Homework 2 — Step-by-step study notes with sources Summary & Study Notes

❓ User request (context)

What you asked: a brief request: “can you explain how I get to these answers from the pdf.” These notes show the step‑by‑step logic and arithmetic you should follow to reproduce the answers in the PDF.

🔍 How to approach the PDF solutions

• Identify the utility function and the probabilities for each state of the world. In this problem the PDF uses the utility function $U(I)=I^{1/3}$ and two states (healthy with probability $0.75$, sick with probability $0.25$).
• Compute outcomes (income in each state) for each insurance scheme (no insurance, partial, full).
• Compute expected utility (E[U]) by weighting the utility of each state by its probability: $E[U]=p_H\cdot U(I_H)+p_S\cdot U(I_S)$.
• For actuarially fair premiums, multiply the benefit by the probability of the bad state: $\text{premium}=r\cdot \text{benefit}$.
• For full insurance, set post‑insurance incomes equal across states and solve for the benefit (so sick income after benefit = healthy income after premium).
• To get the certainty equivalent (CE): given $E[U]$, solve $U(CE)=E[U]$, i.e. $CE=(E[U])^{3}$ because $U(I)=I^{1/3}$.

✅ Quick checklist to reproduce numeric answers

1. Compute cube roots of monetary amounts (e.g., $8000^{1/3}=20$, $1000^{1/3}=10$).
2. Multiply by probabilities and sum to get $E[U]$.
3. For partial insurance: compute new incomes (healthy and sick) after subtracting premium and adding benefit; then compute utilities and $E[U]$.
4. For full insurance: find benefit that equalizes incomes, compute actuarially fair premium, then compute $E[U]$.
5. Compute CE by cubing $E[U]$.

🛠 Tips for checking arithmetic

• Use a calculator for cube roots and decimal multiplication. The PDF rounds intermediate cube roots (e.g., $6750^{1/3}\approx18.89$).
• Keep track of signs: premium is always paid, benefit is paid only in sick state.
• Ensure probabilities add to 1 and are applied to the correct states.

📄 Overview of the PDF solutions (Homework2Solutions)

These notes reproduce and explain the calculations from ps2_solutions.pdf question by question. Bolded terms indicate the key concept used in each step.

🧮 Question 1 — Step‑by‑step (parts a–f)

• Part (a): Concavity / diminishing marginal utility: The utility function $U(I)=I^{1/3}$ is concave because the exponent $1/3<1$, so marginal returns are diminishing. The PDF states "Yes. <0 means that marginal returns are diminishing." (Interpretation: second derivative is negative.)

• Part (b): Expected income: With $p_H=0.75$ and $p_S=0.25$, expected income is $E[I]=0.75\times 8000+0.25\times 1000=6000+250=6250$. The PDF reports </span>6,250$.

• Part (c): Expected utility without insurance: Compute cube roots: $8000^{1/3}=20$, $1000^{1/3}=10$. Then $E[U]=0.75\times20+0.25\times10=15+2.5=17.5$.

• Part (d) Partial insurance policy analysis:

  • The policy gives a benefit of </span>5,000$in the sick state and charges premium$r=0.25\times5000=<span class="katex">$1,250$ (actuarially fair premium, since premium = probability of sick $\times$ benefit). The PDF checks whether the policy is full: compare incomes after the premium/benefit.
  • Income when healthy: $8000-1250=6750$.
  • Income when sick: $1000-1250+5000=4750$.
  • Because $6750\neq4750$, the policy is not full. It is actuarially fair because premium equals $0.25\times5000=1250$.

• Part (e) Expected utility: compare partial vs full insurance

  • For partial (part d): compute utilities using cube roots: $6750^{1/3}\approx18.89$, $4750^{1/3}\approx16.81$. So $E[U]_{partial}=0.75\times18.89+0.25\times16.81\approx18.37$ (matches PDF).
  • For full, actuarially fair insurance: find benefit so that post‑insurance incomes equalize. Requirement: $8000-\text{premium}=1000-\text{premium}+\text{benefit}$, so $\text{benefit}=7000$. Premium = 0.25\times7000=</span>1,750$. Post‑premium income (both states): $8000-1750=6250$, and $6250^{1/3}\approx18.42$. Therefore $E[U]_{full}=18.42$ (both states equal, so same utility). The PDF shows $18.42$, which is higher than $18.37$ (partial) and higher than no insurance ($17.5$) — demonstrating risk aversion.

• Part (f) Certainty equivalent (CE) without insurance:

  • We have $E[U]_{no,insurance}=17.5$.
  • Solve $U(CE)=17.5$, i.e. $CE^{1/3}=17.5$ so $CE=(17.5)^{3}=5359.375$.
  • Rounded: CE=</span>5,359.38$(matches PDF's$<span class="katex">$5,359.38$).

🔁 Question 2 — Adverse selection and the death spiral

• Part (a): If the healthiest 10% (lowest marginal healthcare cost) drop out, the insurer's remaining pool has higher average costs. With premiums set earlier, the insurer collects less in premiums than pays in claims — a net loss.
• Part (b): The insurer raises premiums next period to cover higher average costs. That higher price can induce the next‑healthiest group to drop out (if they perceive premiums exceed their expected cost), further worsening the risk pool. This iterative process can continue and is called a "death spiral." The PDF explains this qualitatively.

🧾 Question 3 — Premium calculations and cross‑subsidy

• Part (a) Actuarially fair premiums by group (with claim size </span>10,000$):
  • Group A: 0.20\times10000=</span>2,000$.
  • Group B: 0.02\times10000=</span>200$.
• Part (b) Premium using population average (if insurer believes groups are 50/50): $0.5\times2000+0.5\times200=1100$, so the insurer charges </span>1,100$ per person.
• Part (c) Subsidy/tax interpretation: Group A (high risk) pays $1100$ but their expected cost is $2000$, so they receive an implicit subsidy. Group B (low risk) pays $1100$ but their expected cost is $200$, so they are implicitly taxed relative to actuarially fair pricing.

📊 Question 4 — Welfare loss, subsidy, and demand shift

• Part (a): The PDF notes a diagram (EF C curve) where the shaded area is the social welfare loss (deadweight loss) from underprovision of insurance.
• Part (b): The socially optimal quantity is $Q_{max}$ where demand curve is above marginal cost.
• Part (c): A subsidy shifts the demand curve upward by the subsidy amount because consumers pay less out of pocket (their willingness to pay increases by the subsidy). The new demand is the original willingness plus the subsidy.
• Part (d): With the subsidy, the equilibrium quantity and price increase and social welfare loss decreases. Whether the welfare loss is eliminated depends on the subsidy size — a sufficiently large subsidy can raise equilibrium quantity to the social optimum; smaller subsidies reduce but may not fully eliminate the welfare loss.

⚠️ Final check on numeric rounding

• The PDF rounds cube roots to two decimals (e.g., $6750^{1/3}\approx18.89$). Use a calculator for exact decimals when reproducing the arithmetic.
• Keep probability weights consistent and ensure premiums are computed as $r\times$benefit for actuarially fair pricing.

If you want, I can (a) show the exact calculator steps for each cube root and multiplication, or (b) produce a short annotated calculation sheet replicating each numeric line from the PDF.