Question

Jane has a utility function of the form: U=x y. The price for x is 2 , the price for y is 1 and her income is 40.

i). Derive the marginal utilities of the above function.

ii) Give an expression of the marginal rate of substitution between x and y

iii) use the Lagrange method to find the optimal choice of Jane.

iv) Find the value of & and give the economic interpretation of the Lagrange multiplier.

b) Explain what is monotonic function and give examples of such. show that an even power may change the sign of the monotonic function.

Answer 1

Given that:

$U=xy$

And the budget constraint: $I=P_xX+P_yY$

a)

i) $MU_x=U_x=y$

$MU_y=U_y=x$

ii)$MRS_{xy}=\frac{MU_x}{MU_y}=\frac{y}{x}$

iii) The budget constraint is given as

$40-2x-y$

Forming the Lagrange equation:

$U=xy+\lambda(40-2x-y)$

$U_x=y-2\lambda=0. .......(i)$

$U_y=x-\lambda=0..........(ii)$

$U_{\lambda}=40-2x-y=0......(iii)$

From (ii)

$x=\lambda$

Substituting $\lambda$ into i):

$y-2x=0\implies y=2x$

$2x+y=40$

$2x+2x=40$

$4x=40$

$x^*=10=y^*$

iv)$\lambda=10$

The Lagrange multiplier is used to explain the marginal impact of a small change in the constraint on the objective function. In the utility function above, a unit change in the constraint will cause the $U$ to increase(decrease) by approximately 10-units.

b) A monotonic function is a function that decreases or increases over it's entire domain.

Examples: if $f(x)= x^{-3}+2x^{-2}-4x+5$ at $x=3$

Then$f(3)=3^{-3}+2(3)^{-2}-4(3)+5=-16$

A positive power will increase the function over its domain: $f(3)=3^3+2(3)^2-4(3)+5=38$