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ECC2000/ECC5900 Intermediate Microeconomics
创建日期：2024-06-12 15:53:13
所属分类：经济Economics
标签： ECC2000
Intermediate Microeconomics

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ECC2000/ECC5900 Intermediate Microeconomics Workshop

Problem Set: Class 2
Exercise 2.1 [4 pts] Jacob is on a road trip in a desert. He and his car both need to drink: water costs $2 per gallon, and
gasoline costs $4 per gallon. Jacob brought $80 for that trip. We assume that consumption of negative quantities of
water and gasoline is impossible.
(i) [1 pt] Assigning water to the horizontal axis and gasoline to the vertical, draw Jacob’s budget set. Write down the
equation of budget constraint, and label its intercepts and kinks.
↪→ Solution. It is easy to see that the budget set will be a triangle limited by the x-axis, y-axis, and the line of the
budget constraint. The latter can be written as 2x1 + 4x2 = 80, hence x2 = − 12x1 + 20. The budget constraint
crosses the x and y axes at (40,0) and (0,20) respectively.
water (x1)
gasoline (x2)
x2 = − 12x1 + 20
(0, 20)
(40, 0)
0
10
20
30
10 20 30 40 50
(ii) [1 pt] Suppose the local government in the desert is the only gasoline seller and it wants to generate revenue from
taxation. They impose $1 tax on each gallon of gasoline sold. Draw Jacob’s budget set, write down the equation of
budget constraint, and label its intercepts and kinks.
↪→ Solution. The tax effectively raises the price of gasoline to $5. Since the government is imposing the tax and it
is the only seller, we should expect perfect pass-through. Therefore, the budget constraint becomes 2x1+5x2 = 80,
or alternatively x2 = − 25x1 + 16.
water (x1)
gasoline (x2)
x2 = − 25x1 + 16
(0, 16)
(40, 0)
0
10
20
30
10 20 30 40 50
Filip Premik ECC2000/ECC5900 Intermediate Microeconomics Workshop
(iii) [1 pt] Suppose the tax raised strong concerns among the desert citizens, who claimed taxing goods necessary for
survival is immoral. During the negotiations, the government and desert citizens agreed that it is safe to assume that
10 gallons of gasoline should be enough to survive within a certain period. Therefore, the government imposes a tax
of $1 per gallon on every unit above 10 gallons of gasoline sold. Draw Jacob’s budget set, and write down in detail
the equation of budget constraint, and label its intercepts and kinks.
↪→ Solution. Nothing changes for the first 10 units of gasoline, so the budget constraint remains x2 = − 12x1 + 20
for any x2 ∈ [0, 10]. However, for x2 ≥ 10, it follows that the budget constraint will be described as x2 = − 25x2+b,
where b is some intercept. We can find that b = 18 by noticing that (20, 10) should belong to the budget constraint.
Therefore the budget constraint can be written as:
x2 =
{ − 12x1 + 20 if x2 ∈ [0, 10] (or equivalently x1 ∈ [20, 40] )− 25x1 + 18 if x2 ∈ [10, 20] (or equivalently x1 ∈ [0, 20] )
water (x1)
gasoline (x2)
(0, 18)
(40, 0)
(20, 10)
0
10
20
30
10 20 30 40 50
(iv) Suppose now elections are coming, so the ruling party removes any taxes. Unfortunately, the company responsible
for the water supply made a typo in their spreadsheet, which led to ordering only 30 gallons of water. Draw Jacob’s
budget set, and write down in detail the equation of budget constraint, and label its intercepts and kinks.
↪→ Solution. In this case, the budget constraint is as in the original problem for any x1 ∈ [0, 30]. Since there is no
more than 30 gallons of water, Jacob cannot consume more than 30 even if he lowers his consumption of gasoline.
The budget constraint is:
x2 = −1
2
x1 + 20 if x1 ∈ [0, 30]
water (x1)
gasoline (x2)
(0, 20)
0
10
20
30
10 20 30 40 50
Filip Premik ECC2000/ECC5900 Intermediate Microeconomics Workshop
Exercise 2.2 [4 pts] Suppose a consumer consumes positive amounts of goods x1 and x2, and her preferences are represented
by utility function U(x1, x2) = x1 · x2.
(i) [1pt] Consider bundle x = (1, 2). Define and draw the indifference curve at x and the set of bundles weakly
preferred to x.
↪→ Solution. The indifference curve at x is a set of all points that give the consumer the same utility. That is:
IC(x) = {y ∈ R2++ : u(x) = u(y)}
In turn, the set of bundles weakly preferred to x is:
W (x) = {y ∈ R2++ : u(x) ≤ u(y)}
To draw the IC(1,2), we write that u(1, 2) = 2, hence the indifference curve is given by x2 =
2
x1
. W (x) is the
upper contour set of the indifference curve.
x1
x2
0
1
1
(ii) [1pt] Define the Marginal Rate of Substitution, MRS12(x), and provide its interpretation. Calculate MRS12(x)
at x∗ = (1, 2). Suppose x∗ is the consumer’s optimal choice, maximizing her utility at a given set of prices and
income. Derive the ratio of prices. Do we have enough information to infer what is p1 and what is p2? If we know
w = 12, do we have enough information to infer p1 and p2? Justify in details each of your answers.
