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Microeconomics Worksheet: Consumer Choice

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Worksheet 2 – solutions
Part A
1. Equation of BL:
π‘š = 𝑝𝐡 𝐡 + 𝑝𝐸 𝐸
𝐡=
π‘š 𝑃𝐸
βˆ’ 𝐸
𝑃𝐡 𝑃𝐡
2. MRS:
𝑀𝑅𝑆 =
βˆ’π‘€π‘ˆπΈ βˆ’2𝐡
=
π‘€π‘ˆπ΅
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 = βˆ’
π‘€π‘ˆπ· βˆ’πΉ
=
π‘€π‘ˆπΉ
𝐷
𝑀𝑅𝑇 = βˆ’
𝑝𝐷 βˆ’4
=
𝑃𝐹
5
4
Tangency condition: 𝐹 = 5 𝐷
Budget constraint: 160𝐷 + 200𝐹 = 8,000
π·βˆ— = 25; 𝐹 βˆ— = 20
π‘ˆ(25,20) = 500
2. Change in price of D
Budget constraint: 250𝐷 + 200𝐹 = π‘₯
π·βˆ— =
π‘₯
π‘₯
; πΉβˆ— =
500
400
3. Keep utility constant
βˆ—
π‘ˆ(𝐷 , 𝐹
βˆ—)
π‘₯2
=
= 500
500 βˆ— 400
π‘₯ = √108 = 10,000
4. Compensated bundle
π·πΆβˆ— =
π‘₯
10,000
π‘₯
10,000
=
= 20; 𝐹 πΆβˆ— =
=
= 25
500
500
400
500
5. Final optimal bundle
π·βˆ—βˆ— =
π‘₯
8,000
π‘₯
8,000
=
= 16; 𝐹 βˆ—βˆ— =
=
= 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 π‘ž1 + 𝑝2 π‘ž2 = π‘Œ
2. 𝑀𝑅𝑆 = 𝑀𝑅𝑇
Using the tangency condition first:
π‘€π‘ˆ1 =
πœ•π‘ˆ
= π‘Ž. π‘ž1π‘Žβˆ’1 . π‘ž21βˆ’π‘Ž
πœ•π‘ž1
π‘€π‘ˆ2 =
πœ•π‘ˆ
= (1 βˆ’ π‘Ž). π‘ž1π‘Ž . π‘ž2βˆ’π‘Ž
πœ•π‘ž2
𝑀𝑅𝑆 =
π‘€π‘ˆ1
π‘Ž π‘ž2
=
.
π‘€π‘ˆ2 1 βˆ’ π‘Ž π‘ž1
𝑀𝑅𝑇 =
𝑀𝑅𝑆 = 𝑀𝑅𝑇
𝑝1
𝑝2
π‘Ž π‘ž2 𝑝1
. =
1 βˆ’ π‘Ž π‘ž1 𝑝2
≑
(1 βˆ’ π‘Ž). π‘ž1 . 𝑝1 = π‘Ž. π‘ž2 . 𝑝2
Second, we substitute using the budget line:
BL: 𝑝2 π‘ž2 = π‘Œ βˆ’ 𝑝1 π‘ž1
Quantity demanded for each good:
π‘ž1 =
π‘ž2 =
π‘Žπ‘Œ
𝑝1
(1 βˆ’ π‘Ž)π‘Œ
𝑝2
Note the steps! They will always be the same.
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