A Summary on Microeconomics
Let 𝐶 be a consumer and let 𝒙 and 𝒚 be bundles of 𝑙 products where the prices of the products are
given by 𝒑. Now let 𝐶 have a budget 𝑚 then we set up the budget constraint as following:
𝑝1 𝑥1 + ⋯ + 𝑝𝑙 𝑥𝑙 = 𝑚 where 𝒙, 𝒚 ∈ ℝ𝑙 +
Definition Preference Relation
Consider the same variables given above. The Consumer preference relation is given by:
𝒙 ≻ 𝒚 Means that 𝐶 strictly prefers 𝒙 to 𝒚
𝒙 ∼ 𝒚 Means that 𝐶 is indifferent between 𝒙 and 𝒚
𝒙 ≿ 𝒚 Means that 𝐶 weakly prefers 𝒙 to 𝒚
There are five axioms in Consumer Theory:
• (C ) Completeness: “ Consumers can always compare “
For any 𝒙, 𝒚 ∈ ℝ𝑙 + we have that 𝒙 ≿ 𝒚 or 𝒚 ≿ 𝒙
• (R ) Reflexivity: “Any bundle is at least as good as itself”
For any 𝒙 ∈ ℝ𝑙 + we have 𝒙 ≿ 𝒙
• (T ) Transitivity: “Ordering is consistent”
For any 𝒙, 𝒚, 𝒛 ∈ ℝ𝑙 + if 𝒙 ≿ 𝒚 and 𝒚 ≿ 𝒛 ,then 𝒙 ≿ 𝒛
• (M ) Monotonicity: “More is Better”
∀𝒙, 𝒚 ∈ ℝ𝑙 + if 𝑥𝑗 ≥ 𝑦𝑗 for all 𝑗 = 1, … , 𝑙 and 𝒙 ≠ 𝒚 then 𝒙 ≻ 𝒚
• (CONV ) Convexity: “Averages are preferred to extremes”
∀𝒙, 𝒚 ∈ ℝ𝑙 + if 𝒙 ∼ 𝒚 then for any 𝜆 ∈ [0,1] ∶ 𝜆𝒙 + (1 − 𝜆)𝒚 ≿ 𝒙
Definition Strict Convexity
∀𝒙, 𝒚 ∈ ℝ𝑙 + if 𝒙 ∼ 𝒚 and 𝒙 ≠ 𝒚 then for any 𝜆 ∈ [0,1] ∶ 𝜆𝒙 + (1 − 𝜆)𝒚 ≻ 𝒙
Note: The indifference curves with flat spots aren’t strict convex indifference curves
Definition Indifference Curves
The theory on consumer choice can be formulated in terms of preferences that satisfy the axioms
above. These preferences can be described graphically by using indifference curves.
Indifference curves show the preferences in consumption bundles.
Formula Indifference curve 𝑈(𝒙) = 𝑐 Here 𝒙 is a bundle of goods an 𝑐 = 𝑢𝑡𝑖𝑙𝑖𝑡𝑦
An important characteristic of indifference curves representing distinct levels of preference is that
they can’t cross.
Proof
Let 𝑋, 𝑌, 𝑍 be bundles of goods such that 𝑋 and 𝑍 lie only on one indifference curve and 𝑌 lies on the
intersection of the curves. So 𝑋~𝑌 and 𝑍~𝑌 ,therefore, by the axiom of Transitivity we have that
𝑋~𝑍. However 𝑋 and 𝑍 have a strictly preference relation as they don’t lie on the same curve, which
contradicts 𝑋~𝑍. So the curves can’t intersect.
,Indifferent curves have the following properties:
• Because of (M):
o For each 𝑥1 there can be at most one 𝑥2 such that (𝑥1 , 𝑥2 ) ∼ 𝒙∗
o Indifference curves must be downward sloping
o Indifference curves cannot intersect
• Because of (CONV)
o Indifference curves must be curved towards the origin, hence the set of bundles on
or above an indifference curve is convex. {𝒙 ∈ 𝑅+2 : 𝒙 ≿ 𝒙∗ }
Definition Perfect Substitutes
If the consumer is willing to substitute one good for the other at a constant rate, the goods are called
perfect substitutes. That means the consumer doesn’t care about the proportions of the products
and only cares that he gets his total amount.
