Worksheet 2 – solutions
Part A
1. Equation of BL:
𝑚 = 𝑝𝐵 𝐵 + 𝑝𝐸 𝐸
𝐵=
𝑚 𝑃𝐸
− 𝐸
𝑃𝐵 𝑃𝐵
2. MRS:
𝑀𝑅𝑆 =
−𝑀𝑈𝐸 −2𝐵
=
𝑀𝑈𝐵
2𝐸
3. B=10 books and E=20 units of entertainment
Be consistent with the orientation of your graph and equations…
Part B
1.
Initial situation
𝑀𝑅𝑆 = −
𝑀𝑈𝐷 −𝐹
=
𝑀𝑈𝐹
𝐷
𝑀𝑅𝑇 = −
𝑝𝐷 −4
=
𝑃𝐹
5
4
Tangency condition: 𝐹 = 5 𝐷
Budget constraint: 160𝐷 + 200𝐹 = 8,000
𝐷∗ = 25; 𝐹 ∗ = 20
𝑈(25,20) = 500
2. Change in price of D
Budget constraint: 250𝐷 + 200𝐹 = 𝑥
𝐷∗ =
𝑥
𝑥
; 𝐹∗ =
500
400
3. Keep utility constant
∗
𝑈(𝐷 , 𝐹
∗)
𝑥2
=
= 500
500 ∗ 400
𝑥 = √108 = 10,000
4. Compensated bundle
𝐷𝐶∗ =
𝑥
10,000
𝑥
10,000
=
= 20; 𝐹 𝐶∗ =
=
= 25
500
500
400
500
5. Final optimal bundle
𝐷∗∗ =
𝑥
8,000
𝑥
8,000
=
= 16; 𝐹 ∗∗ =
=
= 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. 𝑝1 𝑞1 + 𝑝2 𝑞2 = 𝑌
2. 𝑀𝑅𝑆 = 𝑀𝑅𝑇
Using the tangency condition first:
𝑀𝑈1 =
𝜕𝑈
= 𝑎. 𝑞1𝑎−1 . 𝑞21−𝑎
𝜕𝑞1
𝑀𝑈2 =
𝜕𝑈
= (1 − 𝑎). 𝑞1𝑎 . 𝑞2−𝑎
𝜕𝑞2
𝑀𝑅𝑆 =
𝑀𝑈1
𝑎 𝑞2
=
.
𝑀𝑈2 1 − 𝑎 𝑞1
𝑀𝑅𝑇 =
𝑀𝑅𝑆 = 𝑀𝑅𝑇
𝑝1
𝑝2
𝑎 𝑞2 𝑝1
. =
1 − 𝑎 𝑞1 𝑝2
≡
(1 − 𝑎). 𝑞1 . 𝑝1 = 𝑎. 𝑞2 . 𝑝2
Second, we substitute using the budget line:
BL: 𝑝2 𝑞2 = 𝑌 − 𝑝1 𝑞1
Quantity demanded for each good:
𝑞1 =
𝑞2 =
𝑎𝑌
𝑝1
(1 − 𝑎)𝑌
𝑝2
Note the steps! They will always be the same.