↪→ Solution. MRS12(x) tells us how many units of good 2 we need to provide to the customer to keep their utility
levels unchanged despite a marginal change in the quantity of good 1, given the initial bundle x. The MRS(x)
happens to be the ratio of derivatives:
MRS12(x) = −∂x2
∂x1
= − ∂u
∂x1
(x)
/ ∂u
∂x2
(x)
Since this is the case, we calculate that ∂u∂x1 (x) = x2 and
∂u
∂x2
(x) = x1, so MRS12(x) = −x2x1 and hence
MRS12(1, 2) = −2. Also, in the optimum, −MRS(x∗) = p1p2 , so
p1
p2
= 2: the good 1 must be twice as ex-
pensive as good 2. However, we cannot say anything about these prices in separation. To see this, consider
another set of prices p˜1 = α · p1 and p˜2 = α · p2 for some α > 0. It follows that p˜1p˜2 =
p1
p2
= 2 for any α > 0.
(p˜1, p˜2) can also be equilibrium prices. Note that knowing the income doesn’t give any additional information
needed to separate the prices.
(iii) [1pt] Propose another utility function that represents the consumer’s preferences and call it V (x). Formally prove
your answer or provide detailed explanations to obtain full credit.
↪→ Solution. Suppose V (x) = U(x)2. To see that V represents the same preferences as U we need to show that for
every x and y in the choice set, it follows that U(x) ≥ U(y)⇔ V (x) ≥ V (y). Let’s take x and y from the choice
set and without loss of generality assume u(x) ≥ u(y). This means that x1 ·x2 ≥ y1 · y2. Since we are considering
strictly positive bundles, it follows that (x1 ·x2)2 ≥ (y1 · y2)2, which is equivalent to V (x) ≥ V (y). The key aspect
of choosing V is that it has to be a monotone increasing transformation of U .
(iv) [1pt] Calculate MRS12(x) for at x
∗ = (1, 2) for the new utility function V suggested in the previous point. Is x∗ still
an optimal bundle given the prices from the previous point? Is it a coincidence of a rule? Prove formally or provide
a detailed intuition for the full mark.
↪→ Solution. Suppose V (x) = U(x)2. Then ∂V∂x1 (x) = 2x1x22 and ∂V∂x2 (x) = 2x21x2, so MRS12(x) = −x2x1 , and finally
MRS12(1, 2) = −2. This is in fact a rule. Consider a continuously differentiable function h : R++ → R++, such
that V (x) = h(U(x)). In our example, h(u) = u2. By using the differentiation rule for composition of functions,
we obtain that ∂V∂x1 (x) = h
′(U(x)) · ∂U∂x1 (x) and ∂V∂x2 (x) = h′(U(x)) · ∂U∂x2 (x). Hence the term h′(U(x)) cancels out
when we take the ratio of derivatives to calculate MRS.
Filip Premik ECC2000/ECC5900 Intermediate Microeconomics Workshop
Exercise 2.3 [4 pts] Consider two goods with prices p = (p1, p2) = (3, 1), and a consumer with income w = 20.
(i) [1pt] Suppose the goods are perfect complements. Write down the utility function. Draw the set of indifference
curves, remembering to mark the direction of the increase in utility. Solve for the optimal bundle/bundles.
↪→ Solution. The utility function for perfect complements in R2 can be written as:
U(x1, x2) = min{x1, x2}
And the indifference curves are:
x1
x2
U(x) = 2
U(x) = 1
0
1
1
Note that the maximal level of utility attainable at income w = 20 is 5. The only affordable bundle bringing the
customer 5 utils, and hence the optimal bundle, is (5,5).
(ii) [1pt] Suppose the goods are perfect substitutes. Write down the utility function. Draw the set of indifference curves,
remembering to mark the direction of the increase in utility. Solve for the optimal bundle/bundles.
↪→ Solution. The utility function for perfect substitutes in R2 can be written as:
U(x1, x2) = x1 + x2
And the indifference curves are:
x1
x2
U(x) = 2
U(x) = 3
0
1
1
Since goods are perfect substitutes and one is cheaper, only the cheaper good is consumed at the optimum. Since
w = 20 and p2 = 1, the optimal bundle is (0, 20).
(iii) [2pts] Suppose consumer’s preferences are represented by the utility function u(x1, x2) = x1 ·x2. Solve for the optimal
bundle. Hint: express x2 as a function of x1 and known objects using the budget constraint. Plug it back to the utility
so that now it depends only on x1. Does u(x1) belong to a class of functions that you are familiar with? What do we
know about the extrema of functions within this class?