Definition Perfect Complements
Products that are always consumed together in fixed proportions are called perfect complements
(shoes for example).
Definition Well-behaved preference relation
A preference relation ≿ is called well-behaved if it satisfies:
• The five axioms
• All indifference curves of ≿ are smooth. This allows us to talk about the slope.
A utility function that is differentiable and quasi-concave is called well-behaved
,Definition Marginal Rate of Substitution (MRS)
The marginal rate of substitution is the slope of an indifference curve and measures the rate at
which the consumer is willing to substitute good 𝑗 for good 𝑖 (if 𝑀𝑅𝑆𝑖𝑗 (𝒙)). Therefore the MRS also
measures the marginal willingness to pay.
∂𝑈
∂𝑥𝑖 ∂𝑥
𝑗
In general between goods 𝑖 and 𝑗 we have: 𝑀𝑅𝑆𝑖𝑗 (𝒙) = − ∂𝑈 (𝒙) (= − ∂𝑥 (𝒙))
𝑖
∂𝑥𝑗
𝜕𝑈
Here ∂𝑥 = 𝐷𝑗 𝑈(𝒙) which is the partial derivative of the utility function with resp to the 𝑗-th product
𝑗
Let 𝐶 be a consumer and let 𝒙 (blue) and 𝒚 (orange) be bundles of 𝑙 products where the prices of the
products are given by 𝒑. Now let 𝐶 have a budget 𝑚 then we set up the budget constraint as
following:
𝑝1 𝑥1 + ⋯ + 𝑝𝑙 𝑥𝑙 = 𝑚 where 𝒙, 𝒚 ∈ ℝ𝑙 +
So 𝐶 has the budget set 𝐵(𝒑, 𝑚), where the bundle on
the highest possible indifference curve is the best. Let 𝒙∗
denote the optimal bundle for 𝐶. Also assume that the
preferences are well-behaved (follow the five axioms) and
strictly convex.
𝑚
𝑝2 −𝑝2
The slope of the budget line is: 𝑚 = 𝑝1
𝑝1
𝑓(∆𝑥1 +𝒙∗ )−𝑓(𝒙∗ ) 𝑓(∆𝑥1 +𝒙∗ )−𝑓(𝒙∗ )
Slope of indifference curve (function of 𝑥1 ) = − lim = − lim
∆𝑥1 →0 (∆𝑥1 +𝒙∗ )−𝒙∗ ∆𝑥1 →0 (∆𝑥1 )
Here 𝑓(𝒙∗ ) = 𝑥2 is the indifference curve and 𝑓(𝒙∗ ) = 𝑥2 ⇔ 𝑥2 (𝑥1 ) = 𝑥2
Definition Utility Function
Let ≿ be a preference relation on ℝ𝑙 +. A function 𝑢 ∶ ℝ𝑙 + → 𝑅 is a utility function that represents
≿, if: ∀𝒙, 𝒚 ∈ ℝ𝑙 + , 𝒙 ≿ 𝒚 if and only if 𝑢(𝒙) ≥ 𝑢(𝒚)
So to derive which bundle is preferred, we have to calculate which bundle has the highest utility.
Utility can be seen as a number that describes the level of satisfaction.
