Part 3: Market Failures Practice Problem Solutions
Econ 502: Advanced Microeconomics
Problem 1: Adverse Selection in Insurance
Part (c): Range of \(\lambda\) for unraveling
The pooled premium as a function of the high-risk share is
\[P_{\text{pool}}(\lambda) = (0.4 \lambda + 0.2) \cdot 36 = 14.4 \lambda + 7.2\]
Low-risk types drop out when \(P_{\text{pool}}(\lambda) > P_L^{\max}\):
\[14.4 \lambda + 7.2 > 7.84 \;\;\Longrightarrow\;\; \lambda > \frac{0.64}{14.4} = \frac{2}{45} \approx 0.044\]
\[\boxed{\text{Market unravels for } \lambda > 2/45 \approx 4.4\%}\]
Logic. At the pooled premium, low-risk types subsidize high-risk types. The maximum subsidy a low-risk type tolerates equals their risk premium (\(P_L^{\max} - p_L L = 0.64\)). As soon as the pool contains enough high-risk types to push the pooled premium above \(P_L^{\max}\), low-risk consumers prefer to self-insure. Their exit raises the average risk in the pool, forcing the premium even higher - the classic adverse-selection death spiral. With only two types here, the dynamic stops once all low-risk are out and only high-risk remain (paying near \(p_H L\)).
Part (d): Compulsory insurance at \(P_{\text{pool}} = 14.40\)
Wealth under insurance is \(w - P_{\text{pool}} = 85.60\) in both states, giving utility \(\sqrt{85.60} \approx 9.252\).
| Type | \(EU\) no insurance | \(u\) with mandate | Change |
|---|---|---|---|
| Low-risk | \(9.600\) | \(9.252\) | \(-0.348\) |
| High-risk | \(8.800\) | \(9.252\) | \(+0.452\) |
- Low-risk are worse off (forced to subsidize the bad pool).
- High-risk are better off (get coverage at a price below their willingness to pay).
The insurer collects \(14.40\) per policy and pays out \(0.5(0.6) + 0.5(0.2) = 0.4\) in probability \(\times\) \(L = 36\) on average \(= 14.40\), so it breaks even.
Why the policy can still be desirable. A utilitarian planner adds the changes: net gain \(= -0.348 + 0.452 = +0.104\) utils per person - total welfare rises. More fundamentally, the mandate solves the adverse-selection failure: high-risk types receive coverage they could otherwise obtain only at much higher prices, while low-risk types lose only their small risk-aversion benefit. There are also equity arguments: people generally do not choose their risk type, and pooling spreads the cost of bad luck. This logic underlies, e.g., the individual mandate in the Affordable Care Act.
Problem 2: Pigouvian Tax and Cap-and-Trade
Part (a): Unregulated equilibrium
Each plant ignores pollution damage and maximizes \(y_i (P - c) = 2 y_i\) subject to \(y_i \leq 250\). The per-gallon profit is positive, so each plant produces at capacity:
\[y_1 = y_2 = 250\]
Total smog: \(s_1 + s_2 = 250^2 + \tfrac{1}{2}(250^2) = 62{,}500 + 31{,}250 = 93{,}750\) cubic feet.
\[\boxed{\text{Total damage} = 0.01 \times 93{,}750 = \$937.50}\]
Part (c): Pigouvian tax
With tax \(t = 0.01\) per cubic foot of smog, each plant now solves
\[\max_{y_i} \; (P - c) y_i - t \cdot s_i(y_i)\]
This is identical to the planner’s problem in part (b), so each plant chooses the efficient output \(y_1 = 100\), \(y_2 = 200\). \(\checkmark\)
Total tax revenue: \(t \times \text{total smog} = 0.01 \times 30{,}000 = \$300\).
The tax decentralizes the efficient outcome with minimal information: the regulator only needs to know the marginal damage of smog, not each plant’s pollution function.
