Back to the on-screen lesson ·
Derive a price-setting outcome and identify best responses in a supplied strategic interaction.
Paper packet. Every task here also exists on screen, where it is checked automatically; answers written on paper are not assessed by Nydus. When you are back at a device, enter your answers there.
You will derive a price-setting outcome and identify best responses in a supplied strategic interaction, showing the calculation and stating the assumptions that make the conclusion valid.
Use the supplied definitions and units. Separate an accounting identity, a behavioral assumption, and a normative criterion before drawing conclusions.
| Term | What it means |
|---|---|
| Marginal revenue | Additional revenue from a small output increase, including any price effect on existing sales. |
| Best response | A payoff-maximizing strategy given the other players' specified strategies. |
| Strict dominance | A strategy yields a higher own payoff for every possible strategy of the other player. |
| Pure Nash equilibrium | A deterministic strategy pair with no profitable unilateral deviation. |
| Mixed strategy | A probability distribution over a player's pure actions. |
A price-taking firm treats the price of its output as fixed when choosing how much to sell. A price-setting firm instead faces a demand relationship: selling more may require a lower price. This difference changes marginal revenue. It does not remove the need to consider costs or prove that the firm can charge any price it wishes. Buyers' alternatives and willingness to pay constrain the available revenue combinations.
Suppose inverse demand is P(q)=a-bq, with b positive, over the relevant nonnegative price region. Total revenue is price multiplied by quantity, so R(q)=aq-bq squared. Differentiating gives marginal revenue MR(q)=a-2bq. The derivative includes both the revenue from another unit and the reduction in receipts on units that would otherwise have sold at a higher uniform price.
With constant marginal cost c and no capacity limit, profit apart from a fixed term is (a-c)q-bq squared. An interior optimum satisfies MR=c, giving q=(a-c)/(2b). Because b is positive, this objective is strictly concave. If a exceeds c, the candidate is positive; substituting it into inverse demand gives price (a+c)/2. If a does not exceed c, the nonnegative output boundary must be considered instead.
For a thirty, b one and c six, output is twelve and price eighteen. Setting price equal to marginal cost would instead produce twenty-four units in the corresponding competitive benchmark. The difference comes from the revenue assumption, not from a change in production technology. A calculation should say whether the firm takes price as fixed or recognizes its effect on the price of its own sales.
Another way: A markup is not itself a complete welfare calculation
In this simple uniform-price monopoly model, the chosen price exceeds marginal cost when output is positive. Some units between monopoly and competitive output would have willingness to pay above marginal resource cost, so their absence creates a surplus loss under the usual partial-equilibrium assumptions. Transfers from buyers to the seller are different from this loss of mutually beneficial trades.
The seller's operating profit is (P-c)q when marginal cost is constant. A fixed cost must still be subtracted to obtain the corresponding total profit. A firm can charge above marginal cost and nevertheless earn a negative total profit if a sufficiently large fixed commitment is unavoidable. Entry, shutdown and long-run exit require comparisons with the relevant alternatives, just as they did in the earlier cost lesson.
An industry can contain scale economies, innovation incentives, quality differences or network effects absent from the simple model. These features may complicate the benchmark used for policy analysis. They do not make the derivation incorrect; they make its scope limited. A report should identify which benefits and costs the exercise includes before turning its surplus comparison into a recommendation about a particular institution.
The model also assumes one uniform price and no strategic response by another seller. Price discrimination, bargaining and product differentiation can change the revenue function. With several firms, each firm's best choice may depend on what others do. That dependence motivates a game: a representation of players, their available strategies and the payoffs resulting from each combination of choices.
Another way: Read a payoff table from each player's perspective
Consider two players, a row player and a column player. Each chooses strategy one or strategy two simultaneously. A cell in the resulting two-by-two table contains an ordered payoff pair: the first number belongs to the row player and the second to the column player. A payoff represents the player's ranking in the supplied game; it need not be money and should not be compared across people without additional justification.
A best response is a strategy that gives a player the highest payoff given the other player's strategy. To find the row player's response to column strategy one, compare the first numbers in the two cells of that column. To find the column player's response to row strategy one, compare the second numbers across that row. Mixing the two payoff coordinates is a common error that changes whose incentives are being analyzed.
For a concrete table, let the cells at row-one/column-one, row-one/column-two, row-two/column-one and row-two/column-two be respectively (4,4), (1,5), (5,1) and (2,2). If the column player chooses one, the row player gets four from one and five from two, so two is better. If the column player chooses two, the row player gets one from one and two from two, so two is again better.
The column player's comparisons give the same ranking by symmetry. Strategy two is strictly dominant for each player: it gives a strictly higher payoff regardless of the other's strategy. Dominance is stronger than being a best response to one particular choice. Many games have no dominant strategy, so do not assume the method of eliminating a dominated option will always finish the analysis.