Theorem
If the consumer preference relation ≿ on ℝ𝑙 + satisfies the axioms: Completeness, Reflexivity,
Transitivity and Monotonicity (all except Convexity), then there exists a utility function 𝑢 ∶ ℝ𝑙 + → 𝑅
that represents ≿
Proof
Let 𝑙 = 2 and ≿ be monotonic. Then for any 𝒙 ∈ ℝ2 + we have that its indifference curve must be
monotonic so it cuts the diagonal only once. Let (𝑡, 𝑡) ∈ ℝ2 be the intersection point with the
diagonal and define 𝑢(𝒙) = 𝑡. Now if 𝒚 is on a higher indifference curve we have that 𝒚 ≻ 𝒙 and also
that the intersection point of the indifference curve of 𝒚 is higher so 𝑢(𝒚) > 𝑢(𝒙)
,Theorem
Let 𝑓 be a strictly increasing function on ℝ. If 𝑢 is a utility function representing ≿ then also 𝑓 ∘ 𝑢 is a
utility function representing ≿ (𝑓 ∘ 𝑢)(𝒙) = 𝑓(𝑢(𝒙))
Note before the following theorem: Recall the axiom of Convexity
Theorem
𝑢 represents convex preferences if and only 𝑢 is quasi-concave.
Definition Quasi-concave
∀𝒙, 𝒚 ∈ ℝ𝑙 + if 𝑢(𝒙) = 𝑢(𝒚) then for any 𝜆 ∈ [0,1] ∶ 𝑢( 𝜆𝒙 + (1 − 𝜆)𝒚) ≥ 𝑀𝑖𝑛{ 𝑢(𝒙), 𝑢(𝒚) }
A function being quasi-concave solely means that it doesn’t have to be exactly concave so
Theorem Finding the optimal bundle
Let 𝑢(. ) be a well behaved strictly quasi-concave utility function. Let 𝐶’s optimal interior bundle be
denoted by 𝒙∗ (𝑥𝑖∗ , 𝑥𝑗∗ > 0) then we have:
∂𝑢 ∗
(𝒙 ) 𝑝
∂𝑥 𝑖
𝑀𝑅𝑆 𝑖𝑗 = 𝑖 =
∂𝑢 ∗ 𝑝𝑗
(𝒙 )
∂𝑥𝑗
Proof
1
Assume that 𝒙∗ is interior (𝑥1∗ , 𝑥2∗ > 0) by monoticity, 𝒑 ∙ 𝒙∗ = 𝑚 ⇔ 𝑥2 = 𝑝 (𝑚 − 𝑝1 𝑥1 )
2
1
Therefore 𝑢 (𝑥1 , 𝑝 (𝑚 − 𝑝1 𝑥1 )) and the goal is to maximize the utility. Now because 𝑢 is
2
1
one-dimensional we have that we can solve Max𝑥1 𝑈(𝑥1 ) = Max𝑥1 𝑢 (𝑥1 , 𝑝 (𝑚 − 𝑝1 𝑥1 ))
2
∂𝑢 ∗
∂𝑢 𝑝 ∂𝑢 (𝒙 ) 𝑝
1 ∂𝑥 1
𝑈′(𝑥1∗ ) = (𝒙∗ ) − (𝒙∗ ) = 0 ⇔ 1 =
∂𝑥1 𝑝2 ∂𝑥2 ∂𝑢 ∗ 𝑝2
(𝒙 )
∂𝑥2
, Theorem Engel’s Law
As the income of the consumer increases, the proportion of income spent on necessary goods
decreases and luxury goods increases
The picture on the left displays the
indifference curves for different
budgets. The ICC (Income
Indifference Curve) is the curve
through all the different optimal
bundles that correspond to a
certain budget. (Engel curve)
Figure 1 Engel curve of luxury good
Definition Normal and Inferior good
𝛿𝑥1 (𝑝,𝑚)
If good 1 is a normal good, it means that the good gets consumed more as income rises. 𝛿𝑚
>0
𝛿𝑥1 (𝑝,𝑚)
If good 1 is an inferior good, it means that the good gest consumed less as income rises. 𝛿𝑚
<0
The pictures on the right
display the ICC and Engel
curve for an inferior good.
Definition Demand Function
The demand function for good 1 is given by: 𝑝 → 𝑥1 (𝒑, 𝑚) which is a function 𝑅+ →𝑅+ which tells us
how 𝐶’ s demand for a good changes if the price of the good changes.
