Question 11.4: The utility a consumer derives from consuming the two goods ......
The utility a consumer derives from consuming the two goods A and B can be assumed to be determined by the utility function U = 40 A^{0.25}B^{0.5} . If A costs £4 a unit and B costs £10 a unit and the consumer’s income is £600, what combination of A and B will maximize utility?
The marginal utility of A is
MU_A = \frac{\partial U}{\partial A} = 10 A^{-0.75}B^{0.5}
The marginal utility of B is
MU_B = \frac{\partial U}{\partial B} = 20 A^{0.25}B^{-0.5}
Consumer theory tells us that total utility will be maximized when the utility derived from the last pound spent on each good is equal to the utility derived from the last pound spent on any other good. This optimization rule can be expressed as
\frac{MU_A}{P_A} = \frac{MU_B}{P_B}
Therefore, substituting the above MU functions and the given prices of £4 and £10, this condition becomes
\frac{10 A^{-0.75}B^{0.5}}{4} = \frac{20A^{0.25}B^{-0.5}}{10}
100B = 80A
B = 0.8A (1)
Substituting (1) for B in the budget constraint
4A + 10B = 600
gives
A + 10(0.8A) = 600
4A + 8A = 600
12A = 600
A = 50
Thus from (1)
B = 0.8(50) = 40