Example: How Many Songs to Download?
Suppose we have a consumer that only cares about consuming music (𝑀) and all other goods
(𝐴) with the following utility function
1 1
𝑈(𝑀, 𝐴) = 𝑀2 𝐴2
M is the number of songs you download (which can take any real number). This consumer has
income 𝑌 and is deciding how much money to allocate between purchasing music which costs
𝑃𝑀 per song and purchasing all other goods at a price 𝑃𝐴 . The consumer’s choice problem is
then
1
1
max{𝑀,𝐴} 𝑀2 𝐴2 𝑠. 𝑡. 𝑃𝑀 𝑀 + 𝑃𝐴 𝐴 = 𝑌
You can easily verify that
𝜕𝑈
𝐴
𝜕𝑀
𝑀𝑅𝑆 = −
=−
𝜕𝑈
𝑀
𝜕𝐴
Setting this equal to the price ratio gives
−
𝐴
𝑃𝑀
= −
𝑀
𝑃𝐴
so that the “bang for the buck” is equalized. Re-arranging to 𝑃𝐴 𝐴 = 𝑃𝑀 𝑀 and plugging this into
the budget constraint gives us the consumer’s demand functions for music and all other goods:
𝑀=
1 𝑌
1𝑌
,𝐴 =
2 𝑃𝑀
2 𝑃𝐴
For example, if 𝑃𝑀 = 1 , 𝑃𝐴 = 1 and 𝑌 = 30 then, plugging these values into the above
expressions, the optimal allocation of 𝑀∗ = 15, 𝐴∗ = 15.
Kinked Budget Constraint
Now suppose the online music site is running a promotion which gives a volume discount when
a customer purchases more than 18 songs. Specifically, the unit price of the first 18 songs is
𝑃𝑀 = 1 and the unit price of each additional song beyond 18 is 𝑃𝑀′ = 0.5.
Active Learning Exercise #1: Provide an expression for the budget constraint and graph it. Put
AOG on the vertical axis and # of songs on the horizontal axis.
Answer: With this discount deal the vertical intercept is still 30. The slope is -1 up to M = 18. At
that point, the slope becomes -.50 as the price of another song drops to .5 and the price of AOG
̅ denote the maximum number of songs that can be purchased, it is
remains at 1. If we let 𝑀
defined by the following equation:
̅ − 18) = 30 → 𝑀
̅ = 42,
1 × 18 + .5 × (𝑀
where $1 is spent on each of the first 18 songs and the remaining income is used to buy songs at
the discounted price of $0.50 per song.
The budget constraint is
{
1 × 𝑀 + 1 × 𝐴 = 30
18 + 0.5 × (𝑀 − 18) + 1 × 𝐴 = 30
𝑖𝑓 0 ≤ 𝑀 ≤ 18
𝑖𝑓 18 ≤ 𝑀 ≤ 42
We want to find the optimal bundle for this consumer and will do so in two steps.
Active Learning Exercise #2: Consider a bundle at the kink so that the consumer is buying 18
songs and determine whether she would prefer to buy more songs or less songs. (Use the “bang
for the buck” principle.)
Answer: Consider the consumer buying more songs when the bundle is at the kink ( 𝑀 =
18, 𝐴 = 12) which means paying a price of $0.50 for another song. At the discounted price, the
bang for the buck is higher for music when:
1 1
1
1
1
1
𝜕𝑈
𝜕𝑈
(2) 𝑀−2 𝐴2 (2) 𝑀2 𝐴−2
𝑀 18
𝜕𝑀 > 𝜕𝐴 →
>
→
2
>
=
𝑃𝑀′
0.50
1
𝐴 12
𝑃𝐴
which is true; hence, marginal utility per dollar for a song (when the price is $0.50) exceeds the
marginal utility for AOG when the bundle is at the kink. This property tells us that a consumer
prefers to buy more than 18 songs than to buy 18 songs as her utility rises from doing so.
However, that does not imply the optimal solution has her buying more than 18 songs. We still
need to consider the option of buying fewer than 18 songs.
Consider the consumer buying less songs when at the kink which means the price is $1.00 per
song. The bang for the buck is lower for music when:
1 1
1
1
1
1
𝜕𝑈
𝜕𝑈
(2) 𝑀−2 𝐴2 (2) 𝑀2 𝐴−2
𝑀 18
𝜕𝑀 < 𝜕𝐴 →
<
→1< =
′
𝑃𝑀
1
1
𝐴 12
𝑃𝐴
which is true. This property tells us that a consumer prefers to buy fewer than 18 songs than to
buy 18 songs.
Intuition: What we know thus far is that buying 18 songs is not optimal as the consumer can
raise her utility either by buying one more song or buying one less song. In order to determine
her optimal bundle, we need to compare the best solution assuming the consumer buys fewer
than 18 songs (that is, the marginal price she faces is $1) with the best solution assuming the
consumer buys more than 18 songs (that is, the marginal price she faces is $0.50).
Active Learning Exercise #3: Find the optimal bundle.
In the problem we originally reviewed without the discount, it was shown that the optimal bundle
is M = 15, A = 15 in which case the associated utility is
1
1
𝑈(15,15) = 152 152 = 15
Thus, if the consumer chooses to buy fewer than 18 songs, the optimal number is 15 and her
utility is 15.
Now suppose the consumer buys more than 18 songs so that the marginal price of another song
is $0.50. The solution is given by equating the MRS and the price ratio:
𝜕𝑈/𝜕𝑀 𝑃𝑀′
𝐴
=
→
= 0.5 → 𝐴 = 0.5𝑀
𝜕𝑈/𝜕𝐴 𝑃𝐴
𝑀
and satisfying the budget constraint:
1 × 18 + 0.5 × (𝑀 − 18) + 𝐴 = 30
Substitute the first equation into the second one and solve for M:
18 + 0.5 × (𝑀 − 18) + 0.5 × 𝑀 = 30 → 𝑀 = 21
Take the solution for M and plug it into the budget constraint to solve for A:
1 × 18 + 0.5 × (21 − 18) + 𝐴 = 30 → 𝐴 = 10.5.
If the consumer chooses to purchase more than 18 songs then it is optimal for her to buy 21
songs, which will cost her 1 × 18 + 0.5 × 3 = 19.5, and spend the remaining $10.50 on AOG.
Her utility from that bundle is
1
1
(21)2 (10.5)2 = 14.85
Given that this utility is less than that from buying 15 songs, the optimal bundle is 𝑀 = 15, 𝐴 =
15.