Microeconomics: Consumer Theory, Production, Markets, Games and Welfare

This chapter answers "Section XH1.1: Microeconomics" of the Humanities and Social Sciences (XH) paper. It is part of Section XH1.1 of XH1 Economics, one of the six optional sections of XH, chosen at the time of application. It teaches the cardinal and ordinal theories of consumer behaviour, Marshallian and Hicksian demand with duality and the Slutsky equation, revealed preference, homothetic utility and Euler's theorem, production and cost in the short and long run, the existence and stability of market equilibrium (Walrasian, Marshallian and the cobweb), decision under uncertainty and risk, asymmetric information, agency and search, non-cooperative games, market structures and factor pricing, and general-equilibrium and welfare economics with Pareto optimality. The numericals here are elasticities, the Slutsky decomposition, mixed-strategy equilibria and firm output optima.

1. Consumer behaviour: cardinal, ordinal, utility and demand

The cardinal approach (Marshall) treats utility as a measurable quantity in units called utils, so differences in satisfaction have a size. It rests on the law of diminishing marginal utility and derives the equi-marginal rule: a consumer maximises utility when the marginal utility per rupee is equal across goods, MUx/Px = MUy/Py. The ordinal approach (Hicks and Allen) drops measurability: the consumer only ranks bundles. Preferences are assumed complete, transitive and monotonic, and are shown by indifference curves, which are downward-sloping and convex to the origin. The slope is the marginal rate of substitution MRS = MUx/MUy, which diminishes along the curve.

Maximising utility subject to the budget line gives the tangency condition MRS = Px/Py: the consumer equates the subjective rate of exchange to the market rate. Solving this with the budget constraint yields the Marshallian (uncompensated) demand x(Px, Py, M), a function of prices and money income. Its properties follow from the problem: it is homogeneous of degree zero in prices and income (doubling all prices and income leaves demand unchanged), and it satisfies the budget identity (Walras' law for one consumer).

⚠️ Diminishing MU is not diminishing MRS
The cardinal law of diminishing marginal utility is a statement about one good. The ordinal convexity of indifference curves is the diminishing MRS, a statement about the trade-off between two goods. Convex preferences can hold even where marginal utility of a single good is not falling, so do not quote one law to prove the other.

2. Hicksian demand, duality, Slutsky, revealed preference and homotheticity

The dual problem is expenditure minimisation: reach a target utility u at least cost. Its solution is the Hicksian (compensated) demand h(Px, Py, u), a function of prices and utility, and the minimised cost is the expenditure function e(Px, Py, u). The duality theorem links the two problems: the indirect utility function v(p, M) and e(p, u) are inverses in the relevant argument, Shephard's lemma gives h_i = ∂e/∂p_i, and Roy's identity recovers Marshallian demand as x_i = −(∂v/∂p_i)/(∂v/∂M). The two demands agree at the optimum for the reference utility.

The Slutsky equation splits a Marshallian price response into two parts: ∂x_i/∂p_j = (∂h_i/∂p_j) − x_j(∂x_i/∂M). The first term is the substitution effect (movement along an indifference curve); the second is the income effect (the shift to a new curve). The own-price substitution effect is always non-positive, so demand slopes down unless the income effect is both positive (an inferior good) and large. A Giffen good is the extreme inferior case where the positive income effect outweighs the substitution effect and demand slopes up.

Revealed preference rebuilds demand theory from choices. If bundle A is chosen when B was affordable, A is revealed preferred to B. The Weak Axiom (WARP) forbids a direct contradiction: B must never be chosen when A is affordable. The Strong Axiom (SARP) extends this to all indirect chains, ruling out cycles, and is what guarantees choices come from a well-behaved (transitive) preference. A homothetic utility is a monotonic transform of a homogeneous function; its indifference curves have the same MRS along any ray from the origin, so demand is proportional to income and every income elasticity equals one. Euler's theorem states that for a function homogeneous of degree k, Σ x_i(∂f/∂x_i) = k·f; with constant returns (k = 1) in production, paying each factor its marginal product exactly exhausts the product.

