Composit Goods: Food and Clothing
In these notes we show that x1 and x2 can be collapsed into a composite good, called food, y1 , and that x3 and
x4 can be collapsed into a composite good, called clothing, y2. A utility function for y1 and y2,
υ is upsilon.)
υ(y1, y2), exists
with the usual characteristics of utility functions. (
Below, we first examine the constrained utility maximization problem using a Cobb-Douglas utility function. Then
we use a constrained maximization problem to collapse the problem into 2 composite goods and generate
y2).
The original constrained utility maximization problem:
Max:
α
β
γ
δ
A. x 1 . x 2 . x 3 . x 4
U x 1, x 2,x 3, x 4
Subject to:
p 1 .x 1
p 2.x 2
p 3.x 3
p 4 .x 4
M 0
L x 1, x 2, x 3, x 4, λ , p 1, p 2, p 3, p 4, M
α
β
γ
δ
A . x 1 . x 2 . x 3 . x 4 ...
+ λ . M p 1 .x 1 p 2 . x 2
p 3.x 3
The necessary conditions:
d
dλ
L x 1, x 2, x 3, λ , p 1, p 2, p 3, M
M
p 1.x 1
p 2 .x 2
p 3 .x 3
p 4.x 4 0
d
L x 1, x 2, x 3, λ , p 1, p 2, p 3, M
dx 1
α α
β
γ
δ
A.x 1 . . x 2 .x 3 . x 4
x1
λ .p 1 0
d
L x 1, x 2, x 3, λ , p 1, p 2, p 3, M
dx 2
β β
γ
δ
α
A.x 1 .x 2 . .x 3 . x 4
x2
λ .p 2 0
d
L x 1, x 2, x 3, λ , p 1, p 2, p 3, M
dx 3
β
γ γ
δ
α
A.x 1 .x 2 . x 3 . . x 4
x3
λ .p 3 0
d
L x 1, x 2, x 3, x 4, λ , p 1, p 2, p 3, p 4, M
dx 4
α
β
γ
δ δ
A.x 1 .x 2 . x 3 . x 4 .
x4
λ .p 4
p 4 .x 4
υ(y1,
The demand curves derived from these 5 implicit functions are:
x' 1 p 1 , p 2 , p 3 , p 4 , M
M.
x' 3 p 1 , p 2 , p 3 , p 4 , M
M.
α
p 1 .( α
β
γ
δ)
x' 2 p 1 , p 2 , p 3 , p 4 , M
γ
δ)
x' 4 p 1 , p 2 , p 3 , p 4 , M
γ
p 3 .( α
β
M.
M.
β
p 2 .( α
p 4 .( α
Max:
α
β
γ
δ
A. x 1 . x 2 . x 3 . x 4
Subj to:
p 3 .x 3
p 4.x 4
y2 0
where y2, x1 and x2 are exogenous, in addition, to p3 and p4.
L x 3, x 4, Λ ,y 2, x 1,x 2, p 3,p 4
α
β
γ
δ
A.x 1 .x 2 . x 3 .x 4
Λ. y 2
p 3 .x 3
p 4.x 4
The necessary conditrions:
d
dΛ
L x 3,x 4, Λ , y 2,x 1, x 2,p 3, p 4
y2
p 3 .x 3
p 4 .x 4 0
d
L x 3, x 4, Λ ,y 2, x 1, x 2, p 3, p 4
dx 3
α
β
γ γ
δ
A.x 1 .x 2 . x 3 . . x 4
x3
Λ .p 3 0
d
L x 3, x 4, Λ ,y 2, x 1, x 2, p 3, p 4
dx 4
β
γ
δ δ
α
A.x 1 .x 2 . x 3 .x 4 .
x4
Λ .p 4 0
We desire to combine x3 and x4 into y2, expenditure on clothing. To do this we first solve for x3.
δ
x4
γ . δ
A. x 1 x 2 x 3
x
x3 4
Λ .p 3 0
Λ A. x 1 x 2 x 3 γ .
p 3 .x 3
α
β
γ
δ δ
A. x 1 . x 2 . x 3 . x 4 .
x4
Λ .p 4 0
Λ A. x 1α . x 2β . x 3γ . x 4δ .
α.
β.
γ.
α.
β.
γ.
γ
δ)
γ
δ)
δ
Consider the following problem, which is used to collapse the utility function.
U x 1, x 2,x 3, x 4
β
δ
p 4 .x 4
β
δ
α
β
γ
δ
A. x 1 . x 2 . x 3 . x 4 .
p 4 .x 4
δ
x4
α
β
γ
A.x 1 . x 2 . x 3 . γ .
p 3 .x 3
x4
x 3 γ .p 4 .
δ .p 3
Substitute this into the budget constraint and solve for x4.
p 3 . γ .p 4 .
x4
x4
p 4.x 4
δ .p 3
y2
p 4.( γ
δ)
.δ
y2 0
This gives us x4 as a function of the exogenous variables of this problem;
however, only y2 and p4 actually show up in the result.
Solve for x4 first then x 3.
