BRMP COMPLETE EXAM QUESTIONS AND
DETAILED ANSWERS COMPREHENSIVE
REVIEW RESOURCE
●● DD model — three things you can do with your endowment
Answer: 1. Store: 1 unit in → 1 unit out at t=1 or t=2 (safe, zero return).
2. Invest long: 1 unit at t=0 → C₂ > 1 at t=2 (productive but illiquid). 3.
Liquidate early: only recovers c₁ < 1 at t=1 (costly early exit). Hierarchy:
c₁ < 1 < C₂.
●● DD model — source of liquidity risk
Answer: At t=1 each agent discovers whether they are impatient (must
consume at t=1) or patient (can wait until t=2). This type is unknown at
t=0. Fraction λ are impatient. Uncertainty about future type is what
makes insurance valuable.
●● DD utility function
Answer: U = λu(x₁) + ρ(1−λ)u(x₂). Weighted average of utility from
early consumption (x₁) and late consumption (x₂). u′>0 (more is better),
u″<0 (concave — diminishing marginal utility, implies risk aversion).
●● Why does concavity (u″<0) matter in DD?
Answer: Concave utility means diminishing marginal returns to
consumption — each extra unit gives less additional satisfaction. This
, makes agents risk-averse and means they prefer a smooth consumption
profile to a volatile one. It is why liquidity insurance has value.
●● DD autarky — how it works
Answer: Each agent decides alone how much to invest (α) vs store
(1−α). If impatient, must liquidate early at cost c₁. If patient, waits and
gets C₂. No risk sharing — impatient consumers always suffer
liquidation losses.
●● DD autarky — FOC
Answer: u′(x₁*) = [(1−λ)/λ] · [(C₂−1)/(1−c₁)] · u′(x₂*). Invest more when
c₁ is high (early liquidation less costly) or C₂ is high (long return better).
Invest less when λ is high (more likely to be impatient).
●● DD bank — how it works
Answer: Bank pools all endowments. Holds (1−αB) as reserves for
impatient depositors; invests αB in the long-term asset for patient
depositors. Offers deposit contract: withdraw x₁ at t=1 or x₂ at t=2.
●● DD bank — budget constraints
Answer: Period 1: (1−αB) = λx₁ — reserves exactly cover impatient
withdrawals. Period 2: αB·C₂ = (1−λ)x₂ — long-term returns pay patient
depositors.
●● DD bank — optimality condition
DETAILED ANSWERS COMPREHENSIVE
REVIEW RESOURCE
●● DD model — three things you can do with your endowment
Answer: 1. Store: 1 unit in → 1 unit out at t=1 or t=2 (safe, zero return).
2. Invest long: 1 unit at t=0 → C₂ > 1 at t=2 (productive but illiquid). 3.
Liquidate early: only recovers c₁ < 1 at t=1 (costly early exit). Hierarchy:
c₁ < 1 < C₂.
●● DD model — source of liquidity risk
Answer: At t=1 each agent discovers whether they are impatient (must
consume at t=1) or patient (can wait until t=2). This type is unknown at
t=0. Fraction λ are impatient. Uncertainty about future type is what
makes insurance valuable.
●● DD utility function
Answer: U = λu(x₁) + ρ(1−λ)u(x₂). Weighted average of utility from
early consumption (x₁) and late consumption (x₂). u′>0 (more is better),
u″<0 (concave — diminishing marginal utility, implies risk aversion).
●● Why does concavity (u″<0) matter in DD?
Answer: Concave utility means diminishing marginal returns to
consumption — each extra unit gives less additional satisfaction. This
, makes agents risk-averse and means they prefer a smooth consumption
profile to a volatile one. It is why liquidity insurance has value.
●● DD autarky — how it works
Answer: Each agent decides alone how much to invest (α) vs store
(1−α). If impatient, must liquidate early at cost c₁. If patient, waits and
gets C₂. No risk sharing — impatient consumers always suffer
liquidation losses.
●● DD autarky — FOC
Answer: u′(x₁*) = [(1−λ)/λ] · [(C₂−1)/(1−c₁)] · u′(x₂*). Invest more when
c₁ is high (early liquidation less costly) or C₂ is high (long return better).
Invest less when λ is high (more likely to be impatient).
●● DD bank — how it works
Answer: Bank pools all endowments. Holds (1−αB) as reserves for
impatient depositors; invests αB in the long-term asset for patient
depositors. Offers deposit contract: withdraw x₁ at t=1 or x₂ at t=2.
●● DD bank — budget constraints
Answer: Period 1: (1−αB) = λx₁ — reserves exactly cover impatient
withdrawals. Period 2: αB·C₂ = (1−λ)x₂ — long-term returns pay patient
depositors.
●● DD bank — optimality condition
