Answer to Question #242936 in Microeconomics for mia

Question

Think about a consumer with a utility function given by U=x₁x₂, he is facing a budget constraint: p1x1+p2x2<= I.

(a)   Can you decide if the indifference curve is convex?

(b)  Please derive the Marshallian demand functions for x1 and x2.

(c)   How much of each good will the consumer purchase if I =8, p1=1 and p2=4? 

Answer 1

Solution:

a.). Yes, you can decide if the indifference curve is convex. This is because the shape of an indifference curve depicts a consumer’s willingness to substitute one product for another, which is measured as the MRS. As you consume more of a good, the marginal utility from each additional unit becomes lower due to the diminishing marginal utility. The MRS diminishes the further along the curve we move, thus the slope of an indifference curve is convex to the origin.

 

b.). Derive the Marshallian demand functions for x1 and x2:

Maximize Lagrangian function (ℒ):

ℒ(x1,x2,λ) = x1x2 + λ (I – p1x1 + p2x2)

Set the partial derivatives equal to zero:

$\frac{\partial \mathcal{L} } {\partial x1}$ = x2 – λp1 = 0

$\frac{\partial \mathcal{L} } {\partial x2}$ = x1 – λp2 = 0

$\frac{\partial \mathcal{L} } {\partial \lambda }$ = I – p1x1 – p2x2 = 0

Solve for x1 and x2:

x1= λp2

x2= λp1

I = p1x1 + p2x2

$\frac{x1}{x2} = \frac{\lambda p2}{\lambda p1} = \frac{p1}{p2}$

x1 = $\frac{x2p2}{p1}$

x2 = $\frac{p2}{x1p1}$

Substitute the values of x1 and x2 in budget constraint to derive the Marshallian demand functions for x1 and x2:

I = p1x1 + p2x2

I = $p1(\frac{x2p2}{p1}) + p2x2 = x2p2 + p2x2$

I = $x2(p2 + p2)$

x2 = $\frac{I}{2p2}$

I = p1x1 + p2x2

I = $p1x1 + p2(\frac{p2}{x1p1}) = p1x1 + x1p1$

I = $x1 (p1 + p1)$

x1 = $\frac{I}{2p1}$

c.). I = 8

p1 = 1

p2 = 4

Units of x1:

x1 = $\frac{I}{2p1} = \frac{8}{2\times 1} = \frac{8}{2} = 4 \;units$

Units of x2:

x2 = $\frac{I}{2p2} = \frac{8}{2\times 4} = \frac{8}{8} = 1 \;unit$