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 \(-| ;\) and b. The cross-price elasticity of a change in the price of cheese on ham consumption is also - - c. How would your answers to (a) and (b) change if a slice of ham cost twice as much as a slice of cheese? \(\\{\text {Hint}\) : Use the Slutsky equation - what is the substitution elasticity here?)

Short answer

Answer: The own-price elasticity of demand for ham is -1, and the cross-price elasticity between ham and cheese is also -1.

Step-by-step solution

Understand the Relationship Between Ham and Cheese in Sandwiches

Since ham and cheese are pure complements, we know that they are always used in a 1:1 ratio to make sandwiches. That is, for every slice of ham, we need one slice of cheese to make a complete sandwich. This implies that if the demand for ham increases, the demand for cheese should increase as well (and vice versa).

Calculate the Own-Price Elasticity of Demand for Ham

Own-price elasticity of demand is given by the formula: $$E_d = \frac{\% \ ΔQ_d}{\% \ ΔP}$$, where \(E_d\) is the own-price elasticity, \(Q_d\) is the quantity demanded, and \(P\) is the price. Since ham and cheese are always used in a 1:1 ratio in the sandwiches, we have: $$E_d = \frac{\% \ ΔQ_d}{\% \ ΔP}=-1$$ Thus, the own-price elasticity of demand for ham is -1.

Determine the Cross-Price Elasticity Between Ham and Cheese

Cross-price elasticity of a change in the price of cheese on ham consumption is given by the formula: $$E_{hc} = \frac{\% \ ΔQ_d^h}{\% \ ΔP_c}$$, where \(E_{hc}\) is the cross-price elasticity, \(Q_d^h\) is the quantity demanded of ham, and \(P_c\) is the price of cheese. Since ham and cheese are used in a 1:1 ratio to make sandwiches, when the price of cheese increases, ham consumption will decrease proportionally: $$E_{hc} = \frac{\% \ ΔQ_d^h}{\% \ ΔP_c}=-1$$ So, the cross-price elasticity of a change in the price of cheese on ham consumption is also -1.

Analyze the Case Where the Price of a Ham Slice is Twice the Price of a Cheese Slice

If the price of a ham slice is twice the price of a cheese slice, the Slutsky equation can be used to find the substitution elasticity. The Slutsky equation is given by: $$\frac{\% \ ΔQ_d^h}{\% \ ΔP} = E^H - E^C$$, where \(E^H\) is the substitution elasticity between ham and cheese, and \(E^C\) is the cross-price elasticity between ham and cheese. Since both the own-price elasticity and cross-price elasticity of demand for ham are -1, we have: $$-1 = E^H - 1 \Rightarrow E^H = 0$$ Therefore, the substitution elasticity is 0, which indicates that there is no substitution effect in this case; consumers will continue to buy ham and cheese in the same proportion, regardless of the price change ratio.

Key concepts

Own-Price Elasticity

The own-price elasticity of demand represents how sensitive the quantity demanded of a good is to a change in its own price. In simpler terms, it tells us how much the quantity consumers buy will change if the price goes up or down.
For ham in the given scenario, since it is consumed in a fixed 1:1 ratio with cheese, if the price of ham increases by a certain percentage, the quantity demanded will decrease by that same percentage. This results in an own-price elasticity of \(E_d = -1\). This value indicates perfect inelasticity under the given conditions, meaning consumers will precisely adjust their consumption to keep buying one slice of ham for each slice of cheese.

Cross-Price Elasticity

Cross-price elasticity measures the responsiveness of the quantity demanded of one good when the price of another related good changes. In our example, it looks at how ham consumption is affected by a change in the price of cheese.
Because ham and cheese are perfect complements, the cross-price elasticity between them provides important insights. When the price of cheese rises, ham consumption responds equivalently, adhering to the perfect 1:1 ratio.
If cheese becomes more expensive, people cut back proportionally on buying ham too. Therefore, the cross-price elasticity: \(E_{hc} = -1\) shows that the two goods are tightly linked, and an increase in the price of one sees an equivalent fall in the demand for both.

Substitution Elasticity

Substitution elasticity explores the idea that consumers might switch between different goods if prices change, except here, substitution doesn't occur under the specified conditions. Even if the price of a ham slice is doubled compared to cheese, they remain perfect complements in consumption.
Utilizing the Slutsky equation, we analyze the situation:\[-1 = E^H - 1\]which implies the substitution elasticity \[E^H = 0\]. This zero value reveals no substitution effect, meaning no amount of price differential between ham and cheese will change the necessity of using both in fixed proportions. Consumers do not adjust the ratio and simply follow the same consumption pattern.