Here are the final numerical results for each section of the exam. You can use them to check your work if you do the exam for practice. If you have trouble with the problems, or don't get the answers shown here, stop by during office hours or make and appointment and we can go over them.
New quantity: 70k; `\Delta CS`: -$85k; `\Delta PS`: +$25k; `\Delta SS`: -$60k. The policy is pretty effective at reducing teen smoking, which drops by 30%. A potentially undesirable side effect is that it raises profits for cigarette makers by $25k.
Extra credit: the policy causes `P^d` to rise to $5 and `P^s` to fall to $1. Since a tax `T` causes `P^d=P^s+T`, it would take a $4 tax to cause the same $4 gap between `P^d` and `P^s`. In that situation, `\Delta Rev`=+$280k and `\Delta PS`=-$255k. The tax would be better for the government and much worse for the firms. However, it would require a very large tax rate, which might not be politically feasible.
Step 1: insert the demand equations into the utility function to give:
`U=( 10+(0.5(M - 10 P_x + 10 P_y ))/P_x -10)^0.5(-10+ (0.5(M - 10 P_x + 10 P_y ))/P_y +10)^0.5`
Since the numerators are the square roots of the same thing, this can be written:
`U=(0.5(M - 10 P_x + 10 P_y ))/(P_x^0.5 P_y^0.5)`
Step 2: solve for M:
`U P_x^0.5 P_y^0.5 = 0.5(M - 10 P_x + 10 P_y )`
`2U P_x^0.5 P_y^0.5 = M - 10 P_x + 10 P_y `
`M = 2U P_x^0.5 P_y^0.5 + 10 P_x - 10 P_y`
Evaluating the function when U=100, `P_x`=$16 and `P_y`=$25 gives $3910.
(a) Household B is CD and `b` is 0.45. Diagram omitted but it should contain the budget constraint, the equilibrium, and at least one indifference curve.
(b) `M_2` = $3600; `X_2` = 135; `Y_2` = 220; `\Delta Rev` = $+720; yes, the policy meets the revenue target; `U_1` = 222.236; `M_3` = $4530; CV = $930; household is worse off; `\Delta SS` = -$210.
(a) Derivation omitted; result is:
`Y = M/(h P_x + P_y)`
`X = (hM)/(h P_x + P_y)`
Household D is PC; h = 1.5.
(b) `M_2` = $1674; `X_2` = 93; `Y_2` = 62; `\Delta Rev` = $704; meets the revenue target but just barely; `M_3` = $2538; CV = $864; household is worse off. Diagram omitted but should show the BC, the equilibrium, and at least one IC.
`X_1` = 2500; `Y_1` = 12,500; `X_2` = 1600; `\Delta Rev` = $3200; `U_1` = 10,000; `M_3` = $64,000; CV = $4k; household is worse off; `X_3` = 1600; substitution effect, `\Delta X_s` = -900; income effect, `\Delta X_i` = 0; unusual because there's no income effect: the entire change is the substitution effect.
PVI of endowment: $136k; PVI of A: $139k; PVI of B: $146k. B is best. `C_0`: $87.6k; `C_1`: $73k; borrows $85.6k. Diagram omitted but should show the budget constraint, the endowment point, point A, the consumption bundle, and an indifference curve.