3. Production and costs: short run and long run

In the short run at least one factor is fixed. As the variable factor rises, total product grows, then the law of diminishing marginal returns sets in and marginal product (MP) falls. MP cuts average product (AP) at AP's maximum, and when MP exceeds AP, AP rises. The firm uses stage II of production, where both AP and MP are positive but falling. In the long run all factors are variable. Isoquants (equal-output curves) have slope the marginal rate of technical substitution MRTS = MPL/MPK, and the least-cost input mix sets MRTS equal to the input-price ratio w/r.

Short-run and long-run cost relationships
ConceptShapeKey relationship
AFC = TFC/QFalls, approaches zeroA rectangular hyperbola; never negative
AVC and ATCU-shapedMC cuts each at its own minimum
AVC minimum vs ATC minimumAVC bottoms out firstATC min is at larger Q because AFC keeps falling
LACEnvelope of SAC curvesTangent to each SAC; LMC = LAC at min LAC

The long-run average cost (LAC) is the lower envelope of the short-run average cost (SAC) curves: for each output the firm picks the plant with the lowest average cost. LAC is tangent to each SAC, but only at the minimum efficient scale (the lowest point of LAC) is the tangency at the SAC's own minimum, where LMC = LAC. The falling part of LAC is economies of scale (increasing returns), the flat part constant returns, and the rising part diseconomies (decreasing returns).

4. Market equilibrium: existence, stability and the cobweb

Equilibrium is a price at which quantity demanded equals quantity supplied. Existence needs demand and supply to cross; uniqueness fails when curves are not monotone (for example a backward-bending supply). Two dynamics test stability. Walrasian adjustment moves the price in response to excess demand: price rises when demand exceeds supply and falls when it is short. Marshallian adjustment moves the quantity in response to the gap between the demand price and supply price: output expands when buyers will pay more than suppliers require.

For normally sloped curves the two criteria agree and the equilibrium is stable. They can disagree only when a curve has an unusual slope. The cobweb model adds a one-period lag: supply responds to last period's price, S_t = f(Pt−1), while demand responds to the current price. Market clearing each period generates P_t = A − (d/b)Pt−1, where b and d are the slopes of demand and supply in quantity. The path converges when, with both curves drawn with price on the vertical axis, the supply curve is steeper than the demand curve (|d| < |b|, i.e. supply less elastic than demand), produces a constant cycle when slopes are equal, and diverges when supply is flatter (more elastic) than demand.

🎯 Why the lag makes a cycle
Because producers plan output on yesterday's price, a high price today brings a glut tomorrow, which drops the price, which cuts planned output, which raises the price again. Whether these swings shrink or grow depends only on the ratio of slopes, which is why elasticities decide convergence.

5. Uncertainty, asymmetric information, agency and search

Choice under risk uses the expected-utility (von Neumann–Morgenstern) rule: rank gambles by the expected value of a utility-of-wealth function. A strictly concave utility means risk aversion: by Jensen's inequality the utility of the expected value exceeds the expected utility, so the certainty equivalent (the sure amount giving the same utility) is below the expected value, and the gap is the risk premium. A risk-averse person rejects an actuarially fair gamble and will pay to insure. Linear utility is risk-neutral; convex utility is risk-loving.

  • Adverse selection is hidden information before the contract: when the insurer cannot tell high risks from low, mostly high risks buy, as in Akerlof's market for lemons. Remedies are screening (the uninformed party offers a menu) and signalling (the informed party sends a costly signal, as in Spence's education model).
  • Moral hazard is hidden action after the contract: once insured, people take less care. It is controlled by deductibles, co-payments and monitoring.
  • Agency costs (Jensen and Meckling) arise when an agent acts for a principal with divergent interests: they are the sum of monitoring costs, bonding costs and the residual loss. The theory of search treats costly information gathering: a searcher sets a reservation wage or price and stops when the expected gain from one more search equals its marginal cost.

6. Non-cooperative games

A game lists players, strategies and payoffs. A Nash equilibrium is a strategy profile where no player can gain by a unilateral deviation; each strategy is a best response to the others. A dominant strategy is best whatever others do; the prisoner's dilemma is the classic case where dominant strategies give a Pareto-inferior outcome. A constant-sum (zero-sum) game has strictly opposed interests and is solved by the minimax rule. Nash proved that every finite game has at least one equilibrium in mixed strategies, where players randomise. In a mixed equilibrium each player randomises so as to make the opponent indifferent between the strategies the opponent mixes over.

