Question

Suppose that a person regards ham and cheese as pure complements-he or she will always use one slice of ham in combination with one slice of cheese to make a ham and cheese sandwich. Suppose also that ham and cheese are the only goods that this person buys and that bread is free. a. If the price of ham is equal to the price of cheese, show that the own- price elasticity of demand for ham is -0.5 and that the cross-price elasticity of demand for ham with respect to the price of cheese is also -0.5 b. Explain why the results from part (a) reflect only income effects, not substitution effects. What are the compensated price elasticities in this problem? c. Use the results from part (b) to show how your answers to part (a) would change if a slice of ham cost twice the price of a slice of cheese. d. Explain how this problem could be solved intuitively by assuming this person consumes only one good-a ham and cheese sandwich.

Short answer

The own-price elasticity of demand for ham is -0.5, and the cross-price elasticity of demand for ham with respect to the price of cheese is -0.5. b. Why do these results only show income effects? What are the compensated price elasticities? These results only show income effects because there are no substitution effects since the consumer only consumes ham and cheese together and cannot substitute one good for the other. The calculated elasticities only reflect income effects, as an increase in the price of one good affects the consumer's overall purchasing power, causing them to reduce their consumption of ham and cheese. The compensated price elasticities would be zero, as there would be no change in consumption if the consumer was compensated for the price increase. c. How would the results change if a slice of ham costs twice the price of a slice of cheese? If a slice of ham costs twice the price of a slice of cheese, the own-price and cross-price elasticities of demand for ham would both be -0.75. d. How can this problem be solved intuitively by considering a single good - a ham and cheese sandwich? When considering only one good - a ham and cheese sandwich, the own-price elasticity of demand for the sandwich is -1. This simplified and intuitive solution indicates that if the price of a ham and cheese sandwich increases by 1%, the consumption of sandwiches decreases by 1%.

Step-by-step solution

a. Own-price and cross-price elasticities of demand

To find the own-price elasticity of demand for ham, we use the formula: Own-price elasticity of demand (E) = (% change in quantity demanded) / (% change in price) Let's denote the price of ham as P_h, the price of cheese as P_c, and the quantity demanded of both ham (Q_h) and cheese (Q_c). Since the consumer always uses one slice of ham with one slice of cheese, we know that Q_h = Q_c. Now, if the price of ham increases by 1%, the cost of making a sandwich increases by 0.5% (since bread is free). The consumer's budget constraint is now tighter, and they must decrease their consumption by 0.5% to maintain the same-level consumption. Thus, the percentage change in quantity demanded of ham is -0.5%. E_own = (-0.5%) / (1%) = -0.5 Similarly, for the cross-price elasticity of demand for ham with respect to the price of cheese, we follow the same process and find that when the price of cheese increases by 1%, consumers reduce their demand for ham by 0.5%. This gives us: E_cross = (-0.5%)/ (1%) = -0.5

b. Income effects and compensated price elasticities

In this problem, there are no substitution effects, because the consumer only consumes ham and cheese together. Thus, they cannot substitute one good for the other. The calculated elasticities only reflect income effects, as an increase in the price of one good affects the consumer's overall purchasing power, causing them to reduce their consumption of ham and cheese. The compensated price elasticities in this problem would be zero, as there would be no change in consumption if the consumer was compensated for the price increase. They would continue to consume the same amount of ham and cheese sandwiches as before.

c. Changes when a slice of ham costs twice the price of cheese

If a slice of ham costs twice the price of a slice of cheese, we need to revisit the budget constraint. If we let P_c = 2P_h, the cost of making a sandwich is now 1.5 times more expensive when the price of ham increases by 1%. The percentage change in quantity demanded of ham would be now -0.75%. Own-price elasticity of demand: E_own = (-0.75%) / (1%) = -0.75 Cross-price elasticity of demand: E_cross = (-0.75%) / (1%) = -0.75 In this case, both the own-price and cross-price elasticities of demand for ham are -0.75.

d. Intuitive solution considering one good

We can simplify the problem by considering only one good: a ham and cheese sandwich. In this case, we only need to find the own-price elasticity of demand for the sandwich. Let the new prices be P_h_new = 2P_c_new. With this constraint, if the price of a ham and cheese sandwich increases by 1%, the consumer's purchasing power is reduced, and they must reduce their consumption of sandwiches. Assuming linear demand, this will result in a decrease in sandwich consumption by 1%. Own-price elasticity of demand for the combined good (sandwich): E_s = (-1%) / (1%) = -1 This intuitive solution demonstrates that when considering only a single good (the ham and cheese sandwich), the own-price elasticity of demand is -1, which is different from the previous parts of the exercise, where we calculated it separately for ham and cheese.