α.
β.
γ.
δ.
A. x 1 x 2 x 3 x 4
δ
p 4 .x 4
α.
β.
γ.
δ
x4
A.x 1 x 2 x 3 γ .
p 3 .x 3
x3
.
.
x4 δ p3
γ .p 4
p 3 .x 3
x3
x3
.
.
.
p4 δ p3
γ .p 4
y2
p 3.( γ
δ)
y2 0
.γ
This gives us x3 as a function of the exogenous variables of this problem;
however, only y2 and p3 actually show up in the result.
Substitute these solution functions for x3 and x4 into the utility function to form V(x 1, x2, y2)
V x 1, x 2,y 2
α.
A.x 1 x 2
β.
γ
y2
p 3.( γ
δ)
.γ .
δ
y2
p 4.( γ
δ)
.δ
V x 1, x 2,y 2
γ
1
α
β
γ δ. 1
.
A.x 1 .x 2 . y 2
p3
(γ δ)
γ
δ
δ
.γγ . 1
p4
.δδ
We now combine x1 and x2 into y1.
Max:
U x 1, x 2,y 2
γ
1
α
β
γ δ. 1
.
A.x 1 .x 2 . y 2
p3
(γ δ)
γ
δ
.γγ . 1
p4
δ
.δδ
Subj to:
p 1 .x 1
p 2.x 2
y1 0
where y1, y2, the p's are exogenous.
L x 1, x 2, Λ ,y 1, y 2, p 1, p 2, p 3, p 4
d
dΛ
γ
L x 1,x 2, Λ , y 1,y 2, p 1,p 2, p 3,p 4
d
L x 1, x 2, Λ ,y 1, y 2, p 1, p 2, p 3, p 4
dx 1
d
L x 1, x 2, Λ ,y 1, y 2, p 1, p 2, p 3, p 4
dx 2
Solve for x1:
y1
p 1.x 1
1 .
1
Λ A. x 1α . α . x 2β . y 2( γ δ ) .
p3
(γ δ)
δ
. γγ .
δ
1 . δ
δ ...
p4
p 2.x 2
γ
1
(γ δ). 1
β
α α
.
A.x 1 . .x 2 .y 2
x1
p3
(γ δ)
+ Λ .p 1
(γ
δ)
δ
. γ γ . 1 . δδ ...
p4
γ
(γ δ)
δ
1
γ. 1
(γ δ). 1
β. β .
α.
.
.
.
. δδ ...
γ
Ax1 x2
y2
x2
p3
p4
(γ δ)
+ Λ .p 2
γ
(γ δ)
δ
1
(γ δ). 1
α α
β
.
. γγ . 1 . δδ ... 0
A. x 1 . . x 2 . y 2
x1
p3
p4
(γ δ)
+ Λ .p 1
γ
γ
1
α
β
γ δ. 1
.
A.x 1 . x 2 . y 2
p3
(γ δ)
+ Λ . y 1 p 1 .x 1 p 2 .x 2
(γ
δ)
δ
δ
. γγ . 1 . δ
p4
p 1.x 1
(γ δ)
γ
δ
1
γ. 1
(γ δ). 1
α.
β. β .
.
.
.
. δδ ... 0
γ
Ax1 x2
y2
x2
p3
p4
(γ δ)
+ Λ .p 2
γ
1
(γ δ). 1
α
β
.
Λ A. x 1 . x 2 . β . y 2
p3
(γ δ)
(γ
δ)
δ
δ
. γγ . 1 . δ
p 2.x 2
p4
Solve these for the x1 and x2 demand curves:
x 1 α .p 2.
p 1 . α .p 2.
x2
x 2 β .p 1.
x2
β .p 1
x2
p 2 .x 2
β .p 1
y1
p 2.( α
β)
p 1 .x 1
y1 0
.β
x1
α .p 2
p 2 . β .p 1 .
y1
x1
p 1.( α
β)
Substitute these demand curves into the utility function V(x1, x2, y2 )to form
υ y1,y 2
A.
α
y1
p 1.( α
β)
A.
υ y1,y 2
A. y 1
p 1.( α
α
β.
.
α
y1
υ y1,y 2
.α
β)
1
p1
.α
α
.
p 2.( α
.y β . 1
1 p
2
α
1
(α
β
y1
β)
β)
β
β
.
.β
(α
.α α .
1
p2
.α
υ(y1, y2).
γ
β)
β
y1 0
α .p 2
1
.y γ δ . 1 .
2
p3
(γ δ)
β
1
x1
γ
δ
. γγ . 1
p4
γ
γ
δ
γ
γ
δ
1
.β β .y γ δ . 1 .
2
p3
(γ δ)
1
.β β .y γ δ . 1 .
2
p3
(γ δ)
δ
.δδ
. γγ . 1
p4
. γγ . 1
p4
δ
δ
. δδ
.δδ
This is the desired result. We can analyze this composite good problem just like any other two good problem, just as
long as the relative prices of x1 and x2 and the relative prices of x3 and x4 remain constant.