Question

Suppose that ham and cheese are pure complements-they will always be used in the ratio of one slice of ham to one slice of cheese to make a sandwich. Suppose also that ham and cheese sandwiches are the only goods that a consumer can buy and that bread is free. Show that if the price of a slice of ham equals the price of a slice of cheese, a. The own-price elasticity of demand for ham is \(-\frac{1}{2} ;\) and b. The cross-price clasticity of a change in the price of checse on ham consumption is also \(-\frac{1}{2}\) c. How would your answers to (a) and (b) change if a slice of ham cost twice as much as a slice of cheese? (Hint: Use the Slutsky equation-what is the substitution elasticity here?)

Short answer

Based on the given information, the own-price elasticity of demand for ham is -1/2, which indicates that ham is inelastic, meaning that a percentage change in the price of ham results in a proportionally smaller change in the quantity demanded for ham. Similarly, the cross-price elasticity of demand between ham and cheese is also -1/2, which implies that they are complementary goods since the value is negative. This negative value indicates that an increase in the price of cheese would lead to a proportionally smaller decrease in the demand for ham. When the price ratio between ham and cheese changes, with the price of a slice of ham now being twice the price of a slice of cheese, the own-price and cross-price elasticities remain the same at -1/2. This highlights the fact that purely complementary goods, like ham and cheese in this case, do not experience a substitution effect when the price ratio changes.

Step-by-step solution

Understand the complementary goods and given conditions

From the given exercise, we understand that ham and cheese are pure complements - they are always consumed together in a fixed ratio to make a sandwich. Also, ham and cheese sandwiches are the only goods the consumer can buy, and bread is free. Since 1 slice of ham is used with 1 slice of cheese, we have the constraint: q_ham = q_cheese.

Finding the own-price elasticity of demand for ham

The own-price elasticity of demand is defined as: \(E_h = \frac{dQ_h}{dP_h}\frac{P_h}{Q_h}\) Let the price of ham (P_h) and price of cheese (P_c) be equal, and let the consumer's income be denoted by I. The consumer will spend I on a combination of ham and cheese as: \(I = P_hq_h + P_cq_c = q_h(P_h + P_c) \Rightarrow q_h = \frac{I}{(P_h + P_c)}\) Since P_h = P_c, we have: \(q_h = \frac{I}{2P_h}\) Now find the derivative of demand with respect to price and putting the values in the formula for elasticity: \(E_h = \frac{d(\frac{I}{2P_h})}{dP_h}\frac{P_h}{\frac{I}{2P_h}} = (-\frac{1}{2}\frac{I}{P_h^2})\frac{P_h}{\frac{I}{2P_h}} = -\frac{1}{2}\)

Finding the cross-price elasticity of a change in the price of cheese on ham consumption

The cross-price elasticity of demand is defined as: \(E_{hc} = \frac{dQ_h}{dP_c}\frac{P_c}{Q_h}\) Since P_h = P_c, we have: \(E_{hc} = \frac{d(\frac{I}{2P_c})}{dP_c}\frac{P_c}{\frac{I}{2P_c}}\) Therefore: \(E_{hc} = (-\frac{1}{2}\frac{I}{P_c^2})\frac{P_c}{\frac{I}{2P_c}} = -\frac{1}{2}\)

Analyzing the result when the price ratio between ham and cheese changes

Suppose the price of a slice of ham is now twice the price of a slice of cheese, i.e., P_h = 2P_c. Based on the hint provided, we should use the Slutsky equation, but given that we're looking at purely complementary goods, the substitution effect will be zero since there are no other ways to create a sandwich. As a result, changing the price ratio between ham and cheese will not affect the own-price and cross-price elasticities, meaning they remain the same: \(E_h = -\frac{1}{2}\) and \(E_{hc} = -\frac{1}{2}\).

Key concepts

Complementary Goods

Understanding complementary goods is key when examining how the price of one item affects the consumption of another. Complementary goods are products that are typically consumed together, like the ham and cheese in our exercise. Consumers use these goods in conjunction to fulfill a particular desire or need, and they have a unique relationship: when the price of one complementary good changes, it directly affects the demand for its partner good.

For instance, when the price of ham increases and they are a pure complement to cheese (as in our exercise scenario), fewer sandwiches are made, hence the demand for cheese would naturally decrease. In a scenario where these goods are to be consumed in a fixed ratio, such as a 1:1 slice of ham to cheese for sandwiches, their demand levels are intrinsically linked.

Slutsky Equation

The Slutsky equation is an important concept in consumer theory that separates the effect of a price change into two parts: the substitution effect and the income effect.

The substitution effect occurs as consumers react to a price change by substituting the good that has become relatively more expensive with a cheaper alternative. The income effect happens because a price change effectively changes the consumer's purchasing power.

In our exercise, however, the Slutsky equation simplifies things. Since ham and cheese are consumed together and no substitutes are available (bread is free and not substitutable), the substitution effect is zero. This means that the demand response is due to the income effect alone. When the prices change, consumers' ability to purchase ham and cheese is affected, but they would still maintain the same consumption ratio of these complementary goods.

Own-Price Elasticity

The concept of own-price elasticity of demand deals with how the quantity demanded of a good responds to a change in its own price, holding all else constant. It is expressed mathematically as the percentage change in quantity demanded divided by the percentage change in price. A negative own-price elasticity indicates that the demand for a product decreases as its price increases, which is typical for most goods.

As calculated in the textbook solution, if the price of ham increases while the price of cheese remains constant, the quantity demanded for ham would fall. Since ham and cheese are used in a 1:1 ratio, the demand elasticity for ham is -1/2, reflecting a decrease in demand for ham (and therefore cheese) when the price of ham increases.

Cross-Price Elasticity

The cross-price elasticity of demand measures how the quantity demanded of one good, say ham, changes in response to a change in the price of another good, like cheese. This metric can indicate whether two goods are substitutes or complements based on whether the elasticity is positive or negative, respectively.

In a situation where the goods are complements, as in our exercise, an increase in the price of cheese would lead to a decrease in the demand for ham, hence the negative cross-price elasticity. Specifically, the cross-price elasticity of -1/2 implies that for any percentage increase in the price of cheese, the demand for ham decreases by half that percentage. This reinforces the direct relationship between the price of ham and cheese in the context of constructing sandwiches, consistent with their role as complementary goods.