta p_j}{I} $$ #tag_title# Step 5: Show that uncompensated cross-price effects are equal #tag_content# In order to determine if the uncompensated cross-price effects are equal, we need to compare the change in fraction of income spent on good \(i\) when the price of good \(j\) changes: $$ \frac{ds_i}{dp_j} = \frac{ds_k}{dp_j} $$ We can plug in the expressions derived in Step 4: $$ \frac{x_i \cdot \delta_{ij} + p_i \cdot h_{ij} \cdot \Delta p_j}{I} = \frac{x_k \cdot \delta_{kj} + p_k \cdot h_{kj} \cdot \Delta p_j}{I} $$ Since the individual spends a constant fraction of income on each good, the terms \(x_i \cdot \delta_{ij}\) and \(x_k \cdot \delta_{kj}\) cancel out. What remains is: $$ p_i \cdot h_{ij} \cdot \Delta p_j = p_k \cdot h_{kj} \cdot \Delta p_j $$ Cancel out the common term \(\Delta p_j\) on both sides: $$ p_i \cdot h_{ij} = p_k \cdot h_{kj} $$ Thus, we have shown that the uncompensated cross-price effects are equal if the individual spends a constant fraction of income on each good, regardless of relative prices. #Short Answer# The uncompensated cross-price effects are equal if the individual spends a constant fraction of income on each good, regardless of relative prices. This is demonstrated by using the generalized Slutsky equation and differentiating the constant fraction with respect to the price of another good. After some simplification and assuming a constant fraction of income, we can show that the uncompensated cross-price effects are equal in this situation.

Step-by-step solution

Review the generalized Slutsky equation

The generalized Slutsky equation can be written as: $$ \Delta x_i = \sum_j{h_{ij} \cdot \Delta p_j} + \frac{I}{p_i} \cdot \Delta I $$ Where: - \(\Delta x_i\) is the change in the quantity demanded of good \(i\) - \(h_{ij}\) is the element in the matrix of compensated price elasticities - \(\Delta p_j\) is the change in the price of good \(j\) - \(I\) is the consumer's income - \(p_i\) is the price of good \(i\) - \(\Delta I\) is the change in income

Identify the uncompensated cross-price effect

The uncompensated cross-price effect refers to the change in the quantity demanded of one good when the price of another good changes. In the generalized Slutsky equation, this effect is represented by the term \(h_{ij} \cdot \Delta p_j\).

Assume constant fraction of income spent on each good

We are asked to show that the uncompensated cross-price effects are equal if the individual spends a constant fraction of income on each good, regardless of relative prices. Let \(s_i\) be the constant fraction of income spent on good \(i\). We can write this as: $$ s_i = \frac{p_i \cdot x_i}{I} $$ Now let's differentiate this equation with respect to the price of good \(j\): $$ \frac{

Key concepts

Cross-Price Effect

Understanding how the price of one good affects the demand for another is central to analyzing market dynamics. The cross-price effect is a concept that measures this relationship. Essentially, it refers to the change in the quantity demanded of Good A (x_i) when the price of Good B (p_j) changes, without adjusting for changes in income.

Compensated Price Elasticity

When the price of a good changes, consumers' purchasing power is affected. To understand the pure effect of a price change, economists use the concept of compensated price elasticity. This elasticity measures the change in demand for a good when its price changes, assuming that the consumer is compensated for the change in purchasing power (hence 'compensated'). This allows us to isolate the substitution effect, which occurs when consumers switch from a more expensive good to a less expensive substitute.

Income Effect

Lastly, we must consider what happens to the consumer's demand for goods when their income changes. This is known as the income effect. It represents the change in demand that occurs because the consumer feels richer or poorer as a result of the income change. Distinguishing the income effect from the substitution effect helps economists predict how changes in prices will influence the quantity of goods that consumers demand, both in terms of which goods they can now afford and how they perceive their overall wealth.