𝛼 𝛼 𝛼1 𝛼
An example is the Cobb-Douglas utility function 𝑢(𝒙) = 𝑥1 1 𝑥2 2 Then 𝐷1 𝑢(𝑥) = 𝑥1
and 𝐷2 𝑢(𝑥) = 𝑥 2
2
∂𝑢 ∗ 𝛼1
(𝒙 ) 𝑝 𝑝1 𝛼1 𝑥2 𝑝1 𝛼2
∂𝑥1 1 𝑥
= ⇔ 𝛼1 = ⇔ = ⇔ 𝑥1 𝛼2 𝑝1 = 𝛼1 𝑥2 𝑝2 ⇔ 𝑥2 𝑝2 = 𝑝1 𝑥1
∂𝑢 ∗
(𝒙 ) 𝑝2 𝑝2 𝑥1 𝛼2 𝑝2 𝛼1
2
∂𝑥2 𝑥2
𝛼2 𝛼1 𝛼2 𝑚 𝛼1
𝑝1 𝑥1 + 𝑝2 𝑥2 = 𝑚 ⇔ 𝑝1 𝑥1 + 𝑝1 𝑥1 = 𝑚 ⇔ 𝑝1 𝑥1 ( + ) = 𝑚 ⇔ 𝑥1 =
𝛼1 𝛼1 𝛼1 𝑝1 𝛼1 + 𝛼2
𝛼1 𝑚 𝛼 𝑚
So 𝒙(𝒑, 𝑚) = (𝛼 , 2 )
hence (by making 𝑝2 numeraire) the demand function for good 1:
1 +𝛼2 𝑝1 𝛼1 +𝛼2 𝑝2
𝛼1 𝑚 𝛼1 𝑚
𝒑→ ⇔ 𝑥1 =
𝛼1 + 𝛼2 𝑝 𝛼1 + 𝛼2 𝑝
Let 𝐶 be a consumer and let 𝒙 and 𝒚 be bundles of 𝑙 products where the prices of the products are
given by 𝒑. Now let 𝐶 have a budget 𝑚 then we set up the budget constraint as following:
𝑝1 𝑥1 + ⋯ + 𝑝𝑙 𝑥𝑙 = 𝑚 where 𝒙, 𝒚 ∈ ℝ𝑙 +
Definition Preference Relation
Consider the same variables given above. The Consumer preference relation is given by:
𝒙 ≻ 𝒚 Means that 𝐶 strictly prefers 𝒙 to 𝒚
𝒙 ∼ 𝒚 Means that 𝐶 is indifferent between 𝒙 and 𝒚
𝒙 ≿ 𝒚 Means that 𝐶 weakly prefers 𝒙 to 𝒚
There are five axioms in Consumer Theory:
• (C ) Completeness: “ Consumers can always compare “
For any 𝒙, 𝒚 ∈ ℝ𝑙 + we have that 𝒙 ≿ 𝒚 or 𝒚 ≿ 𝒙
• (R ) Reflexivity: “Any bundle is at least as good as itself”
For any 𝒙 ∈ ℝ𝑙 + we have 𝒙 ≿ 𝒙
• (T ) Transitivity: “Ordering is consistent”
For any 𝒙, 𝒚, 𝒛 ∈ ℝ𝑙 + if 𝒙 ≿ 𝒚 and 𝒚 ≿ 𝒛 ,then 𝒙 ≿ 𝒛
• (M ) Monotonicity: “More is Better”
∀𝒙, 𝒚 ∈ ℝ𝑙 + if 𝑥𝑗 ≥ 𝑦𝑗 for all 𝑗 = 1, … , 𝑙 and 𝒙 ≠ 𝒚 then 𝒙 ≻ 𝒚
• (CONV ) Convexity: “Averages are preferred to extremes”
∀𝒙, 𝒚 ∈ ℝ𝑙 + if 𝒙 ∼ 𝒚 then for any 𝜆 ∈ [0,1] ∶ 𝜆𝒙 + (1 − 𝜆)𝒚 ≿ 𝒙
Definition Strict Convexity
∀𝒙, 𝒚 ∈ ℝ𝑙 + if 𝒙 ∼ 𝒚 and 𝒙 ≠ 𝒚 then for any 𝜆 ∈ [0,1] ∶ 𝜆𝒙 + (1 − 𝜆)𝒚 ≻ 𝒙
Note: The indifference curves with flat spots aren’t strict convex indifference curves
Definition Indifference Curves
The theory on consumer choice can be formulated in terms of preferences that satisfy the axioms
above. These preferences can be described graphically by using indifference curves.
