Mock Midterm 2
Econ 366-Markets with Frictions
Spring 2024
1. Equilibria in the market for lemons. Consider the product market model of Akerlof (1970). The market is populated by a continuum of sellers. Each seller is endowed with a unit of the good. If the quality of his good is θ ∈ [0; 1], the seller gets utility v(θ) = θ2 from consuming the good and utility p from selling the good at the price p. The fraction of sellers endowed with a good of quality θ ≤ θ is F (θ) = θ . The market is also populated by a continuum of buyers. Each buyer gets utility θ -p from purchasing a good of quality θ at the price p. Information is asymmetric. Sellers know the quality of their own good. Buyers only know the average quality of the goods that are traded in equilibrium.
a. Write down and solve the problem of the seller.
b. Compute the reservation quality function R(p), i.e. the highest quality of the goods that sellers choose to trade when the price is p ∈ [0; 1].
c. Compute the average quality function μ(p), i.e. the average quality of the goods that sellers choose to trade when the price is p ∈ [0; 1].
d. Plot the reservation quality function, R(p), the average quality function, μ(p), and identify the set of equilibria of the economy.
e. Compute the reservation quality, R* , the average quality, μ* , the price, p* , and the number of goods traded for all of the equilibria of the economy
2. Sales taxes in frictional product markets. Consider the product market model of Burdett and Judd (1983). The buyersíutility from consuming a unit of the good is u > 0 and the disutility from paying p dollars for the good is -p. The sellers can produce the good at the unit cost c, where c > 0. Each buyer meets one seller with probability α 1 > 0 and two sellers with probability α2 > 0. The government taxes consumption of the good at the rate t ∈ (0; 1).
a. Write down and solve the problem of a buyer who meets a seller charging the price p. Remember to take into account that the buyer has to pay the sales tax tp.
b. Write down and solve the problem of a buyer who meets a seller charging the price p1 and a seller charging the price p2 .
c. Write down the proÖts of a Örm that charges the price p.
d. What is the highest price on the equilibrium price distribution F?
e. Compute the maximized proÖts of a seller Π*
f. Compute the equilibrium price distribution F.
g. Does the price distribution increase or decrease if the tax rate t increases? Provide an intuition for your Öndings.
3. Local prices and gentriÖcation. Using the search-theoretic model of price dispersion of Burdett and Judd (1983), we want to understand the e§ect of gentriÖcation on local prices. Consider a neighborhood populated by a measure 1 of local buyers and a measure 1 of sellers. Each local buyer wants a unit of some good, which gives him a utility of u > 0. Each seller has the technology to produce the good at the unit cost c ∈ (0; u). Buyers can only shop from a small number of sellers on a given day. SpeciÖcally, a buyer can shop from 1 randomly-selected seller with probability αL;1 = λ ∈ (0; 1) and from 2 randomly-selected sellers with probability αL;2 = 1 - λ .
a. Write down the equilibrium distribution of posted prices F (p). What is the key property of F?
b. Write down the equilibrium distribution of transaction prices G(p). Explain the formula.
c. What is the sign of the di§erence between F and G? Explain.
Now, suppose that a fraction 1=2 of the local buyers are displaced by newcomers. Each newcomer can shop from 1 randomly-selected seller with probability αN;1 = μ and from 2 randomly-selected sellers with probability αN;2 = 1 - μ . Newcomers are busy relative to locals, in the sense that μ > λ .
d. After the displacement of locals, what is the fraction ^(α)1 = ^(α) of buyers who can shop from only one seller? What is the fraction ^(α)2 = 1 -^(α) of buyers who can shop from multiple sellers? Compare ^(α) to λ .
e. After the displacement of locals, what is the equilibrium distribution of posted prices F(^)(p)? And the distribution of transaction prices G(^)(p)?
f. What happens to posted prices because of the displacement of locals? [Hint: compute the di§erence between F(^) and F] Explain.
g. Do locals pay more or less after the arrival of the newcomers? Explain.
4. Return to investment and herding. Consider the herding model of Bikhchandani, Hirshleifer and Welch (1992). In the economy, there is a unique investment opportunity. The cost of the investment is C = 1=2. The return of the investment V is equal to R with probability 1=2 and is equal to 0 with probability 1=2. Agent t = 1; 2; .... does not observe the value of V. However, he observes the investment decision of agents j = 1; 2...t - 1 and a private signal σt. If V = R, the signal is equal to H with probability α ∈ (1=2; 1) and to L with probability 1 - α . If V = 0, the signal is equal to H with probability 1 - α and equal to L with probability α .
a. Write down and solve the investment problem for an agent who believes that the project is successful with probability πt ∈ [0; 1]. How does the solution depend on R?
b. What probability does agent 1 assign to the event V = R after receiving the signal σ 1 = L? What probability does agent 1 assign to the event V = R after receiving the signal σ 1 = H?
c. Suppose that σ 1 = H. Under what condition on R does the agent Önd it optimal to invest? Under what condition on R does the agent Önd it optimal not to invest? Now, suppose that σ 1 = L. Under what condition on R does the agent Önd it optimal to invest? Under what condition on R does the agent Önd it optimal not to invest?
d. Find a condition on R under which the agent invests in the project irrespective of σ 1 .
e. Under the condition derived in (d), what do agents 2, 3, ... do?