Here’s the breakdown of the problems:

1. True/False/Uncertain and Why

• Statement: If the demand for a good, $Z$, can be described as relatively elastic at its current level of consumption, then a decrease in the price of $Z$ will cause total spending on good $Z$ to decrease.

  Solution Approach:

  • Elastic Demand: If demand is relatively elastic, the percentage change in quantity demanded is greater than the percentage change in price ($|E_d| > 1$).
  • When the price decreases, the increase in quantity demanded outweighs the price reduction, so total revenue (or spending) on $Z$ increases, not decreases.
  • Answer: False. When demand is elastic, a price decrease leads to an increase in total spending because the rise in quantity demanded is proportionally larger than the price reduction.

2. True/False/Uncertain and Why

• Statement: If an increase in the price of $X$ causes more $Y$ to be consumed, then the two goods, $X$ and $Y$, are substitutes, and their cross-price elasticity coefficient is negative.

  Solution Approach:

  • Substitutes: When two goods are substitutes, an increase in the price of one leads to an increase in the quantity demanded of the other.
  • Cross-Price Elasticity ($E_{xy}$): For substitutes, this value is positive, not negative, since quantity demanded of $Y$ increases as the price of $X$ increases.
  • Answer: False. While $X$ and $Y$ may be substitutes, the cross-price elasticity would be positive, not negative.

3. Income Elasticity of Demand for Fava Beans

• Function: $F(p_F, p_C, M) = \frac{3M}{p_F - \frac{1}{4}p_C}$, where:

  • $p_F$: Price of fava beans.
  • $p_C$: Price of chianti wine.
  • $M$: Income (budget) of the consumer.

  Income Elasticity Formula:

  • Income elasticity of demand, $E_M$, is given by: $E_M = \frac{\partial F}{\partial M} \cdot \frac{M}{F}.$ Step 1: Compute $\frac{\partial F}{\partial M}$:
  • Differentiate $F$ with respect to $M$: $\frac{\partial F}{\partial M} = \frac{3}{p_F - \frac{1}{4}p_C}.$

  Step 2: Substitute $F$ and simplify $E_M$:

  • Substituting $F$ into the formula: $E_M = \frac{\frac{3}{p_F - \frac{1}{4}p_C} \cdot M}{\frac{3M}{p_F - \frac{1}{4}p_C}}.$
  • Simplify: $E_M = 1.$

  Answer: The income elasticity of demand for fava beans is 1, meaning demand is unitary elastic with respect to income.

Follow-Up Questions:

1. What happens to total revenue if the price of a relatively inelastic good is decreased?
2. Why is the cross-price elasticity of substitutes always positive?
3. How would the income elasticity of fava beans change if the function were modified to include a diminishing effect of $M$?
4. Explain the significance of unit income elasticity in terms of consumer preferences.
5. What would the demand for $F$ look like if $p_F$ were to approach $\frac{1}{4}p_C$?

Tip:

Elasticity concepts always revolve around percentage changes. Focus on interpreting the direction and magnitude of these changes to predict outcomes accurately.