Indifference curves show the preferences in consumption bundles.
Formula Indifference curve 𝑈(𝒙) = 𝑐 Here 𝒙 is a bundle of goods an 𝑐 = 𝑢𝑡𝑖𝑙𝑖𝑡𝑦
An important characteristic of indifference curves representing distinct levels of preference is that
they can’t cross.
Proof
Let 𝑋, 𝑌, 𝑍 be bundles of goods such that 𝑋 and 𝑍 lie only on one indifference curve and 𝑌 lies on the
intersection of the curves. So 𝑋~𝑌 and 𝑍~𝑌 ,therefore, by the axiom of Transitivity we have that
𝑋~𝑍. However 𝑋 and 𝑍 have a strictly preference relation as they don’t lie on the same curve, which
contradicts 𝑋~𝑍. So the curves can’t intersect.
,Indifferent curves have the following properties:
• Because of (M):
o For each 𝑥1 there can be at most one 𝑥2 such that (𝑥1 , 𝑥2 ) ∼ 𝒙∗
o Indifference curves must be downward sloping
o Indifference curves cannot intersect
• Because of (CONV)
o Indifference curves must be curved towards the origin, hence the set of bundles on
or above an indifference curve is convex. {𝒙 ∈ 𝑅+2 : 𝒙 ≿ 𝒙∗ }
Definition Perfect Substitutes
If the consumer is willing to substitute one good for the other at a constant rate, the goods are called
perfect substitutes. That means the consumer doesn’t care about the proportions of the products
and only cares that he gets his total amount.
Definition Perfect Complements
Products that are always consumed together in fixed proportions are called perfect complements
(shoes for example).
Definition Well-behaved preference relation
A preference relation ≿ is called well-behaved if it satisfies:
• The five axioms
• All indifference curves of ≿ are smooth. This allows us to talk about the slope.
A utility function that is differentiable and quasi-concave is called well-behaved
,Definition Marginal Rate of Substitution (MRS)
The marginal rate of substitution is the slope of an indifference curve and measures the rate at
which the consumer is willing to substitute good 𝑗 for good 𝑖 (if 𝑀𝑅𝑆𝑖𝑗 (𝒙)). Therefore the MRS also
measures the marginal willingness to pay.
∂𝑈
∂𝑥𝑖 ∂𝑥
𝑗
In general between goods 𝑖 and 𝑗 we have: 𝑀𝑅𝑆𝑖𝑗 (𝒙) = − ∂𝑈 (𝒙) (= − ∂𝑥 (𝒙))
𝑖
∂𝑥𝑗
𝜕𝑈
Here ∂𝑥 = 𝐷𝑗 𝑈(𝒙) which is the partial derivative of the utility function with resp to the 𝑗-th product
𝑗
Let 𝐶 be a consumer and let 𝒙 (blue) and 𝒚 (orange) be bundles of 𝑙 products where the prices of the
products are given by 𝒑. Now let 𝐶 have a budget 𝑚 then we set up the budget constraint as
following:
𝑝1 𝑥1 + ⋯ + 𝑝𝑙 𝑥𝑙 = 𝑚 where 𝒙, 𝒚 ∈ ℝ𝑙 +
So 𝐶 has the budget set 𝐵(𝒑, 𝑚), where the bundle on
the highest possible indifference curve is the best. Let 𝒙∗
denote the optimal bundle for 𝐶. Also assume that the
preferences are well-behaved (follow the five axioms) and
strictly convex.
