Price a shared network and account for its funding

One network, three price rules

A town is considering a fictional service network. A single provider can avoid duplicating pipes or cables, but that does not tell us what price it should charge. Your job is to advise a mock public hearing: compare use, cost recovery and who pays under three contracts. You do not need to contact an official or take a political position.

Try and predict

Read the technology and contract records in the source notebook. Predict whether setting the service price equal to the cost of one extra unit will also pay the fixed network cost. Save that prediction before opening the answer panels below.

Quantity Q means divisible service units supplied in one period, not people. Price P means credits per service unit. Demand is P=14−Q: the last unit’s willingness to pay falls as more service is supplied. An operating network’s total economic cost is TC=32+2Q, in credits per period. It pays 32 to make service possible, then 2 for each unit. These costs include opportunity costs and normal returns. They are invented teaching values.

Before committing to this period, supplying nothing avoids the fixed cost too and gives zero net benefit. Once a commitment has become unavoidable, a short-run shutdown comparison would differ. Keep that timing distinct from the earlier cost-table lesson.

Why one provider can cost less

Suppose the town wants 8 units. One network costs 32+2×8=48. Two separately built networks supplying 4 each cost 2×(32+2×4)=80. The extra 32 buys a duplicated fixed facility in this model, with no extra service benefit. More generally, splitting a positive total quantity across two operating networks costs 64+2Q instead of 32+2Q.

Average total cost, ATC, is total cost divided by output. For positive Q, ATC=32/Q+2. Spreading the same fixed cost over more units lowers ATC throughout this model’s range. ATC is undefined at zero; never plot it as zero. This cost advantage does not establish that every actual network industry has the same technology or that monopoly pricing is desirable.

Construct the three operating plans

Use a table with columns for contract, quantity, price, sales revenue, total economic cost and profit before public payment. Revenue is P×Q; economic profit is revenue minus TC. Compare each plan with not operating before committing resources.

Contract M: unregulated single-price profit choice. The operator chooses one price for all service. The supplied marginal-revenue rule is MR=14−2Q: selling more requires lowering the price on earlier units too. Marginal cost is MC=2. Set MR=MC to find quantity, then read its price from demand. Do not read price from the MR curve.

Contract A: specified break-even service. The operator must serve demand at a price covering total economic cost. Choose the lower-price, higher-output intersection of demand and ATC. There can be two intersections: a graph or equation must not silently discard one.

Contract C: marginal-cost service with committed funding. Price is 2 and the operator must serve all demand at that price. A separate lump-sum payment covers the verified cost gap. The contract, service obligation and funding are all needed: a low posted price alone does not compel an unsubsidized firm to incur a loss.

Check quantities, prices and operating ledgers

For M, 14−2Q=2 gives Q=6. Demand then gives P=8. Revenue is 48, cost is 44 and profit is 4. As an algebraic check, profit for positive output simplifies to 4−(Q−6)², so 6 is the best operating quantity; its profit also exceeds the zero from not committing to operate.

For A, P=ATC requires 14−Q=32/Q+2. Multiply by positive Q and rearrange to (Q−4)(Q−8)=0. Q=4 at P=10 and Q=8 at P=6 both break even. Our contract specifies Q=8, the greater service quantity. It has revenue 48 and cost 48. A bare instruction to “set price equal to average cost” is incomplete without the intended branch and service obligation.

For C, P=MC=2 gives Q=12. Revenue is 24; cost is 32+24=56. Before the payment the operator loses 32. The stipulated payment of 32 makes its final economic profit zero. Zero economic profit already includes the normal return embedded in economic cost; it does not mean the provider’s resources receive no compensation.

Read and reconstruct the graph

Demand, marginal revenue and average cost are separate relationships. The horizontal marginal-cost line is at 2. Mark M at (6,8), A at (8,6), and C at (12,2) on demand. Mark the other break-even point (4,10) separately. Each ordered pair gives (quantity, price). Reconstruct these points from the ledger before looking at the diagram.

On narrow screens, focus the graph region and scroll horizontally. The formulas and labeled points above are its text equivalent.

The average-cost curve is shown only from Q=3 onward to keep the displayed segment within the vertical range. At very small positive quantities it rises above that range; it does not begin at zero. MR is drawn only where nonnegative. Solid demand, dashed MR, dotted ATC and the labeled horizontal MC line distinguish curves without color. The numerical points and formulas above provide the same evidence without inspecting the graph. These are assumed continuous relationships, unlike the discrete workshop table.

Count benefits without counting a transfer twice

Consumer surplus is willingness to pay above what users actually pay. In this linear example it is a triangle: half × quantity × (14−price), in credits per period. It is not revenue available to pay the network’s bills. Use the following ledger to check your calculations, then explain each column in your own words.

On narrow screens, focus the table region and use arrow keys to scroll.

For C, the operator’s final profit is −32+32=0 and taxpayers pay 32. Net benefit is therefore 72+0−32=40. Adding the subsidy to user surplus without subtracting the payment invents a benefit. Alternatively, add user surplus directly to profit before payment: 72−32=40. These two accounting routes agree.