Dynamic and incomplete-information games need refinements. A subgame-perfect Nash equilibrium (SPNE), found by backward induction, requires optimal play in every subgame, which rules out equilibria that rest on non-credible threats. A Bayesian game has players of different types drawn by nature; a Bayesian Nash equilibrium has each type best-responding given beliefs about others' types. For dynamic games of incomplete information a Perfect Bayesian Equilibrium (PBE) pairs a strategy profile with beliefs at each information set, requiring sequential rationality and that beliefs follow Bayes' rule wherever possible.

7. Market structure, factor pricing, general equilibrium and welfare

Every firm maximises profit where MR = MC. Under perfect competition the firm is a price-taker, so P = MR = MC, long-run economic profit is zero, and the outcome is both allocatively (P = MC) and productively (min AC) efficient. A monopoly sets MR = MC with MR below price, so P > MC and output is restricted, creating a deadweight loss. Monopolistic competition has differentiated products, P > MC, zero long-run profit and excess capacity. Oligopoly is modelled by Cournot (firms choose quantities), Bertrand (prices; with identical goods this drives P to MC) and Stackelberg (a leader moves first). The Structure–Conduct–Performance (SCP) paradigm reads market structure as shaping firms' conduct and hence performance.

The marginal-productivity theory of distribution says that in perfectly competitive factor and product markets, a factor is paid the value of its marginal product VMP = P × MP. If the product market is a monopoly, the factor is paid its marginal revenue product MRP = MR × MP, which is below VMP; this shortfall is monopolistic exploitation (a term associated with Pigou and Joan Robinson). A monopsony in the factor market pays a wage below MRP, a further exploitation. General-equilibrium analysis (Walras) studies all markets at once; in the Edgeworth box, mutually beneficial trade continues until no further gains exist, tracing the contract curve of Pareto-efficient allocations.

Welfare economics judges allocations by Pareto optimality: no one can be made better off without making someone worse off. Its conditions are efficiency in exchange (MRS equal across consumers), in production (MRTS equal across firms) and in the product mix (MRS = MRT). The First Fundamental Theorem says a competitive equilibrium is Pareto efficient (given no externalities); the Second says any Pareto-efficient allocation can be reached as a competitive equilibrium after suitable lump-sum transfers (given convex preferences and technologies). Pareto efficiency is silent on equity — there are many efficient allocations — so a Bergson–Samuelson social welfare function is needed to rank them, and Arrow's impossibility theorem shows no voting rule can build one satisfying all of a short list of reasonable conditions.

Key takeaways

  • The cardinal approach measures utility in utils and uses MUx/Px = MUy/Py; the ordinal approach ranks bundles with indifference curves and MRS = Px/Py. Utility maximisation gives Marshallian demand (prices and income), which is homogeneous of degree zero.
  • Expenditure minimisation gives Hicksian demand (prices and utility); duality links them through Shephard's lemma and Roy's identity. The Slutsky equation splits a price change into a non-positive substitution effect and an income effect; a Giffen good is inferior with income effect dominating. WARP and SARP, and homothetic preferences with unit income elasticity (Euler's theorem), complete the theory.
  • In the short run MP cuts AP at AP's maximum and MC cuts AVC and ATC at their minima; AFC falls toward zero. In the long run LAC is the envelope of SAC curves, with LMC = LAC at minimum efficient scale, and returns to scale set economies and diseconomies.
  • Walrasian stability adjusts price to excess demand, Marshallian adjusts quantity to the demand-price gap; they agree for normal curves. The cobweb converges when supply is steeper (less elastic) than demand, cycles when slopes are equal, and diverges when supply is flatter.
  • Concave utility means risk aversion: certainty equivalent below expected value, the gap being the risk premium. Adverse selection is pre-contract hidden information (lemons, screening, signalling); moral hazard is post-contract hidden action; agency costs are monitoring plus bonding plus residual loss; search sets a reservation value.
  • A Nash equilibrium has no profitable unilateral deviation; mixed strategies make the opponent indifferent, and every finite game has one. SPNE rules out non-credible threats; Bayesian NE and PBE handle incomplete information. Perfect competition gives P = MC and efficiency; monopoly restricts output; the two welfare theorems and Pareto optimality judge allocations, with a social welfare function for equity.