𝑚
𝑝2 −𝑝2
The slope of the budget line is: 𝑚 = 𝑝1
𝑝1
𝑓(∆𝑥1 +𝒙∗ )−𝑓(𝒙∗ ) 𝑓(∆𝑥1 +𝒙∗ )−𝑓(𝒙∗ )
Slope of indifference curve (function of 𝑥1 ) = − lim = − lim
∆𝑥1 →0 (∆𝑥1 +𝒙∗ )−𝒙∗ ∆𝑥1 →0 (∆𝑥1 )
Here 𝑓(𝒙∗ ) = 𝑥2 is the indifference curve and 𝑓(𝒙∗ ) = 𝑥2 ⇔ 𝑥2 (𝑥1 ) = 𝑥2
Definition Utility Function
Let ≿ be a preference relation on ℝ𝑙 +. A function 𝑢 ∶ ℝ𝑙 + → 𝑅 is a utility function that represents
≿, if: ∀𝒙, 𝒚 ∈ ℝ𝑙 + , 𝒙 ≿ 𝒚 if and only if 𝑢(𝒙) ≥ 𝑢(𝒚)
So to derive which bundle is preferred, we have to calculate which bundle has the highest utility.
Utility can be seen as a number that describes the level of satisfaction.
Theorem
If the consumer preference relation ≿ on ℝ𝑙 + satisfies the axioms: Completeness, Reflexivity,
Transitivity and Monotonicity (all except Convexity), then there exists a utility function 𝑢 ∶ ℝ𝑙 + → 𝑅
that represents ≿
Proof
Let 𝑙 = 2 and ≿ be monotonic. Then for any 𝒙 ∈ ℝ2 + we have that its indifference curve must be
monotonic so it cuts the diagonal only once. Let (𝑡, 𝑡) ∈ ℝ2 be the intersection point with the
diagonal and define 𝑢(𝒙) = 𝑡. Now if 𝒚 is on a higher indifference curve we have that 𝒚 ≻ 𝒙 and also
that the intersection point of the indifference curve of 𝒚 is higher so 𝑢(𝒚) > 𝑢(𝒙)
,Theorem
Let 𝑓 be a strictly increasing function on ℝ. If 𝑢 is a utility function representing ≿ then also 𝑓 ∘ 𝑢 is a
utility function representing ≿ (𝑓 ∘ 𝑢)(𝒙) = 𝑓(𝑢(𝒙))
Note before the following theorem: Recall the axiom of Convexity
Theorem
𝑢 represents convex preferences if and only 𝑢 is quasi-concave.
