Assume that the preferences can be represented by the utility function u = x1x2 and the budget constraint is given as p1x1 + p2x2 = m. Derive the Marshallian demand functions for x1 and x2.

Answer 1

Marshallian demand is essentially a particular type of demand function where demand for one good is given as a function of it's own price, the price of another good, and the income of the consumer. In order to derive it, we need to maximize the consumer's utility function subject to their budget constraint.

We are given:

utility function U = x1*x2

budget constraint: m = p1*x1 + p2*x2

We can solve this using Lagrangian optimization. We set up the Lagrangian as the utility function plus lambda times the budget constraint set equal to zero. So, the budget constriant becomes m - p1*x1 - p2*x2

= 0.

L = x1*x2 + 𝜆(m - p1*x1 - p2*x2)

Now, we take the partial derivatives of L with respect to x1, x2, and 𝜆. Set each of them equal to 0.

w.r.t. x1: x2 + 𝜆(-p1) = 0

w.r.t. x2: x1 + 𝜆(-p2) = 0

w.r.t. 𝜆: m - p1x1 - p2x2 = 0

Now, we can solve for 𝜆 in the first two equations and set them equal to each other.

𝜆 = x2/p1 and 𝜆 = x1/p2

So, x2/p1 = x1/p2

Solve for x1 or x2 with algebra: x2 = (x1p1)/p2

Now that we have x2 in terms of x1, we can plug it bag into the partial derivative with respect to 𝜆 from before. Then we solve the equation for x1, which gives us the demand.

m - p1x1 - p2(x1p1/p2) = 0

x1* = m/2p1

Plug this value of x1 back into the equation x2 = x1p1/p2, to get:

x2* = m/2p2

These are the respective Marshallian demand functions for x1 and x2.