EC 313 Spring 2000 Problem Set 3

Define CRRA utility in terms of wealth and the risk aversion parameter.

crra[w_, ρ_] := ((w^(1 - ρ))/(1 - ρ))

1. Find the risk premium, γ, that the consumer would be willing to pay to avoid a loss, by numerical methods.

rho = 3;
loss = 2;
FindRoot[
      crra[10 - γ, 
          rho] ==  .5*crra[10, rho] +  .5*
            crra[10 - loss, rho], {γ, 1}]

2. Find the risk premium by solving the equation analytically and verify that it agrees with the numerical solution.

rp[w_, ρ_, loss_] := 
      w - (0.5 (w^(1 - ρ) + (w - 
                        loss)^(1 - ρ)))^(1/(1 - ρ))

rp[10, 3, 2]

3. Graph the risk premium function for ρ=3 for a 50% probability of loss.

Plot[rp[10, 3, L], {L, 0, 10}]

4. Define expected utility for a consumer who invests in a risky asset, and graph it versus the amount invested.

eu[inv_, winpr_] := 
      winpr  crra[10 + 0.7 inv, ρ] + (1 - winpr) crra[
            10 - 0.3 inv, ρ]

ρ = 3;
Plot[eu[inv, 0.50], {inv, 0, 10}]

5. Find the optimal investment for this consumer.

FindMinimum[-eu[inv, 0.50], {inv, 2}]

6. Recalculate the optimal investment if the odds of gain vs. loss are altered to 55-45%.

FindMinimum[-eu[inv, 0.55], {inv, 2}]