Definition Quasi-concave
∀𝒙, 𝒚 ∈ ℝ𝑙 + if 𝑢(𝒙) = 𝑢(𝒚) then for any 𝜆 ∈ [0,1] ∶ 𝑢( 𝜆𝒙 + (1 − 𝜆)𝒚) ≥ 𝑀𝑖𝑛{ 𝑢(𝒙), 𝑢(𝒚) }
A function being quasi-concave solely means that it doesn’t have to be exactly concave so
Theorem Finding the optimal bundle
Let 𝑢(. ) be a well behaved strictly quasi-concave utility function. Let 𝐶’s optimal interior bundle be
denoted by 𝒙∗ (𝑥𝑖∗ , 𝑥𝑗∗ > 0) then we have:
∂𝑢 ∗
(𝒙 ) 𝑝
∂𝑥 𝑖
𝑀𝑅𝑆 𝑖𝑗 = 𝑖 =
∂𝑢 ∗ 𝑝𝑗
(𝒙 )
∂𝑥𝑗
Proof
1
Assume that 𝒙∗ is interior (𝑥1∗ , 𝑥2∗ > 0) by monoticity, 𝒑 ∙ 𝒙∗ = 𝑚 ⇔ 𝑥2 = 𝑝 (𝑚 − 𝑝1 𝑥1 )
2
1
Therefore 𝑢 (𝑥1 , 𝑝 (𝑚 − 𝑝1 𝑥1 )) and the goal is to maximize the utility. Now because 𝑢 is
2
1
one-dimensional we have that we can solve Max𝑥1 𝑈(𝑥1 ) = Max𝑥1 𝑢 (𝑥1 , 𝑝 (𝑚 − 𝑝1 𝑥1 ))
2
∂𝑢 ∗
∂𝑢 𝑝 ∂𝑢 (𝒙 ) 𝑝
1 ∂𝑥 1
𝑈′(𝑥1∗ ) = (𝒙∗ ) − (𝒙∗ ) = 0 ⇔ 1 =
∂𝑥1 𝑝2 ∂𝑥2 ∂𝑢 ∗ 𝑝2
(𝒙 )
∂𝑥2
, Theorem Engel’s Law
As the income of the consumer increases, the proportion of income spent on necessary goods
decreases and luxury goods increases
The picture on the left displays the
indifference curves for different
budgets. The ICC (Income
Indifference Curve) is the curve
through all the different optimal
bundles that correspond to a
certain budget. (Engel curve)
Figure 1 Engel curve of luxury good
Definition Normal and Inferior good
𝛿𝑥1 (𝑝,𝑚)
If good 1 is a normal good, it means that the good gets consumed more as income rises. 𝛿𝑚
>0
𝛿𝑥1 (𝑝,𝑚)
If good 1 is an inferior good, it means that the good gest consumed less as income rises. 𝛿𝑚
<0
The pictures on the right
display the ICC and Engel
curve for an inferior good.
Definition Demand Function
The demand function for good 1 is given by: 𝑝 → 𝑥1 (𝒑, 𝑚) which is a function 𝑅+ →𝑅+ which tells us
how 𝐶’ s demand for a good changes if the price of the good changes.
𝛼 𝛼 𝛼1 𝛼
An example is the Cobb-Douglas utility function 𝑢(𝒙) = 𝑥1 1 𝑥2 2 Then 𝐷1 𝑢(𝑥) = 𝑥1
and 𝐷2 𝑢(𝑥) = 𝑥 2
2
∂𝑢 ∗ 𝛼1
(𝒙 ) 𝑝 𝑝1 𝛼1 𝑥2 𝑝1 𝛼2
∂𝑥1 1 𝑥
= ⇔ 𝛼1 = ⇔ = ⇔ 𝑥1 𝛼2 𝑝1 = 𝛼1 𝑥2 𝑝2 ⇔ 𝑥2 𝑝2 = 𝑝1 𝑥1
∂𝑢 ∗
(𝒙 ) 𝑝2 𝑝2 𝑥1 𝛼2 𝑝2 𝛼1
2
∂𝑥2 𝑥2
𝛼2 𝛼1 𝛼2 𝑚 𝛼1
𝑝1 𝑥1 + 𝑝2 𝑥2 = 𝑚 ⇔ 𝑝1 𝑥1 + 𝑝1 𝑥1 = 𝑚 ⇔ 𝑝1 𝑥1 ( + ) = 𝑚 ⇔ 𝑥1 =
𝛼1 𝛼1 𝛼1 𝑝1 𝛼1 + 𝛼2
𝛼1 𝑚 𝛼 𝑚
So 𝒙(𝒑, 𝑚) = (𝛼 , 2 )
hence (by making 𝑝2 numeraire) the demand function for good 1:
1 +𝛼2 𝑝1 𝛼1 +𝛼2 𝑝2
𝛼1 𝑚 𝛼1 𝑚
𝒑→ ⇔ 𝑥1 =
𝛼1 + 𝛼2 𝑝 𝛼1 + 𝛼2 𝑝
