A consumer's total utility (TU) from consuming 3 units of a commodity is 29 utils. Consumi… — JAMB UTME

A consumer's total utility ( $TU$ ) from consuming $3$ units of a commodity is $29$ utils. Consuming the $4^{\text{th}}$ unit yields a marginal utility ( $MU$ ) of $15$ utils. If consuming the $5^{\text{th}}$ unit results in average utility ( $AU$ ) being equal to marginal utility ( $MU$ ), what is the total utility ( $TU$ ) derived from consuming $5$ units of the commodity?

$55$ utilsAnswer
B
$59$ utils
C
$44$ utils
D
$47$ utils

Answer

The total utility (

TU

) derived from consuming

5

units of the commodity is

55

utils.

The correct answer is derived by first finding total utility at 4 units (

TU_4 = 29 + 15 = 44

utils). Applying the condition that average utility equals marginal utility at 5 units gives

\frac{TU_5}{5} = TU_5 - 44

. Solving this linear equation yields

TU_5 = 55

utils.

Step-by-Step Solution

Calculate the Total Utility at 4 units (

TU_4

)

TU_4 = TU_3 + MU_4 = 29 + 15 = 44

utils

Total utility is the cumulative sum of marginal utilities up to that quantity.

Express Average Utility (

AU_5

) and Marginal Utility (

MU_5

) in terms of

TU_5

AU_5 = \frac{TU_5}{5}

and

MU_5 = TU_5 - TU_4 = TU_5 - 44

By definition,

AU = \frac{TU}{Q}

and

MU = \Delta TU

for a unit increase.

Set

AU_5 = MU_5

as specified by the condition and solve for

TU_5

\begin{aligned} \frac{TU_5}{5} &= TU_5 - 44 \\ TU_5 &= 5(TU_5 - 44) \\ TU_5 &= 5TU_5 - 220 \\ 4TU_5 &= 220 \\ TU_5 &= 55 \text{ utils} \end{aligned}

Equating the average utility and marginal utility expressions allows solving for the unknown total utility.

Key Concept

Mathematical Relationships between Total Utility (TU), Average Utility (AU), and Marginal Utility (MU)

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