Question

A small organization is responsible for distributing two craft products made in a small community, these products are jam and bread. The marginal utility function for the jam is f(x) = 55–4x, and the marginal utility for the package of bread is given by g(x) = 37–x.
Find:
a) The total utility function of the jam.
b) The total utility function of bread.
c) If the consumer wants to buy 2 jars of jam and 3 packages of bread, which of the items will produce more utility.

Answer 1

3 packages of bread would produce more utility than 2 jars of jam

The total utility function of the jam

The marginal utility function of jam is given as:

f(x) = 55 - 4x

Integrate the above function to determine the total utility function

f'(x) = 55x - 2x^2

Hence, the total utility function of the jam is f'(x) = 55x - 2x^2

The total utility function of bread

The marginal utility function of bread is given as:

g(x) = 37 - x

Integrate the above function to determine the total utility function

g'(x) = 37x - 0.5x^2

Hence, the total utility function of the bread is g'(x) = 37x - 0.5x^2

The items that produce more utility

We have:

Jam = 2 jars

Bread = 3 packages

Substitute 2 for x in f'(x) = 55x - 2x^2

f'(2) = 55 * 2 - 2 * 2^2

f'(2) = 102

Substitute 3 for x in g'(x) = 37x - 0.5x^2

g'(3) = 37 * 3 - 0.5 * 3^2

g'(3) = 106.5

106.2 is greater than 102

Hence, 3 packages of bread would produce more utility

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