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?

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

Cevap

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.

Adım Adım Çözüm

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.

Anahtar Kavram

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

Soru

Cevap

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