Current Situation:

Price and Quantity:

Price $($PPP$)=$ $$5$Quantity demanded $($QQQ$)=95$ individuals annually

Demand Curve:

The demand curve is P$=100$QP $=100-$ QP$=100$Q$,$ or rearranged, Q$=100$PQ $=100-$ PQ$=100$P$.$

Total Revenue $($TR$)$:

TR$=$P$$Q$=595=475$TR $=$ P $\backslash$times Q $=5\backslash$times $95=475$TR$=$P$$Q$=595=475$

Cost of Maintaining the Pool:

Total cost $=$ $$1000$

Current Loss:

Loss$=$Total$$Cost$$Total$$Revenue$=1000475=525\backslash$text$\{$Loss$\}=\backslash$text$\{$Total Cost$\}-\backslash$text$\{$Total Revenue$\}=1000-475=525$Loss$=$Total$$Cost$$Total$$Revenue$=1000475=525$

Chad is correct that the pool is losing $$525$ annually in monetary terms. However, this does not consider consumer surplus $($CS$),$ which accounts for the benefit pool users receive beyond the price they pay.

Consumer Surplus $($CS$)$ at P$=5$P $=5$P$=5$:

Consumer surplus is the area under the demand curve and above the price line:

CS$=12$Base$$HeightCS $=\backslash$frac$\{1\}\{2\}\backslash$times $\backslash$text$\{$Base$\}\backslash$times $\backslash$text$\{$Height$\}$CS$=21$Base$$Height

Base $=$ Quantity $=959595$Height $=1005=95100-5=951005=95$

CS$=129595=4512\mathrm{.}5$CS $=\backslash$frac$\{1\}\{2\}\backslash$times $95\backslash$times $95=4512\mathrm{.}5$CS$=219595=4512\mathrm{.}5$

Net Benefit at P$=5$P $=5$P$=5$:

Net$$Benefit$=$CS$$Total$$Cost$\backslash$text$\{$Net Benefit$\}=$ CS $-\backslash$text$\{$Total Cost$\}$Net$$Benefit$=$CS$$Total$$Cost Net$$Benefit$=4512\mathrm{.}51000=3512\mathrm{.}5\backslash$text$\{$Net Benefit$\}=4512\mathrm{.}5-1000=3512\mathrm{.}5$Net$$Benefit$=4512\mathrm{.}51000=3512\mathrm{.}5$

Conclusion: Chad's statement that the pool does not provide a positive net benefit is incorrect because the consumer surplus significantly outweighs the monetary loss.

Scenario $1$: Increasing Price to P$=12$P $=12$P$=12$:

New Quantity Demanded:

From the demand curve, Q$=100$PQ $=100-$ PQ$=100$P:

Q$=10012=88$Q $=100-12=88$Q$=10012=88$

New Consumer Surplus:

Base $=888888$Height $=10012=88100-12=8810012=88$

CS$=128888=3872$CS $=\backslash$frac$\{1\}\{2\}\backslash$times $88\backslash$times $88=3872$CS$=218888=3872$

New Total Revenue $($TR$)$:

TR$=$P$$Q$=1288=1056$TR $=$ P $\backslash$times Q $=12\backslash$times $88=1056$TR$=$P$$Q$=1288=1056$

Net Benefit at P$=12$P $=12$P$=12$:

Net$$Benefit$=$CS$$Total$$Cost$\backslash$text$\{$Net Benefit$\}=$ CS $-\backslash$text$\{$Total Cost$\}$Net$$Benefit$=$CS$$Total$$Cost Net$$Benefit$=38721000=2872\backslash$text$\{$Net Benefit$\}=3872-1000=2872$Net$$Benefit$=38721000=2872$

Scenario $2$: Tax the City Instead:

Revenue Requirement:

Total revenue required $=$ $$1000.$

Marginal Excess Tax Burden $($METB$)$:

For every $$1$ raised in taxes, there is an additional loss of $$0\mathrm{.}25$ in societal surplus.Total METB loss: METB$$Loss$=10000\mathrm{.}25=250\backslash$text$\{$METB Loss$\}=1000\backslash$times $0\mathrm{.}25=250$METB$$Loss$=10000\mathrm{.}25=250$

Net Benefit of Taxation at P$=5$P $=5$P$=5$:

Consumer surplus at P$=5$P $=5$P$=5=4512\mathrm{.}54512\mathrm{.}54512\mathrm{.}5$Net benefit: Net$$Benefit$=$CS$$Total$$Cost$$METB$$Loss$\backslash$text$\{$Net Benefit$\}=$ CS $-\backslash$text$\{$Total Cost$\}-\backslash$text$\{$METB Loss$\}$Net$$Benefit$=$CS$$Total$$Cost$$METB$$Loss Net$$Benefit$=4512\mathrm{.}51000250=3262\mathrm{.}5\backslash$text$\{$Net Benefit$\}=4512\mathrm{.}5-1000-250=3262\mathrm{.}5$Net$$Benefit$=4512\mathrm{.}51000250=3262\mathrm{.}5$

Summary of Net Benefits:

Current Policy $($P$=5$P $=5$P$=5)$: Net Benefit $=3512\mathrm{.}5$

Increased Price $($P$=12$P $=12$P$=12)$: Net Benefit $=2872$

Taxation $($P$=5$P $=5$P$=5)$: Net Benefit $=3262\mathrm{.}5$

Recommendation:

The current pricing policy $($P$=5$P $=5$P$=5)$ maximizes societal net benefits, with a net benefit of $$3512\mathrm{.}5.$ Taxing the city $($P$=5$P $=5$P$=5$ with a tax subsidy$)$ is a second$-$best alternative, as it produces a smaller net benefit due to the marginal excess tax burden. Increasing the price to P$=12$P $=12$P$=12$ reduces societal net benefits further due to a reduction in consumer surplus. Therefore, the city should retain the current pricing policy.