Practice questions (18)

Attempt each one before opening the answer. Every explanation names the tempting wrong option as well as the right one, because that is where marks are lost.

  1. In the ordinal approach to consumer behaviour, utility is treated as:

    1. A ranking of bundles in which only the order of preference matters and the size of any utility difference is meaningless
    2. A quantity measured in units called utils, so that the extra satisfaction from one bundle over another can be stated as a definite number
    3. A magnitude that is identical across all consumers, so that interpersonal comparisons of satisfaction become exact
    4. A cardinal number that necessarily falls at a constant arithmetic rate with each extra unit consumed
    Show answer

    Answer: A — A ranking of bundles in which only the order of preference matters and the size of any utility difference is meaningless

    The ordinal approach of Hicks and Allen requires only that bundles be ranked; the magnitude and comparison of utility differences are not needed. Measuring utility in utils belongs to the cardinal approach, exact comparability across people is a still stronger claim that ordinal theory never makes, and a constant rate of decline is not assumed by either.

  2. Which statement correctly distinguishes the Hicksian (compensated) demand function from the Marshallian (uncompensated) demand function?

    1. The Hicksian demand depends on prices and money income and is obtained from utility maximisation, while the Marshallian demand depends on prices and a fixed level of utility and is obtained from expenditure minimisation
    2. The Hicksian demand holds utility constant and comes from expenditure minimisation, while the Marshallian demand holds money income constant and comes from utility maximisation
    3. Both hold utility constant, but only the Hicksian demand can slope upward for an ordinary normal good
    4. The Marshallian demand is always steeper than the Hicksian demand because it excludes the substitution effect entirely
    Show answer

    Answer: B — The Hicksian demand holds utility constant and comes from expenditure minimisation, while the Marshallian demand holds money income constant and comes from utility maximisation

    Hicksian demand is the solution to expenditure minimisation and is a function of prices and a fixed utility; Marshallian demand solves utility maximisation and is a function of prices and money income. The reversed description swaps the two arguments, and the claims about slopes are wrong: the compensated curve is the one that excludes the income effect, and it never slopes up.

  3. For a Giffen good the quantity demanded rises when its own price rises. Within the Slutsky decomposition, how is this possible?

    1. The own-price substitution effect is positive for this good, and that positive substitution effect is large enough to outweigh a negative income effect working against it
    2. Both the substitution effect and the income effect are positive, so the two reinforce each other
    3. The good is inferior, so the income effect of the price rise is positive, and it is large enough to outweigh the negative substitution effect
    4. The substitution effect is exactly zero, so only the income effect operates and it is always positive
    Show answer

    Answer: C — The good is inferior, so the income effect of the price rise is positive, and it is large enough to outweigh the negative substitution effect

    The own-price substitution effect is always non-positive, so it can never be the source of an upward-sloping demand. A price rise lowers real income; for an inferior good that raises quantity (a positive income effect), and a Giffen good is the case where this positive income effect is large enough to dominate the negative substitution effect.

  4. Consider the following statements about revealed preference and homothetic utility. Which are correct? P: If bundle A was chosen when bundle B was also affordable, WARP forbids B from being chosen in any situation in which A is affordable. Q: SARP rules out cycles and so is consistent with maximisation of a transitive preference. R: Homothetic preferences give straight-line income expansion paths and unit income elasticities. S: Any homothetic utility must itself be homogeneous of degree one.

    1. P
    2. Q
    3. R
    4. S
    Show answer

    Answer: A — P; B — Q; C — R

    WARP is exactly the no-direct-contradiction condition in P, and SARP adds the no-cycle requirement that makes choices rationalisable by a transitive preference, so Q holds; homothetic preferences scale demand with income, giving straight expansion paths and unit income elasticities, so R holds. S is false: a homothetic utility is any increasing transformation of a homogeneous function, and such a transformation need not be homogeneous itself.

  5. A market demand curve is Q = 120 − 4P, where Q is quantity and P is price. Using the point formula e = (dQ/dP)(P/Q), compute the price elasticity of demand at P = 18. Report its magnitude (absolute value) to one decimal place. Give the number only.

    Numerical answer — type the value.

