Quiz 1 Discussion
Question 1
Consider the following consumption data. The table below displays the cost of 4 bundles.
The bundles on the diagonal are the chosen bundles.
Observation
Bundle A
Bundle B
Bundle C
Bundle D
1
$20
$25
$20
$30
2
$10
X
$15
$20
3
$20
$25
$15
$10
4
$25
$10
$30
$20
For which positive values of X, the consumption data satisfies WARP and violates SARP?
Question 1: Solution
Consider the following consumption data. The table below displays the cost of 4 bundles.
The bundles on the diagonal are the chosen bundles.
Observation
Bundle A
Bundle B
Bundle C
Bundle D
1
$20
$25
$20*
$30
2
$10
X
$15
$20
3
$20
$25
$15
$10*
4
$25
$10*
$30
$20
For which positive values of X, the consumption data satisfies WARP and violates SARP?
A is directly revealed preferred to C.
C is directly revealed preferred to D.
D is directly revealed preferred to B.
To satisfy WARP, B shouldn’t be directly revealed preferred to D: 𝑋 < 20.
Since the consumer directly revealed prefers A to C, C to D, and D to B, A is indirectly preferred to B.
SARP is violated when B is directly revealed preferred to A: 𝑋 ≥ 10.
All in all, WARP is satisfied and SARP is not when 10 ≤ 𝑋 < 20.
Question 2
Consider a market with two goods: apple and orange.
Consider a consumer with $80 and a coupon for 5 apples.
The consumer can buy apples and oranges at the following unit prices.
Price of apple: $5
Price of orange: $4
Suppose that the consumer can sell the coupon for $20.
Formulate this consumer's budget set.
Question 2: Solution
Consider a market with two goods: apple and orange.
Consider a consumer with $80 and a coupon for 5 apples.
The consumer can buy apples and oranges at the following unit prices.
Price of apple: $5
Price of orange: $4
Suppose that the consumer can sell the coupon for $20.
Formulate this consumer's budget set.
What is the maximum amount of orange consumption for given level of apple consumption?
Don’t sell: 𝑥𝑎𝑝𝑝𝑙𝑒 ≤ 5: 𝑥𝑜𝑟𝑎𝑛𝑔𝑒 ≤ 20
𝑥𝑎𝑝𝑝𝑙𝑒 > 5: 𝑥𝑜𝑟𝑎𝑛𝑔𝑒 ≤ {80 − 5(𝑥𝑎𝑝𝑝𝑙𝑒 − 5)}⁄4 = 26.25 − 1.25𝑥𝑎𝑝𝑝𝑙𝑒
Sell:
𝑥𝑜𝑟𝑎𝑛𝑔𝑒 ≤ {100 − 5𝑥𝑎𝑝𝑝𝑙𝑒 }⁄4 = 25 − 1.25𝑥𝑎𝑝𝑝𝑙𝑒
Question 2: Solution
Consider a market with two goods: apple and orange.
Consider a consumer with $80 and a coupon for 5 apples.
The consumer can buy apples and oranges at the following unit prices.
Price of apple: $5
Price of orange: $4
Suppose that the consumer can sell the coupon for $20.
Formulate this consumer's budget set.
What is the maximum amount of orange consumption for given level of apple consumption?
Don’t sell: 𝑥𝑎𝑝𝑝𝑙𝑒 ≤ 5: 𝑥𝑜𝑟𝑎𝑛𝑔𝑒 ≤ 20
𝑥𝑎𝑝𝑝𝑙𝑒 > 5: 𝑥𝑜𝑟𝑎𝑛𝑔𝑒 ≤ {80 − 5(𝑥𝑎𝑝𝑝𝑙𝑒 − 5)}⁄4 = 26.25 − 1.25𝑥𝑎𝑝𝑝𝑙𝑒
Sell:
𝑥𝑜𝑟𝑎𝑛𝑔𝑒 ≤ {100 − 5𝑥𝑎𝑝𝑝𝑙𝑒 }⁄4 = 25 − 1.25𝑥𝑎𝑝𝑝𝑙𝑒
𝟒 < 𝒙𝒂𝒑𝒑𝒍𝒆 : Don’t sell
𝒙𝒂𝒑𝒑𝒍𝒆 ≤ 𝟒: Sell
Question 2: Solution
Consider a market with two goods: apple and orange.
Consider a consumer with $80 and a coupon for 5 apples.
The consumer can buy apples and oranges at the following unit prices.
Price of apple: $5
Price of orange: $4
Suppose that the consumer can sell the coupon for $20.
Formulate this consumer's budget set.
What is the maximum amount of orange consumption for given level of apple consumption?