Another way: Nash equilibrium requires mutually compatible best responses
A pure-strategy Nash equilibrium is a strategy pair at which neither player can improve by changing only their own strategy while the other's strategy is held fixed. The definition tests unilateral deviations. It does not ask whether both players could improve by coordinating a joint move, and it does not require that the players like the resulting outcome.
In the preceding table, the pair (2,2) is the unique pure-strategy Nash equilibrium. The row player would fall from payoff two to one by switching alone, and the column player would also fall from two to one. Yet the pair (1,1) gives each player four. Both would prefer that joint outcome, but either has an incentive to deviate from it when the other remains at strategy one.
This conflict illustrates why individually consistent incentives need not maximize the players' combined payoffs. The result depends on the payoffs and the simultaneous one-shot setting. Repetition, enforceable agreements, communication, observability or different objectives can change the game. A lesson about a specified payoff table should not become a claim that cooperation is impossible in every social interaction.
To check all pure equilibria, examine every cell. Mark whether the row player's chosen strategy is a best response to that column, and whether the column player's chosen strategy is a best response to that row. A cell qualifies only when both tests hold. With ties, several strategies can be best responses; the weak inequality in the definition matters. Do not discard a tied best response unless the exercise explicitly imposes a selection rule.
Another way: Multiple equilibria and no pure equilibrium require different reports
A coordination table can have payoffs (3,3), (0,0), (0,0) and (2,2) in the four cells ordered by rows. Each player prefers to match the other's strategy. Both (1,1) and (2,2) are pure Nash equilibria. The first gives both players a higher payoff, but the best-response test alone does not predict that it will be selected. Expectations, history or coordination institutions may matter.
There can also be no pure equilibrium. In a matching-pennies game, one player benefits from matching and the other from mismatching. At every deterministic pair, someone wants to switch. This does not mean the concept of equilibrium is meaningless. A mixed strategy assigns probabilities to pure actions; under the appropriate finite-game framework, equilibrium can exist in those probability distributions. This lesson identifies the pure-strategy limitation rather than deriving mixed probabilities.
A pure-equilibrium count therefore has three distinct possible kinds of report: none, one, or several. None is not the same as no possible play, and several is not the same as a complete prediction of which cell occurs. Even a unique equilibrium depends on the model's assumptions about choices, payoffs, information and rationality. An empirical prediction requires checking whether those assumptions describe the setting well enough.
Timing can change the appropriate analysis. A sequential game includes who moves first, what later players observe, and which actions remain available. Simply reading a simultaneous table as if one player committed first can give a different answer without acknowledging that the game has changed. Commitment can matter precisely because it changes later incentives, but it must be represented rather than inferred from the order in which the analyst writes the table.
Another way: Keep the single-firm and strategic benchmarks distinct
A monopoly output formula and a two-player best-response table answer different questions. The first optimizes one firm's payoff against a supplied demand curve; the second checks mutual incentives among several decision makers. An oligopoly model can connect these ideas by deriving each firm's payoff from demand, costs and rivals' choices, but the assumptions about price or quantity competition must be explicit. Firms choosing quantities simultaneously need not produce the same outcome as firms choosing prices for an identical product. Market concentration alone does not supply all those assumptions. Use the present exercises to identify the missing strategic specification before selecting a formula, rather than treating every market with a few sellers as the same game.
A teaching case first describes a sole provider facing inverse demand P=40-Q and constant marginal cost eight. Under the supplied uniform-price, continuous-output model, marginal revenue is forty minus twice output. Equating it to cost gives output sixteen and price twenty-four. The operating margin is sixteen credits per unit. These values describe the hypothetical demand and cost assumptions, not a recommendation for real pricing.
The case then changes the institution: two providers simultaneously decide whether to use service format one or format two. The supplied payoff cells, ordered (1,1), (1,2), (2,1), (2,2), are (6,6), (2,7), (7,2), (3,3). The analyst compares each player's own payoffs while holding the other player's format fixed. Format two strictly dominates format one for each player, producing the unique pure Nash pair (2,2).
Both providers would earn six at (1,1), more than the three each obtains at equilibrium. That observation identifies a conflict between the joint payoff ranking and unilateral incentives. It does not authorize an analyst to treat coordination between real competitors as automatically desirable; consumer effects and institutional rules are absent from this two-player payoff account.
Finally, the analyst records that the monopoly calculation cannot simply be reused for the two-provider setting. The game has changed who chooses, what choices are available and how payoffs respond. A useful report gives each result under its own assumptions, tests deviations explicitly, and identifies what additional demand, cost and consumer information would be needed for a broader market assessment.
Use demand to recover the price after optimizing marginal revenue. In games compare a player's own payoff coordinate while holding the other choice fixed. A jointly better cell need not be stable against unilateral deviations; multiple equilibria do not supply a selection rule.