    Show answer

    Answer: 1.5

    Here dQ/dP = −4. At P = 18, Q = 120 − 4(18) = 120 − 72 = 48. So e = (−4)(18/48) = (−4)(0.375) = −1.5, whose magnitude is 1.5. Using Q = 120 (ignoring the price) or forgetting the slope −4 are the usual slips.

  6. When the price of good X rises, the quantity demanded of X falls by 3 units in total (total effect = −3). Under the Hicksian substitution experiment, with utility held at its original level, the quantity of X would fall by 7 units (substitution effect = −7). Using total effect = substitution effect + income effect, find the income effect on the quantity of X, in units. Give the number only, including a minus sign if it is negative.

    Numerical answer — type the value.

    Show answer

    Answer: 4

    Income effect = total effect − substitution effect = (−3) − (−7) = +4. The positive income effect (quantity rises as the price rise lowers real income) shows that X is an inferior good; because the substitution effect (−7) still outweighs it, net demand falls, so X is inferior but not Giffen. Answering −10 adds the two effects instead of subtracting.

  7. In the short run, at the level of the variable input where the marginal product (MP) curve crosses the average product (AP) curve:

    1. AP is at its maximum, and MP equals AP at that point
    2. AP is at its minimum, while MP is falling at its steepest rate
    3. MP is at its own maximum, which always coincides with the maximum of AP
    4. Total product reaches its maximum value and then starts to fall
    Show answer

    Answer: A — AP is at its maximum, and MP equals AP at that point

    MP intersects AP at the maximum of AP, and there MP = AP (when MP is above AP, AP rises; when below, AP falls). MP reaches its own maximum earlier, and total product is maximised where MP = 0, not where MP meets AP.

  8. The long-run average cost (LAC) curve is drawn as the lower envelope of the short-run average cost (SAC) curves. At the output where LAC itself is minimised (the minimum efficient scale), which relationship holds?

    1. LAC is tangent to every SAC curve exactly at that SAC's own minimum point, all along its length
    2. The relevant SAC is tangent to LAC at that SAC's own minimum, and long-run marginal cost equals long-run average cost there
    3. Long-run marginal cost lies above long-run average cost at that output, because the firm must be operating under decreasing returns to scale exactly at the minimum of its cost curve
    4. The firm must be earning positive economic profit, since its cost per unit has been minimised
    Show answer

    Answer: B — The relevant SAC is tangent to LAC at that SAC's own minimum, and long-run marginal cost equals long-run average cost there

    Only at the minimum of LAC is the tangency with a SAC at that SAC's own minimum, and there LMC = LAC. At any other output the tangency is to the left or right of the SAC minimum, so LAC is not tangent to every SAC at its lowest point. Returns to scale are constant at the LAC minimum, and profit also depends on price, so minimum cost alone does not imply profit.

  9. Which of the following statements about a firm's short-run cost curves are correct?

    1. Average fixed cost falls continuously as output rises, approaching zero but never becoming negative
    2. The marginal cost curve cuts both the average variable cost and the average total cost curves at their minimum points
    3. Average variable cost reaches its minimum at a larger output than average total cost does
    4. When marginal cost is below average total cost, average total cost must be falling
    Show answer

    Answer: A — Average fixed cost falls continuously as output rises, approaching zero but never becoming negative; B — The marginal cost curve cuts both the average variable cost and the average total cost curves at their minimum points; D — When marginal cost is below average total cost, average total cost must be falling

    AFC = TFC/Q is a falling rectangular hyperbola that tends to zero, MC cuts AVC and ATC at their minima, and MC below ATC pulls the average down. The claim that AVC bottoms out at a larger output than ATC reverses the order: AVC reaches its minimum at a smaller output than ATC, because AFC keeps falling and delays the turning point of ATC.

