Economics 11: Problem set 3 SUMMER 2026 UCLA

Economics 11: Problem set 3
Maurizio Mazzocco


Due: 10/23/2025 by 9:30 am (PST)




Problem 1
Suppose that an individual with income I cares about two goods, X and Y . The prices of
the two goods are pX and pY . The individual has the utility function

2 3
U (X, Y ) = X 5 Y 5 .

(a) Derive the Marshallian demands for the two goods.

(b) Find the indirect utility function.


Solution
(a) To derive the Marshallian demands, we use the Lagrangian method. The budget
constraint is
pX X + pY Y = I,

which we can rewrite as
I − pX X − pY Y = 0.

We set up the Lagrangian function:

2 3
L(X, Y, λ) = X 5 Y 5 + λ(I − pX X − pY Y ).

Now, take the partial derivatives with respect to X, Y , and λ, and set them equal to
zero:

∂L 2 3 3
= X − 5 Y 5 − λpX = 0,
∂X 5

1

, ∂L 3 2 2
= X 5 Y − 5 − λpY = 0,
∂Y 5
∂L
= I − pX X − pY Y = 0.
∂λ
We solve the system of equations. From the first two equations, we can solve for λ:
3 3 2 2
2 X−5 Y 5 3 X 5 Y −5
λ= = .
5 pX 5 pY

Equating the two expressions for λ:
3 3 2 2
2 X−5 Y 5 3 X 5 Y −5
= ,
5 pX 5 pY

we simplify this to:
2Y pX
= .
3X pY
Solving for Y , we get:
3pX
Y = X.
2pY

Substitute this expression for Y into the budget constraint:
 
3pX
pX X + pY X = I,
2pY

which simplifies to:
3
pX X + pX X = I.
2
Simplifying further:
5
pX X = I,
2
and solving for X:
2I
X∗ = .
5pX

Substitute X ∗ into the expression for Y :

3pX ∗ 3pX 2I 3I
Y∗ = X = × = .
2pY 2pY 5pX 5pY




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