Part (d): Cap-and-trade
The regulator issues \(\bar{S} = 30{,}000\) permits. Plant \(i\) holds initial allocation \(e_i\) (with \(e_1 + e_2 = 30{,}000\)) and faces market permit price \(\tau\). Each plant’s profit is
\[\pi_i = (P - c) y_i - \tau \big( s_i(y_i) - e_i \big)\]
(the second term is the cost of using \(s_i\) permits on top of its endowment \(e_i\) - negative if the plant ends up a seller). The FOC is the same as under the tax: \(P - c = \tau \cdot s_i^{\prime}(y_i)\), so
\[2 = 2 \tau y_1 \Longrightarrow y_1 = \frac{1}{\tau}, \qquad 2 = \tau y_2 \Longrightarrow y_2 = \frac{2}{\tau}\]
Market clearing. Total permit demand equals supply:
\[y_1^2 + \tfrac{1}{2} y_2^2 = \frac{1}{\tau^2} + \frac{2}{\tau^2} = \frac{3}{\tau^2} = 30{,}000\]
\[\tau^{*2} = \frac{1}{10{,}000} \;\;\Longrightarrow\;\; \tau^{*} = \$0.01\]
Plugging in: \(y_1 = 100\), \(y_2 = 200\) - same as the efficient allocation, and \(\tau^{*}\) equals the Pigouvian tax rate.
(i) All permits to Plant 1 (\(e_1 = 30{,}000\), \(e_2 = 0\)).
- Plant 1 uses \(s_1 = 100^2 = 10{,}000\) permits and sells the remaining 20{,}000 to Plant 2 at \(\tau = 0.01\).
- Plant 1 profit: \(\pi_1 = 100(2) - 0.01(10{,}000) + 0.01(30{,}000) = 200 - 100 + 300 = \$400\).
- Plant 2 profit: \(\pi_2 = 200(2) - 0.01(20{,}000) + 0 = 400 - 200 = \$200\).
(ii) All permits to Plant 2 (\(e_1 = 0\), \(e_2 = 30{,}000\)).
The first-order conditions and market-clearing equation are unchanged, so \(\tau^{*} = 0.01\) and \(y_1 = 100\), \(y_2 = 200\) as before.
What changes is the trade direction and profits:
- Plant 2 keeps 20{,}000 permits for its own use and sells 10{,}000 to Plant 1.
- Plant 1 profit: \(\pi_1 = 200 - 100 + 0 = \$100\).
- Plant 2 profit: \(\pi_2 = 400 - 200 + 300 = \$500\).
Both plants produce the efficient quantities - only the distribution of profits shifts (totals are \(\$600\) in both cases, redistributed by the permit revenue).
(iii) Connection to the Coase theorem.
With well-defined property rights (permits) and frictionless trade, the efficient allocation arises regardless of who initially holds the rights. The initial allocation only redistributes surplus; it has no effect on which plant pollutes how much. The regulator does not need to know plants’ cost or pollution functions - the market reveals them through trading.
The price-based (Pigouvian tax) and quantity-based (cap-and-trade) instruments are dual: in this stylized environment, \(\tau^{*} = t^{*}\) and the same outputs arise. Cap-and-trade has the additional virtue of fixing the total quantity of pollution (useful when the regulator cares about an absolute pollution ceiling), while the Pigouvian tax fixes the price of pollution (useful when there is uncertainty about quantities).
Problem 3: Coase Theorem
Part (a): Total surplus
| Doctor not SP | Doctor SP | |
|---|---|---|
| Bakery noisy | \(260\) | \(\boxed{300}\) |
| Bakery quiet | \(280\) | \(250\) |
The efficient combination is bakery noisy + doctor soundproofed (total surplus \(300\)). Soundproofing eliminates a \(\$70\) noise damage at a cost of only \(\$30\), while letting the bakery keep its higher-profit noisy machinery.
Part (b): Doctor has the right to silence
Without bargaining. The bakery is forced to be quiet. The doctor compares \(130\) (no soundproofing) with \(100\) (soundproofing) and chooses no soundproofing. Outcome: (quiet, no SP), with payoffs \((\pi_B, \pi_D) = (150, 130)\) and total surplus \(280\).
With bargaining. The parties move to (noisy, SP), where the doctor agrees to soundproof and the bakery pays the doctor \(X\) for permission to be noisy. Joint surplus rises by \(20\).
- Bakery accepts if \(200 - X \geq 150 \Longrightarrow X \leq 50\)
- Doctor accepts if \(100 + X \geq 130 \Longrightarrow X \geq 30\)
\[\boxed{30 \leq X \leq 50}\]
Part (c): Bakery has the right to be noisy
Without bargaining. The bakery chooses to be noisy. The doctor compares \(60\) (no soundproofing) with \(100\) (soundproofing) and chooses to soundproof. Outcome: (noisy, SP), total surplus \(300\) - already efficient. No bargaining is required.
