Question

You like both apples and pears, and you consume nothing but apples and pears. The consumption bundle where you consume xA bushels of apples and xB bushels of pears is written as (xA; xB). You are indi§erent between (10; 0) and the set and the set of consumption bundles that satisÖes xB = 10 x .

(a) (2 POINTS) List 3 consumption bundles that are on the indi§erence curve for (10; 0).

(b) (2 POINTS) What is your marginal rate of substitution between apples and pears?

(c) (3 POINTS) Based on the preferences described above, are apples and pears substitutes, complements, or neither. Explain your reasoning.

(d) (3 POINTS) Now say that apples cost $1 and pears cost $2: Based on the preferences described above, could the consumption bundle of (10; 10) possibly maximize your utility. Explain your reasonin

Answer 1

Feel free to leave a comment below to clarify doubts;
I will update the answer body with whatever explanation you need.

Please leave an upvote if this helps!

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Qa)
The IC for the bundle (10, 0) can be rearranged and written as Xa + Xb = 10;
three consumption bundles that on this indifference are (9, 1), (8, 2) and (7, 3)
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Qb)
Marginal rate of substitution is the slope
of the straight line indifference curve Xa + Xb = 10

Xa + Xb = 10
=> Xb = (-1)Xa + 10
comparing with the generic equation for a straight line Y = MX + C
where M is the slope
M = -1
=> MRS = -1
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Qc) Xa + Xb = 10 is a straight line indifference curve with constant slope; this is sufficient to say that apples and pears are perfect substitutes;

say our original consumption bundle is (7,3) i.e. 7 apples and 3 pears. If you exchange one apple for one pear with a friend, your consumption bundle is (6, 4).
Xa + Xb = 6 + 4 = 10 so the new consumption bundle is also on the same IC.

So whether you start out with 10 apples - i.e. consumption bundle (10, 0)
- or you start out with 10 pears - consumption bundle (0, 10)

- as long as you trade 1 apple for 1 pear you're left off on the same indifference curve; this is what the MRS captures, that is why MRS = -1 for this problem, the magnitude of MRS (=1) represents the exchange rate and the negative sign implies that there's a tradeoff.
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Qd) With perfect substitutes, the rationale is that we spend our entire income on whichever is cheaper - to get the most value for money. (10, 10) is one optimal choice if pears and apples are perfect substitutes and their prices are equal.

1 apple and 1 pear each provide the same increment in utility (because perfect substitutes), so the problem becomes to maximize the number count of fruits (have as many of either apples of pears)

But for the question here prices are not equal, so let's decide whether the agent should give up apples or pears. The agent is willing to give up 1 apple for 1 pear, since it would keep their net utility intact; but since pears cost twice as much as apples ($2 vs $1), the agent can only get 1/2 a pear for an apple. So this trade is unfavorable for the agent given their preferences - it would be acceptable if pear was worth only half as much as an apple to the agent in utility terms, but that's not the case here where a pear and an apple are valued equally.

On the other hand, the agent is willing to give up 1 pear for 1 apple. But because apples are only half as expensive as pears, the agent can get 2 apples for 1 pear. This represents a trade that is more than acceptable to the agent since there were willing to accept just 1 apple for a pear;
they will keep trading units of pears accepting (twice as many) apples in return, moving along points on the budget line as (10, 10), (12, 9), (14, 8), (16, 7) . . . . each subsequent bundle brings more utility than the previous because the total count of juicy fruits in the bundle is more.

continuing by this logic, the agent should give up all 10 pears to get 20 apples - the optimal bundle is (30,0).

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