Worksheet 2 β solutions Part A 1. Equation of BL: π = ππ΅ π΅ + ππΈ πΈ π΅= π ππΈ β πΈ ππ΅ ππ΅ 2. MRS: ππ π = βπππΈ β2π΅ = πππ΅ 2πΈ 3. B=10 books and E=20 units of entertainment Be consistent with the orientation of your graph and equationsβ¦ Part B 1. Initial situation ππ π = β πππ· βπΉ = πππΉ π· ππ π = β ππ· β4 = ππΉ 5 4 Tangency condition: πΉ = 5 π· Budget constraint: 160π· + 200πΉ = 8,000 π·β = 25; πΉ β = 20 π(25,20) = 500 2. Change in price of D Budget constraint: 250π· + 200πΉ = π₯ π·β = π₯ π₯ ; πΉβ = 500 400 3. Keep utility constant β π(π· , πΉ β) π₯2 = = 500 500 β 400 π₯ = β108 = 10,000 4. Compensated bundle π·πΆβ = π₯ 10,000 π₯ 10,000 = = 20; πΉ πΆβ = = = 25 500 500 400 500 5. Final optimal bundle π·ββ = π₯ 8,000 π₯ 8,000 = = 16; πΉ ββ = = = 20 500 500 400 500 6. Conclusion We have decomposed the change in quantity demanded due to a price change into two effects (i) a substitution effect (change in prices: slope of the budget line, but utility is kept constant) and (ii) an income effect (change in purchasing power). Part C Optimality conditions: 1. π1 π1 + π2 π2 = π 2. ππ π = ππ π Using the tangency condition first: ππ1 = ππ = π. π1πβ1 . π21βπ ππ1 ππ2 = ππ = (1 β π). π1π . π2βπ ππ2 ππ π = ππ1 π π2 = . ππ2 1 β π π1 ππ π = ππ π = ππ π π1 π2 π π2 π1 . = 1 β π π1 π2 β‘ (1 β π). π1 . π1 = π. π2 . π2 Second, we substitute using the budget line: BL: π2 π2 = π β π1 π1 Quantity demanded for each good: π1 = π2 = ππ π1 (1 β π)π π2 Note the steps! They will always be the same.