The doctor unilaterally adopts the efficient response because the cost of soundproofing (\(\$30\)) is less than the noise damage avoided (\(\$70\)).
Part (d): Comparison and the Coase theorem
In both regimes, the efficient outcome (noisy, SP) is reached with frictionless bargaining. The assignment of property rights does not affect efficiency, only the distribution of payoffs:
| Regime | \(\pi_B\) | \(\pi_D\) |
|---|---|---|
| Doctor’s right (bargained, \(X = 30\)) | \(170\) | \(130\) |
| Doctor’s right (bargained, \(X = 50\)) | \(150\) | \(150\) |
| Bakery’s right (no bargaining) | \(200\) | \(100\) |
This is the Coase theorem: with well-defined property rights and zero transaction costs, private bargaining yields an efficient allocation regardless of the initial assignment of rights.
Why the assignment of rights still matters in practice:
- Real bargaining has transaction costs (lawyers, time, search) that may exceed the gains from trade.
- Asymmetric information about the parties’ valuations can prevent agreement (each side may misrepresent reservation values).
- Wealth and credit constraints: if the loser cannot afford to pay the winner, the inefficient status quo persists.
- Many parties (multiple residents, multiple polluters) make Coasian bargaining infeasible due to free-riding and coordination costs.
- Distributional / equity considerations: even if efficient, society may prefer one assignment over another.
Problem 4: Public Good Provision
Part (a): Best response and Nash equilibrium
Person \(i\) chooses \(g_i\) to maximize
\[U_i = (Y_i - g_i)(g_i + g_{-i}) = (60 - g_i)(g_i + g_{-i})\]
The FOC is \(-(g_i + g_{-i}) + (60 - g_i) = 0\), giving
\[\boxed{g_i^{*}(g_{-i}) = 30 - \frac{g_{-i}}{2}}\]
In a symmetric Nash equilibrium \(g_1 = g_2 = g_3 = g\) and \(g_{-i} = 2g\):
\[g = 30 - g \;\;\Longrightarrow\;\; g_i^{*} = 15\]
\[\boxed{G^{*} = 45, \quad x_i^{*} = 45}\]
Part (c): Samuelson and the free-rider gap
For \(U_i = x_i G\), \(MRS_i = (\partial U_i/\partial G)/(\partial U_i/\partial x_i) = x_i / G\).
- Nash: \(MRS_i = 45/45 = 1\) for each \(i\). Each person privately equates \(MRS_i\) to the price of the public good, but \(\sum_i MRS_i = 3\) - so collectively they massively under-provide.
- Social optimum: \(MRS_i = 30/90 = 1/3\) for each \(i\), and \(\sum_i MRS_i = 1\). This is the Samuelson condition \(\sum_i MRS_i = 1\) (the relative price of the public good in terms of the private good). \(\checkmark\)
\[\boxed{\frac{G^{*}}{G^{**}} = \frac{45}{90} = \frac{1}{2}}\]
Generalizing. With \(N\) symmetric contributors, the symmetric Nash satisfies \(g_i^{*} = (60 - (N-1) g_i^{*})/2\), giving \(g_i^{*} = 60/(N+1)\) and \(G^{*} = 60 N/(N+1)\). The social optimum gives \(G^{**} = NY/2 = 30 N\). So
\[\frac{G^{*}}{G^{**}} = \frac{2}{N+1}\]
As \(N \to \infty\), this ratio \(\to 0\). The free-rider problem becomes overwhelming as the number of contributors grows.
Part (d): Pigouvian-style subsidy
With subsidy rate \(s\), the consumer’s budget is \(x_i = 60 - (1-s) g_i\) and
\[U_i = (60 - (1-s) g_i)(g_i + g_{-i})\]
FOC: \(-(1-s)(g_i + g_{-i}) + (60 - (1-s) g_i) = 0\), giving \(60 = (1-s)(2 g_i + g_{-i})\).
Symmetric: \(60 = (1-s)\cdot 4 g \Longrightarrow g = 15/(1 - s)\) and \(G = 45/(1-s)\).
Setting \(G = G^{**} = 90\):
\[\frac{45}{1 - s} = 90 \;\;\Longrightarrow\;\; 1 - s = \frac{1}{2} \;\;\Longrightarrow\;\; \boxed{s^{*} = \frac{1}{2}}\]
The government covers half the cost of each contribution, exactly internalizing the externality each contributor imposes on the other two. (More generally, for \(N\) contributors, \(s^{*} = (N-1)/(N+1)\).)