𝑥𝑎𝑝𝑝𝑙𝑒 > 5:
4 < 𝑥𝑎𝑝𝑝𝑙𝑒 ≤ 5:
𝑥𝑎𝑝𝑝𝑙𝑒 ≤ 4:
𝒙𝒐𝒓𝒂𝒏𝒈𝒆 ≤ {𝟖𝟎 − 𝟓(𝒙𝒂𝒑𝒑𝒍𝒆 − 𝟓)}⁄𝟒 = 𝟐𝟔. 𝟐𝟓 − 𝟏. 𝟐𝟓𝒙𝒂𝒑𝒑𝒍𝒆
(Don’t sell the coupon)
𝑥𝑜𝑟𝑎𝑛𝑔𝑒 ≤ {100 − 5(𝑥𝑎𝑝𝑝𝑙𝑒 )}⁄4 = 25 − 1.25𝑥𝑎𝑝𝑝𝑙𝑒
(Sell the coupon)
𝒙𝒐𝒓𝒂𝒏𝒈𝒆 ≤ 𝟐𝟎
(Don’t sell the coupon)
𝑥𝑜𝑟𝑎𝑛𝑔𝑒 ≤ {100 − 5(𝑥𝑎𝑝𝑝𝑙𝑒 )}⁄4 = 25 − 1.25𝑥𝑎𝑝𝑝𝑙𝑒
(Sell the coupon)
𝑥𝑜𝑟𝑎𝑛𝑔𝑒 ≤ 20
(Don’t sell the coupon)
𝒙𝒐𝒓𝒂𝒏𝒈𝒆 ≤ {𝟏𝟎𝟎 − 𝟓𝒙𝒂𝒑𝒑𝒍𝒆 }⁄𝟒 = 𝟐𝟓 − 𝟏. 𝟐𝟓𝒙𝒂𝒑𝒑𝒍𝒆
(Sell the coupon)
Question 2: Solution
Consider a market with two goods: apple and orange.
Consider a consumer with $80 and a coupon for 5 apples.
The consumer can buy apples and oranges at the following unit prices.
Price of apple: $5
Price of orange: $4
Suppose that the consumer can sell the coupon for $20.
Formulate this consumer's budget set.
What is the maximum amount of orange consumption for given level of apple consumption?
𝑥𝑎𝑝𝑝𝑙𝑒 > 5:
𝑥𝑜𝑟𝑎𝑛𝑔𝑒 ≤ {80 − 5(𝑥𝑎𝑝𝑝𝑙𝑒 − 5)}⁄4 = 26.25 − 1.25𝑥𝑎𝑝𝑝𝑙𝑒
4 < 𝑥𝑎𝑝𝑝𝑙𝑒 ≤ 5:
𝑥𝑜𝑟𝑎𝑛𝑔𝑒 ≤ 20
𝑥𝑎𝑝𝑝𝑙𝑒 ≤ 4:
𝑥𝑜𝑟𝑎𝑛𝑔𝑒 ≤ {100 − 5𝑥𝑎𝑝𝑝𝑙𝑒 }⁄4 = 25 − 1.25𝑥𝑎𝑝𝑝𝑙𝑒
Question 3
Good 2
W
X
Y
Z
Good 1
The figure above shows three indifference curves representing different utility levels. Note that bundle W and bundle
Z are on the same indifference curve. Suppose that the preference is monotone.
a) Show that the following claim is incorrect: "The underlying preference represented by these indifference curves
cannot be convex."
b) Consider a convex budget set with a linear budget line under which both X and Y are on the budget line. Explain
why bundle Y cannot be the optimal bundle.
c) Suppose that W is optimal. Explain why Z cannot be optimal.
Question 3: Solution (a)
Good 2
W
A
X
Y
Z
B
Good 1
The figure above shows three indifference curves representing different utility levels. Note that bundle W and bundle
Z are on the same indifference curve. Suppose that the preference is monotone.
Show that the following claim is incorrect: "The underlying preference represented by these indifference
curves cannot be convex."
Note that for every convex combination of two bundles on the same indifference curve, we can find a bundle on the
same indifference curve with the same amount of good 1 and less of good 2. Due to monotonicity, the convex
combination is strictly preferred to the bundles on the indifference curve. Therefore, the preference is strictly convex.
Question 3: Solution (b)
Good 2
W
X
Y
A
Z
B
Good 1
The figure above shows three indifference curves representing different utility levels. Note that bundle W and bundle
Z are on the same indifference curve. Suppose that the preference is monotone.
Consider a convex budget set with a linear budget line under which both X and Y are on the budget line.
Explain why bundle Y cannot be the optimal bundle.
Note that A is on the bundle line and B is on the same indifference curve as Y. Since A has the same amount of good
1 and more of good 2, A is strictly preferred to B. Since B and Y are on the same indifference curve, we can conclude
that A is also strictly preferred to Y. As A is on the budget line and strictly preferred to Y, Y cannot be the optimal
bundle.
Question 3: Solution (c)
Good 2
W
X
Y
Z
Good 1
The figure above shows three indifference curves representing different utility levels. Note that bundle W and bundle
Z are on the same indifference curve. Suppose that the preference is monotone.
Suppose that W is optimal. Explain why Z cannot be optimal.