  10. In the cobweb model, current supply responds to the price of the previous period while demand responds to the current price. Draw both curves with price on the vertical axis. Starting away from equilibrium, the price path converges to equilibrium when:

    1. The demand curve slopes upward, so that buyers purchase more at higher prices
    2. The demand and supply curves have exactly equal slopes, giving a path that dies out to equilibrium
    3. The supply curve is flatter than the demand curve in the price–quantity plane, so that supply is more elastic than demand at the equilibrium point
    4. The supply curve is steeper than the demand curve in the price–quantity plane, so supply is less elastic than demand at equilibrium
    Show answer

    Answer: D — The supply curve is steeper than the demand curve in the price–quantity plane, so supply is less elastic than demand at equilibrium

    The recursion P_t = A − (d/b)Pt−1 converges when |d/b| < 1, that is, when the supply slope is steeper than the demand slope in the price–quantity plane (supply less elastic than demand). Equal slopes give a constant cycle, flatter supply gives divergence, and an upward-sloping demand is not the cobweb condition at all.

  11. Consider the Walrasian and Marshallian approaches to the stability of a single market equilibrium. Which statements are correct?

    1. In Walrasian adjustment, price rises whenever there is excess demand and falls whenever there is excess supply
    2. In Marshallian adjustment, quantity expands whenever the demand price exceeds the supply price
    3. For normally sloped demand and supply curves, the two criteria give opposite verdicts on stability
    4. The two criteria can disagree only when at least one curve has an unusual slope, such as a backward-bending supply curve
    Show answer

    Answer: A — In Walrasian adjustment, price rises whenever there is excess demand and falls whenever there is excess supply; B — In Marshallian adjustment, quantity expands whenever the demand price exceeds the supply price; D — The two criteria can disagree only when at least one curve has an unusual slope, such as a backward-bending supply curve

    Walras adjusts price to excess demand and Marshall adjusts quantity to the demand-price minus supply-price gap, so the statements on Walrasian price adjustment and Marshallian quantity adjustment are correct, and the two criteria agree for normally sloped curves, disagreeing only when a curve is abnormally sloped. The claim that they give opposite verdicts for normal curves is therefore wrong.

  12. An insurer finds that once a household is fully insured against theft, it stops locking its doors and takes less care. This problem is best described as:

    1. Adverse selection, because only the highest-risk households ever choose to buy the insurance in the first place
    2. Moral hazard, because a hidden action taken after the contract is signed changes the insured party's behaviour
    3. A signalling failure, because the insurer is unable to advertise its contract terms credibly to buyers
    4. A pure agency cost, arising because the insurer must monitor the managers who run its own operations
    Show answer

    Answer: B — Moral hazard, because a hidden action taken after the contract is signed changes the insured party's behaviour

    The behaviour changes after the contract and the action (care) is hidden from the insurer, which is moral hazard. Adverse selection is the pre-contract problem of who chooses to buy, signalling concerns the informed party sending a costly message, and an agency cost here would concern the insurer's own managers, not the policyholder.

  13. A decision-maker has a strictly concave von Neumann–Morgenstern utility of wealth and faces a risky gamble with a given expected monetary value. Which statement must be true?

    1. The expected utility of the gamble exceeds the utility of its expected value, so the gamble is always attractive
    2. The certainty equivalent of the gamble equals its expected value, because utility is increasing in wealth
    3. The decision-maker will accept every actuarially fair gamble that is offered to them
    4. The certainty equivalent of the gamble is below its expected value, and the difference is the risk premium
    Show answer

    Answer: D — The certainty equivalent of the gamble is below its expected value, and the difference is the risk premium

    Strict concavity means risk aversion: by Jensen's inequality the utility of the expected value exceeds the expected utility, so the certainty equivalent falls short of the expected value, and that shortfall is the risk premium. A risk-averse person therefore declines an actuarially fair gamble rather than accepting every one.

  14. In a finite dynamic game of perfect information solved by backward induction, a subgame-perfect Nash equilibrium (SPNE) differs from an ordinary Nash equilibrium in that:

    1. It permits players to rely on threats they would not actually wish to carry out once the relevant node is reached
    2. It requires each player's strategy to be a best response only along the equilibrium path, ignoring all off-path nodes
    3. It rules out equilibria that rest on non-credible threats, by requiring optimal play in every subgame
    4. It can be defined only for zero-sum games, where one player's gain is exactly the other player's loss
    Show answer

    Answer: C — It rules out equilibria that rest on non-credible threats, by requiring optimal play in every subgame

    SPNE demands sequential rationality in every subgame, so a strategy must be optimal even at nodes off the equilibrium path, which is exactly what eliminates non-credible threats that an ordinary Nash equilibrium can tolerate. It applies to all finite dynamic games, not only zero-sum ones.