An independent resource check uses the area under demand: total willingness to pay is 14Q−Q²/2. Subtract the actual resource cost 32+2Q, not revenue, to obtain 12Q−Q²/2−32 for positive operation. It gives 22, 32 and 40 at the three quantities. Sales and subsidies redistribute claims; they do not manufacture service benefits or erase construction costs.

Under these assumptions, units up to 12 are worth at least their marginal cost. Contract C maximizes net resource benefit, including comparison with not operating. Its extra benefit over M is 18; over A it is 8. These are model comparisons, not measured policy effects. Raising public funds could impose additional costs; benefits, costs and tax burdens could fall on different people. Neither effect is modeled here.

Repair a proposal and test its boundary

A fictional official says: “Choose C: users gain 72, the firm breaks even, and the subsidy adds another 32 of social benefit.” Write a correction before opening the review. Your recommendation may support C, A or another feasible plan, but it must reconcile the ledger and state the objective you are using.

Review the proposal's evidence and reasoning

The 72 is consumer surplus, not total net benefit. The 32 is a transfer paid by taxpayers and received by the operator. With zero final operator profit, net resource benefit is 72−32=40. A supported recommendation for C states that funding is committed and that the model omits financing distortions, distributional weights and enforcement costs. A supported case for A could instead emphasize the constraint of no public payment, while acknowledging that it yields 8 less modeled net benefit. Calling A efficient merely because its budget balances confuses cost recovery with the marginal-benefit comparison.

Now increase fixed cost from 32 to 80, leaving demand and marginal cost unchanged. Predict whether any single uniform price could cover total cost, and whether an unchanged marginal-cost output still justifies construction.

Check the high-fixed-cost boundary

At Q=12, cost is 104 and revenue 24; the gap is 80. Total willingness to pay is 96, so net resource benefit is −8. Not constructing gives zero and is better under this model. Equating P with MC identifies the best positive operating quantity, not whether paying the fixed cost is worthwhile.

Before fixed cost, the largest contribution to cost recovery under a single uniform price is 36−(Q−6)², at most 36. An 80-credit fixed cost cannot be recovered from any such price. The break-even equation has no real roots. A larger subsidy could pay the bills but would not make a negative resource benefit positive. This conclusion does not assess unmodeled health, access or other external benefits; those would need additional evidence rather than a relabeled transfer.

Apply and continue

Save a three-contract ledger, the repaired proposal and your high-cost conclusion in the source notebook. Cite the technology and contract IDs, and label the 80-credit scenario as your modified teaching assumption. Record whether you used the worked review. The panels do not save a first attempt or certify independent understanding; the separate course application retains its own evidence rules.

For introductory college depth, investigate an attributed real regulatory case. Identify the service-quality obligation, the cost evidence, the pricing rule and the party bearing demand risk. Explain why reimbursement of every reported cost may weaken incentives to control costs, while a fixed price cap can shift risks and quality incentives. Do not claim that this toy ledger estimates those effects.

Return to the production and cost investigation if the explicit/implicit-cost distinction is unfamiliar. Continue to the tax and public-goods investigation to compare a different policy mechanism. The source notebook links to the OpenStax methodological reading; all schedules and calculations on this page are original teaching examples.

Compare a unit charge with a fixed charge

Keep this lesson's original fictional demand P=14−Q and resource cost 32+2Q. Now compare separate levies: t credits per service unit, or F credits per operating period. They are not simultaneously imposed. Demand, technology and single-price monopoly behavior stay fixed. No tax administration cost, externality or public-spending benefit is modeled.

Before commitment, closing avoids both the 32 resource commitment and any fixed levy. After commitment, both remain due even if service stops. The unit levy is paid only on supplied units. These are stated contract assumptions, not claims about actual tax rules.

Predict: which levy changes marginal cost and the best operating quantity? Could a fixed levy change whether the firm operates even if it leaves the operating quantity unchanged?

Initial quantity and operation prediction Unit levy t (credits/unit)00.5124 Fixed levy F (credits/period)048 Decision timingBefore commitment: closure avoids fixed paymentsAfter commitment: fixed payments remain due

Marginal revenue MR=14−2Q. Unit levy raises private marginal cost from 2 to 2+t, so the best positive-output candidate is Q=(12−t)/2. Fixed levy leaves marginal cost 2 and candidate Q=6. Read price from demand, not MR. Average private cost for positive Q is (32+F)/Q+2+t: a fixed levy changes average cost even though it does not change marginal cost.

No levy

Best operating candidate: Q=6, P=8; revenue 48 − resource cost 44 − levy 0 = profit 4. Closure profit is 0. Profit-maximizing choices: operate.

Candidate consumer surplus 18 + seller profit 4 + government receipts 0 = total resource surplus 22. Closure total resource surplus is 0, receipts 0 and no transaction price. Do not count a levy both as a resource cost and as government income.

