Answer to Question #195623 in Calculus for Emmanuella

Question

An individual’s utility function is given by

where is the amount of leisure measured in hours per week and is income earned measured in cedis per week.

Determine the value of the marginal utilities, when = 138 and = 500. Hence estimate the change in utility if the individual works for an extra hour, which increases earned income by GH¢15 per week. Does the law of diminishing utility hold for this function?

Answer 1

An individual’s utility function is given by 

$U=1000x_1+450x_2+5x_1x_2-2x_1^2-x_2^2$

where $x_1$ is the amount of leisure measured in hours per week,

$x_2$ is income earned measured in cedis per week.

$x_1=138,x_2=500$

The value of the marginal utilities:

Marginal Utility = Change In Total Utility / Change In Units

Marginal Utility =$\frac{1000x_1+450x_2+5x_1x_2-2x_1^2-x_2^2}{x_1x_2}=$

$=\frac{1000\cdot138+450\cdot500+5\cdot138\cdot500-2\cdot138^2-500^2}{138\cdot500}=6.09$

The change in utility if the individual works for an extra hour, which increases earned income by GH¢15 per week:

change in utility= $=1000+450\cdot15+5\cdot(139\cdot515-138\cdot500)-2\cdot(139^2-138^2)-$

$-(515^2-500^2)=4896$

Diminishing Marginal Utility is where the consumer values each additional unit less and less the more they consume.

In our case, the law of diminishing utility does not hold for this function since marginal utility is positive value.