  15. A striker and a goalkeeper play a zero-sum game. The striker kicks Left or Right and the goalkeeper simultaneously dives Left or Right. The entries are the striker's probability of scoring, which the goalkeeper tries to minimise: kick Left & dive Left = 0.3; kick Left & dive Right = 0.9; kick Right & dive Left = 0.8; kick Right & dive Right = 0.2. In the mixed-strategy Nash equilibrium the striker kicks Left with a probability chosen to make the goalkeeper indifferent between diving Left and diving Right. Find that probability. Give the number only, as a decimal between 0 and 1.

    Numerical answer — type the value.

    Show answer

    Answer: 0.5

    Let p be the probability the striker kicks Left. If the goalkeeper dives Left, the scoring probability is 0.3p + 0.8(1 − p) = 0.8 − 0.5p; if the goalkeeper dives Right, it is 0.9p + 0.2(1 − p) = 0.2 + 0.7p. Indifference requires 0.8 − 0.5p = 0.2 + 0.7p, giving 0.6 = 1.2p, so p = 0.5. The striker kicks Left with probability 0.5.

  16. A monopolist faces the inverse demand P = 120 − 2Q and has a constant marginal cost MC = 40 with no fixed cost. Using marginal revenue MR = 120 − 4Q, find the profit-maximising output Q. Give the number only.

    Numerical answer — type the value.

    Show answer

    Answer: 20

    Profit is maximised where MR = MC: 120 − 4Q = 40, so 4Q = 80 and Q = 20. The price is P = 120 − 2(20) = 80, above MC, which creates a deadweight loss. A competitive industry would set P = MC, giving 120 − 2Q = 40 and Q = 40, so the monopoly halves output. Setting P = MC on the demand curve (Q = 40) is the usual slip.

  17. A firm sells its product as a monopoly but hires labour in a perfectly competitive labour market. Under the marginal-productivity theory, what wage does the firm pay, and what is the gap called?

    1. It pays the value of the marginal product (P × MP), so there is no gap and no exploitation
    2. It pays the workers more than the value of their marginal product, and this surplus paid above the competitive wage is best understood as a pure monopoly rent handed over to labour
    3. It pays a wage set only by a trade union, so the marginal-productivity theory does not apply at all
    4. It pays the marginal revenue product (MR × MP), which falls short of the value of the marginal product, and the gap is monopolistic exploitation
    Show answer

    Answer: D — It pays the marginal revenue product (MR × MP), which falls short of the value of the marginal product, and the gap is monopolistic exploitation

    A product-market monopolist values an extra unit of output at marginal revenue, not price, so labour is paid its marginal revenue product MR × MP. Because MR is below price for a monopoly, MRP is below the value of the marginal product P × MP, and this gap is called monopolistic exploitation (a term associated with Pigou and Joan Robinson). Only a product-market price-taker pays the full VMP.

  18. Assume the standard conditions (price-taking behaviour, convex preferences and technologies, no externalities). Which of the following statements about the fundamental theorems of welfare economics and Pareto optimality are correct?

    1. The first fundamental theorem states that a competitive (Walrasian) equilibrium is Pareto efficient when there are no externalities
    2. The second fundamental theorem states that any Pareto-efficient allocation can be sustained as a competitive equilibrium after suitable lump-sum transfers
    3. A Pareto-efficient allocation is necessarily the single most equitable allocation and the unique maximiser of social welfare
    4. At a Pareto-efficient allocation in exchange, the marginal rate of substitution between any two goods is equal across all consumers
    Show answer

    Answer: A — The first fundamental theorem states that a competitive (Walrasian) equilibrium is Pareto efficient when there are no externalities; B — The second fundamental theorem states that any Pareto-efficient allocation can be sustained as a competitive equilibrium after suitable lump-sum transfers; D — At a Pareto-efficient allocation in exchange, the marginal rate of substitution between any two goods is equal across all consumers

    The two theorems are stated correctly, and equality of the MRS across consumers is exactly the exchange-efficiency condition. Pareto efficiency is silent on equity: many Pareto-efficient allocations exist, including highly unequal ones, so efficiency does not single out the most equitable or a unique welfare maximiser — that needs a social welfare function.