Per-unit levy

Best operating candidate: Q=5.5, P=8.5; revenue 46.75 − resource cost 43 − levy 5.5 = profit -1.75. Closure profit is 0. Profit-maximizing choices: close.

Candidate consumer surplus 15.125 + seller profit -1.75 + government receipts 5.5 = total resource surplus 18.875. Closure total resource surplus is 0, receipts 0 and no transaction price. Do not count a levy both as a resource cost and as government income.

Fixed levy

Best operating candidate: Q=6, P=8; revenue 48 − resource cost 44 − levy 4 = profit 0. Closure profit is 0. Profit-maximizing choices: operate or close (tie; no unique operation outcome).

Candidate consumer surplus 18 + seller profit 0 + government receipts 4 = total resource surplus 22. Closure total resource surplus is 0, receipts 0 and no transaction price. Do not count a levy both as a resource cost and as government income.

These operating quantities are conditional candidates. When closure is strictly preferred, do not report candidate quantity or candidate receipts as the selected result. When both choices tie, report both; their service and resource outcomes can differ. Existing commitments affect the closure ledger, not demand or the marginal production technology.

Changed case and interpretation

Transfer: fixed levy 8 before commitment

Compare fixed levies 4 and 8 before commitment, then switch timing. Explain why “fixed charges never change output” needs a condition about continued operation. Compare unit levy 0.5 with 1, retaining the actual closure decision. Are these comparisons revenue-neutral? Use the receipt ledgers rather than assuming equal numeric rates collect the same amount.

Revised timing, transfer and receipt explanation

Reasoning guide

Without levies, Q=6/P=8/profit=4 beats closure before or after commitment. At unit levy 1, the best operating candidate Q=5.5/P=8.5 has profit −1.75: before commitment close at 0; after commitment operate rather than close at −32. Fixed levy 4 leaves Q=6/P=8/profit=0: before commitment operating and closure tie; after commitment operation beats −36. Fixed levy 8 gives candidate profit −4: before commitment close; after commitment operate rather than lose −40. Thus fixed charges can affect service through participation even though the conditional interior quantity stays 6.

The unit levy reduces quantity and transfers revenue under these assumptions, but the fixed levy can also eliminate participation. This is not a recommendation or measured effect. Resource surplus adds consumers, the seller and government without weighting distribution, financing or other real-world effects. Antitrust and wage policy require separate institutional evidence.

Method source and model scope

OpenStax Microeconomics 3e section 7.3 supplies context for fixed, variable, average and marginal cost. All charge scenarios and diagrams are original extensions of this lesson's fictional network; no source data or prose is copied. This is a single monopoly comparison, not competitive-industry incidence or actual tax guidance.

Copy the firm-charge report

Read-only JSON report{ "activity": "MIC-05.a firm charge comparison", "fictional": true, "unit": 1, "fixed": 4, "timing": "before", "comparisons": [ { "id": "baseline", "t": 0, "F": 0, "marginalPrivate": 2, "candidate": { "q": 6, "price": 8, "revenue": 48, "resource": 44, "receipts": 0, "profit": 4, "consumer": 18, "benefit": 66, "total": 22 }, "closed": { "q": 0, "price": null, "revenue": 0, "resource": 0, "receipts": 0, "profit": 0, "consumer": 0, "benefit": 0, "total": 0 }, "choices": [ "operate" ] }, { "id": "unit", "t": 1, "F": 0, "marginalPrivate": 3, "candidate": { "q": 5.5, "price": 8.5, "revenue": 46.75, "resource": 43, "receipts": 5.5, "profit": -1.75, "consumer": 15.125, "benefit": 61.875, "total": 18.875 }, "closed": { "q": 0, "price": null, "revenue": 0, "resource": 0, "receipts": 0, "profit": 0, "consumer": 0, "benefit": 0, "total": 0 }, "choices": [ "close" ] }, { "id": "fixed", "t": 0, "F": 4, "marginalPrivate": 2, "candidate": { "q": 6, "price": 8, "revenue": 48, "resource": 44, "receipts": 4, "profit": 0, "consumer": 18, "benefit": 66, "total": 22 }, "closed": { "q": 0, "price": null, "revenue": 0, "resource": 0, "receipts": 0, "profit": 0, "consumer": 0, "benefit": 0, "total": 0 }, "choices": [ "operate", "close" ] } ], "worksheet": { "prediction": "", "revision": "" }, "feedbackSeen": false, "limits": "Original fictional monopoly, unchanged demand and technology. Levies are transfers, no collection costs or benefit assigned to government spending. Fixed resource cost and fixed levy are avoidable before commitment, unavoidable after it. Conditional operating candidates are not automatically selected outcomes. Temporary guided evidence, not policy effectiveness or independent mastery." }

Temporary writing: 3,000 characters per field. Reset restores defaults and blank writing, retaining known guide/report exposure this mount; reload clears it. Guided calculations do not establish independent mastery.

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