As we proved in part (a), the preference is strictly convex. When the preference is strictly convex, we know that the
optimal bundle is unique. Therefore, W and Z cannot be optimal simultaneously.
Question 4
Consider a market with 4 goods: good 1, good 2, good 3, and good 4
The prices of goods are as follows:
Price of good 1: $3
Price of good 3: $2
Price of good 2: $4
Price of good 4: $8
Consider a consumer with $240.
Suppose that the consumer's utility is formulated as follows:
𝑈(𝑥1 , 𝑥2 , 𝑥3 , 𝑥4 ) = 𝑚𝑖𝑛{3𝑥1 , 4𝑥2 + 𝑥3 } + 4𝑥4
Formulate the optimal consumption bundle(s) for this consumer?
Question 4: Solution
Consider a market with 4 goods: good 1, good 2, good 3, and good 4
The prices of goods are as follows:
Price of good 1: $3
Price of good 3: $2
Price of good 2: $4
Price of good 4: $8
Consider a consumer with $240.
Suppose that the consumer's utility is formulated as follows:
𝑈(𝑥1 , 𝑥2 , 𝑥3 , 𝑥4 ) = 𝑚𝑖𝑛{3𝑥1 , 4𝑥2 + 𝑥3 } + 4𝑥4
Formulate the optimal consumption bundle(s) for this consumer?
Good 2 and good 3 are perfect substitutes: 1 utility is cheaper with good 2
𝑥3∗ = 0 and 𝑈(𝑥1 , 𝑥2 , 𝑥4 ) = 𝑚𝑖𝑛{3𝑥1 , 4𝑥2 } + 4𝑥4
Good 1 and good 2 are perfect complements: 1 utility requires 1/3 units of good 1 and 1/4 unit of good 2 (cost: $2)
Good 4 is perfect substitute for good 1 and good 2 combination: cost of 1 utility = $2
Since both combinations cost the same amount for 1 utility, we have infinitely many optimal bundles.
(𝑥1∗ , 𝑥2∗ , 𝑥3∗ , 𝑥4∗ ) = {(𝑥1 , 𝑥2 , 𝑥3 , 𝑥4 ): 3𝑥1 + 4𝑥2 + 8𝑥4 = 240 𝑎𝑛𝑑 3𝑥1 = 4𝑥2 𝑎𝑛𝑑 𝑥3 = 0}
Question 5
Consider a market with three goods: good 1, good 2, and good 3.
The prices of goods are as follows:
Price of good 1: $2
Price of good 3: $4
Price of good 2: $8
Consider a consumer whose preference can be represented by a Cobb-Douglas utility function.
Assume that the consumer has 80 dollars.
Suppose that the consumer’s optimal bundle includes of 4 units of good 1 and 6 units of good 3.
a) What is the optimal level of good 2 consumption?
b) Write a Cobb-Douglas utility function consistent with the optimal choice above.
c) Suppose that the price of good 3 increases from $4 to $8. Calculate the new optimal consumption level of good
3.
Question 5 (a)
Consider a market with three goods: good 1, good 2, and good 3.
The prices of goods are as follows:
Price of good 1: $2
Price of good 3: $4
Price of good 2: $8
Consider a consumer whose preference can be represented by a Cobb-Douglas utility function.
Assume that the consumer has 80 dollars.
Suppose that the consumer’s optimal bundle includes of 4 units of good 1 and 6 units of good 3.
What is the optimal level of good 2 consumption?
4 units of good 1 and 6 units of good 3 cost $8 + $24 = $32.
The consumer spends the remaining $48 on good 2: good 2 consumption = $48/$8 = 6.
Question 5 (b)
Consider a market with three goods: good 1, good 2, and good 3.
The prices of goods are as follows:
Price of good 1: $2
Price of good 3: $4
Price of good 2: $8
Consider a consumer whose preference can be represented by a Cobb-Douglas utility function.
Assume that the consumer has 80 dollars.
Suppose that the consumer’s optimal bundle includes of 4 units of good 1 and 6 units of good 3.
Write a Cobb-Douglas utility function consistent with the optimal choice above.
Spending share of good 1: $8/$80 = 0.1
Spending share of good 2: $48/$80 = 0.6
Spending share of good 3: $24/$80 = 0.3
Utility function: 0.1 ln 𝑥1 + 0.6 ln 𝑥2 + 0.3 ln 𝑥3
Question 5 (c)
Consider a market with three goods: good 1, good 2, and good 3.
The prices of goods are as follows:
Price of good 1: $2
Price of good 3: $4
Price of good 2: $8
Consider a consumer whose preference can be represented by a Cobb-Douglas utility function.
Assume that the consumer has 80 dollars.
Suppose that the consumer’s optimal bundle includes of 4 units of good 1 and 6 units of good 3.
Suppose that the price of good 3 increases from $4 to $8. Calculate the new optimal consumption level of good
3.
Spending share of good 3 is 0.3.
It means that the consumer spends $24 on good 3.
Thus, the new consumption level is $24/$8 = 3 units.