State inverse demand and cost.
P=30-Q; MC=6
The firm recognizes its effect on price.
Write total revenue explicitly.
R=30Q-Q squared
Each unit sells at the same modeled price.
Differentiate the profit objective.
MR=30-2Q; MR-MC=24-2Q
Marginal revenue differs from price.
Solve and check curvature.
Q=12; second derivative=-2
Strict concavity makes the interior candidate a maximum.
Recover price and operating profit.
P=18; (18-6)*12=144
Read price from demand and exclude unspecified fixed cost.
Write the four ordered payoff pairs.
(1,1):(4,4); (1,2):(1,5); (2,1):(5,1); (2,2):(2,2)
The first coordinate always belongs to the row player.
Find row's response to column one.
5>4, so row 2
Compare first coordinates within column one.
Find row's response to column two.
2>1, so row 2
Row 2 strictly dominates row 1.
Check the column player's responses.
Against row 1:5>4; against row 2:2>1; choose column 2
The second coordinates determine column incentives.
Report the unique mutual response.
Equilibrium(2,2), payoffs(2,2)
Neither player gains from switching alone.
State the supplied payoff table.
(1,1):(3,3); (1,2):(0,0); (2,1):(0,0); (2,2):(2,2)
Players choose simultaneously.
Check the first diagonal cell.
At(1,1), either unilateral switch reduces 3 to 0
Both actions are best responses.
Check the second diagonal cell.
At(2,2), either unilateral switch reduces 2 to 0
This cell also meets the Nash condition.
Reject the off-diagonal cells.
At(1,2) and(2,1), matching increases a player's0 payoff
At least one profitable deviation defeats equilibrium.
Count the surviving pairs.
Two pure equilibria:(1,1) and(2,2)
A payoff-superior equilibrium does not erase the other.
State the limit of the prediction.
The table alone supplies no equilibrium-selection rule
Additional coordination assumptions would be needed.
Use P=36-Q and MC 8.
MR=36-2Q=8; Q=14
The firm internalizes its price effect.
Read price from demand.
P=36-14=22
Marginal revenue is not the selling price.
Calculate operating profit.
A uniform-price monopolist has P=30-Q and constant MC 6, no capacity limit, and no fixed cost. Produce optimal output, selling price and operating profit.
| Your result | |
|---|---|
| Output | |
| Price | |
| Operating profit before fixed cost |
Uniform-price demand is P=32-Q and constant marginal cost is 8; no fixed cost or capacity limit applies. Complete the interior solution.
Equate marginal revenue and marginal cost.
output
Differentiate total revenue before solving.
Recover the selling price from demand.
price
Do not substitute output into marginal revenue to obtain price.
Multiply the per-unit margin by output.
profit
This is operating profit before any omitted fixed cost.
A uniform-price monopolist has P=50-2Q and constant MC 10, positive continuous output and no fixed cost. Produce output, price and operating profit.
Output: v0. Price: v1. Operating profit before fixed cost: v2.
Strategies have numeric labels 1,2. Cells (1,1),(1,2),(2,1),(2,2) have row/column payoff pairs (4,4),(1,5),(5,1),(2,2). Produce row's best response to each column and the number of pure Nash equilibria.
Row best response to column 1: v0. Row best response to column 2: v1. Number of pure Nash equilibria: v2.
Strategies are 1,2. In cell order (1,1),(1,2),(2,1),(2,2), payoff pairs are (3,3),(0,0),(0,0),(2,2). Produce row's best response to each column and the pure-equilibrium count; do not assume a selection rule.
Row best response to column 1: v0. Row best response to column 2: v1. Number of pure Nash equilibria: v2.
Two fictional providers choose formats 1,2 simultaneously. Cells (1,1),(1,2),(2,1),(2,2) pay (6,6),(2,7),(7,2),(3,3), with row payoff first. Produce row's best response to each column and the number of pure equilibria, without treating this account as a full consumer-welfare analysis.
Row best response to column 1: v0. Row best response to column 2: v1. Number of pure Nash equilibria: v2.
Lesson test: one question per skill, one attempt each, no hints. Your answers are checked when you submit.
Compare two explicitly separate models. Model A is a uniform-price monopolist with P=44-2Q and MC 12. Model B has strategies 1,2 and payoff cells (1,1):(5,4), (1,2):(1,2), (2,1):(3,6), (2,2):(4,3), row payoff first. Produce A's optimal output and price, then B's unique pure Nash row and column strategies.
A output: v0. A price: v1. B equilibrium row: v2. B equilibrium column: v3.
Reconstruct a fresh case without the worked solution. Explain which assumption would change its conclusion and which result is only an accounting or model condition.
10. Finish a new monopoly calculation, step 3
(22-8)*14=196
The exercise excludes fixed cost.