↪→ Solution. We will use the hint. The budget constraint is 3x1+x2 = 20, hence x2 = 20− 3x1. Now, plug it back
to the utility to make it depend only on x1: u(x1) = x1 · (20−3x1). Note that U is quadratic in x1. Moreover, the
sign at the highest power (second) is negative, so we will have a global maximum. Specific to quadratic functions,
the optimum lies exactly in the middle of zero points. The zeros are obvious from the equation above, as u(x1) = 0
if and only if x1 = 0 or x1 =
20
3 . Hence, the optimum is x
∗
1 =
10
3 and x
∗
2 = 20− 3 103 = 10.
Filip Premik ECC2000/ECC5900 Intermediate Microeconomics Workshop
Exercise 2.4 [4 pts] Consider the following maximization problem:
max
(x1,x2)>0
x
1
3
1 x
2
3
2
such that x1 + 2x2 ≤ 5
of a consumer with the utility function U(x) = x
1
3
1 x
2
3
2 facing prices (p1, p2) = (1, 2) and enjoying the income w = 5. You
will solve it in a few steps.
(i) [1pt] Can we write x1 + 2x2 = 5 instead of x1 + 2x2 ≤ 5? Formally prove or explain in detail for full credit.
↪→ Solution. Yes, we can. The proposition is that the optimal bundle lies at the budget constraint. Suppose not.
Let x∗ = (x∗1, x
∗
2) be such that x
∗
1 + 2x
∗
2 < 5. Consider another bundle x˜ = (x˜1, x˜2) such that x˜1 = x
∗
1 + ϵ where
ϵ = 5−x∗1− 2x∗2 > 0, and x˜2 = x∗2. In other words, x˜ has a little bit more of good 1 and the same amount of good
2. Note also that, it affordable, as 1 · x˜1 + 2 · x˜2 = 5. Since U is increasing in both arguments, U(x˜) > U(x∗).
But this cannot happen, as x∗ is assumed to be the optimal bundle, i.e. yielding the highest utility from among
the affordable bundles. Hence, we conclude that the optimal bundle must lie at the budget constraint.
(ii) [1pt] Write down the Lagrangian function.
↪→ Solution.
L(x1, x2;λ) = x
1
3
1 x
2
3
2 − λ · (x1 + 2x2 − 5)
(iii) [1pt] Write down the first-order conditions.
↪→ Solution.
∂L
∂x1
(x1, x2;λ) =
1
3
(x2
x1
) 2
3 − λ = 0
∂L
∂x2
(x1, x2;λ) =
2
3
(x1
x2
) 1
3 − 2λ = 0
∂L
∂λ
(x1, x2;λ) = x1 + 2x2 − 5 = 0
(iv) [1pt] Solve for optimal (x1, x2).
↪→ Solution. Derive λ as a function of x1 and x2 in the first two equations:
λ =
1
3
(x2
x1
) 2
3
λ =
1
3
(x1
x2
) 1
3
Furthermore, left-hand sides of the two equations above are equal. Thus, since we consider only positive amounts
of goods, we infer that at the optimum x∗1 = x
∗
2. Plugging it into the budget constraint, we obtain that x
∗
1 = x
∗
2 =
5
3
Filip Premik ECC2000/ECC5900 Intermediate Microeconomics Workshop
Exercise 2.5 [4 pts] Preference relation ⪰ defined on R2+ is said to be satiated if there exists a bundle that is preferred to
any other in the choice set. This bundle is called a bliss point.
(i) [1pt] Draw a set of indifference curves for satiated preferences. Remember to mark the direction of an increase in
utility as well as the bliss point.
↪→ Solution. The graph below presents the indifference curves for satiated preferences. Point B = (2, 2) is the bliss
point.
x1
x2
B
0
1
1
Note that the maximal level of utility attainable at income w = 20 is 5. The only affordable bundle bringing the
customer 5 utils, and hence the optimal bundle, is (5,5).
(ii) [1pt] Is ⪰ monotone? Prove or give a counter-example.
↪→ Solution. Of course, this is not a monotone preference. In the example above, point (2,2) is strictly preferred
to (3,3). In general, satiated preferences defined over RN+ will not be monotone.
(iii) [2pts] Give an example of preference relation ⪰ that is both convex and satiated. Explain your reasoning. Hint: it
may be useful to work on indifference curves.
↪→ Solution. The preference relation as illustrated by the indifference curves above is convex. To show this, we
need to show that the weighted average of two bundles for which a consumer is indifferent is preferred over each
of these bundles. Note that the indifference curves are circles. Take x ∈ R2 and y ∈ R2 that are equally preferred.
Then x and y lie on the same circle. Let λ ∈ (0, 1) and consider bundle z = λx+(1−λ)y. Note that geometrically,
points z will lie on a chord connecting x and y, hence in the interior of a circle. Hence, z will be more preferred
than x and y. Since x, y, and λ were arbitrarily chosen, we conclude that the preference relation is convex.
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