NotesAtHKU

When taking an action $n$ , benefit is your gain, and cost is your loss by choosing that action. We ignore all costs that are not related to choosing the action.

If there is X amount of money taht one is willing to pay for an action, it goes towards the benefit of that action.

Economic surplus and opportunity cost

Economic surplus, or net benefit, for taking an action $n$ is given by:

ES(n) = TB(n) - TC(n),\quad T=\text{total}

Opportunity cost is the cost of choosing one action $n$ over another ( $n_2$ ):

OC(n) = C(n) + ES(n_2)

Sunk cost

The cost that has already been incurred and cannot be recovered.

Ignore sunk cost when making decisions, always look forward.

Decision by margin

Marginal benefit/cost ( $MB,MC$ ) means additional gain/loss from taking one more unit of action $n$ .

$MX(n) = TX(n) - TX(n-1),\quad X\in\{B,C\}$ , for integer $n$

If the unit of action is continous, the value $MX$ will be given, or use the derivative: $MX(n) = \frac{dTX}{dn}$ .

Net margin: $NM(n) = MB(n) - MC(n)$

Cost-benefit principle

A rational person:

Only take action when $ES \geq 0$
Only keep taking action when $NM \geq 0$
Best action is when $ES$ is maximized: $ES_1(n) \geq ES_2(n)$

We can make decisions about finding the optimal number of units to take between two or more actions by equalizing the net margins.

2023 Summer Midterm Q3

To allocate 29 hours of tasks between A & B to minimize cost, whose costs are:

Cost	Formula
avg. A	$AC(n) = 330 + 1.5n$
mar. A	$MC(n) = 330 + 3n$
avg. B	$AC(m) = 270 + 2.5m$
mar. B	$MC(m) = 270 + 5m$

We need to find $n$ and $m$ such that $n + m = 29$ and the marginal costs are equalized, i.e., $MC_A(n) = MC_B(m)$ . This is due to the cost-benefit principle: equalizing the net margins.

\begin{align*} MC_A(n) &= MC_B(m) \\ 330 + 3n &= 270 + 5m \\ 3n - 5m &= -60,\quad\text{substitute $n = 29 - m$} \\ 3(29 - m) - 5m &= -60 \\ m = 18.375&,\quad n = 10.625 \end{align*}

Example: Workers

To allocate 4 workers fix items in places A & B, where the number of items repaired:

n. workers	fixed A	fixed B
1	70	40
2	80	60
3	89	79
4	98	97

First convert the benefit table to marginal benefit table:

n. workers	fixed A	fixed B
1	70	40
2	10	20
3	9	19
4	9	18

Then, we simply decide where workers go by choosing highest marginal benefit:

$70\rightarrow 40\rightarrow 20\rightarrow 19\implies n_A = 1, n_B = 3$

Cost and Benefit

Economic surplus and opportunity cost

Sunk cost

Decision by margin

Cost-benefit principle