Question

Problem 5. Let U(x) = xp.. At M-10 and P,-P,-2, what choice will maximize the utility of the consumer?

Answer 1

Utility function - U(x) = x1x2

As total expenditure is equal to income of the consumer, 2x1 + 2x2 = 10 and this will act as a constraint.

We will use Lagrangean approach that transforms a constrained optimization problem to an unconstrained optimization problem. Now we will have to maximize the following function :

L = U(x) + $\lambda$ ( 10 - 2x1 -2x2)

L = x1x2 + $\lambda$ (10 - 2x1 - 2x2)

We will partially differentiate L with respect to x1, x2 and $\lambda$ and setting the derivatives equal to zero. The resulting first order conditions will be -

$\frac{\partial L}{\partial x1}$ = x2 - 2$\lambda$

x2 - 2$\lambda$ = 0

x2 = 2$\lambda$ - Equation 1

$\frac{\partial L}{\partial x2}$ = x1 - 2$\lambda$

x1 - 2$\lambda$ = 0

x1 = 2$\lambda$- Equation 2

$\frac{\partial L}{\partial \lambda }$ = 10 - 2x1 - 2x2

10 - 2x1 - 2x2 = 0

10 = 2x1 + 2x2 - Equation 3

Solving equation 1 and equation 2 we get the following relationship - x1 = x2

Putting this relationship in equation 3 we get -  

10 = 2x1 + 2x1

10 = 4x1

x1 = 2.5

And since x1 = x2, x2 = 2.5

x1 = x2 = 2.5 is the combination that will maximize the utility of the consumer.