ECO2003F: Intermediate Microeconomics Course Notes

Intermediate microeconomics (commonly coded as ECO2003F in South African universities) builds the analytical “engine room” behind consumer choice, firm behaviour, and market outcomes. These notes focus on the core methods typically tested in an intermediate course: utility and preference representation, demand and elasticity, consumer and producer surplus, competitive equilibrium, firm cost and profit maximisation, perfect competition and market power, and the welfare implications of taxes and regulation. Throughout, the emphasis is on exam-ready problem-solving: clear definitions, careful reasoning about assumptions, and step-by-step derivations.

Stellenbosch University — ECO2003F Intermediate Microeconomics Course Notes

Stellenbosch University’s intermediate microeconomics is often assessed through a combination of conceptual questions (e.g., interpreting equilibrium, welfare theorems) and computational exercises (e.g., deriving demand, elasticity, cost functions, and equilibrium prices). While the exact lecture sequence can differ by lecturer and semester, the exam-style structure usually mirrors a standard path: start with consumer theory, move to firm theory, then connect both to market outcomes under different market structures, finishing with welfare analysis and policy.

1) Consumer Theory: Preferences, Utility, and Choice

Preferences and rational choice

A typical exam question begins by specifying preferences over bundles of goods, then asking you to verify whether preferences are complete and transitive, and whether they exhibit continuity and local non-satiation.

Completeness: for any bundles (x) and (y), either (x \succeq y) or (y \succeq x) (or both).
Transitivity: if (x \succeq y) and (y \succeq z), then (x \succeq z).
Continuity: small changes in bundles do not cause “jumps” in preference ordering.
Local non-satiation: there exists a nearby bundle strictly preferred to a given bundle.

These conditions are essential because they justify using a utility function to represent preferences. In exams, you don’t need to recite Debreu’s theorem; you need to know that if preferences satisfy these properties, a utility representation exists (and then you can maximise utility subject to a budget).

Utility functions and common forms

Common utility functions used in exam questions include:

Perfect substitutes:
[
u(x,y)=ax+by
]
where consumption is “all-or-nothing” depending on relative marginal utility.
Perfect complements (Leontief):
[
u(x,y)=\min{x,\ y}
]
implying fixed ratios.
Cobb–Douglas:
[
u(x,y)=x^\alpha y^{1-\alpha},\quad 0<\alpha<1
]
Quasilinear utility: utility linear in one good, e.g.
[
u(x,y)=u(x)+y
]
which simplifies welfare analysis with taxes and transfers.

Optimisation: the constrained maximisation problem

For a standard consumer with income (m) and prices (p_x, p_y), the budget constraint is:
[
p_x x + p_y y \le m
]
To maximise utility:
[
\max_{x,y}\ u(x,y)\quad \text{s.t.}\quad p_x x + p_y y \le m
]

Exam method: set up Lagrangian:
[
\mathcal{L}=u(x,y)+\lambda(m-p_x x-p_y y)
]
FOCs usually yield:

Marginal utility per rand spent equalised across goods:
[
\frac{MU_x}{p_x}=\frac{MU_y}{p_y}
]
plus complementary slackness and positivity constraints when relevant.

Example: Cobb–Douglas demand

Let:
[
u(x,y)=x^{\alpha}y^{1-\alpha}
]
The consumer chooses (x,y) to maximise utility under the budget. The standard result (derivable in exams using FOCs) is:
[
x^=\alpha\frac{m}{p_x},\qquad y^=(1-\alpha)\frac{m}{p_y}.
]
Key interpretation: Cobb–Douglas spending shares are constant. If (p_x) rises, (x^) falls proportionally, while (y^) moves according to the budget share.

Demand curves and the role of elasticities

Once you have (x(p_x,p_y,m)), you can compute:

Price elasticity of demand:
[
\varepsilon_{x,p_x}=\frac{\partial x}{\partial p_x}\cdot \frac{p_x}{x}
]
Income elasticity:
[
\varepsilon_{x,m}=\frac{\partial x}{\partial m}\cdot\frac{m}{x}
]
Cross-price elasticity:
[
\varepsilon_{x,p_y}=\frac{\partial x}{\partial p_y}\cdot\frac{p_y}{x}
]

For Cobb–Douglas with (x^*=\alpha m/p_x):

(\varepsilon_{x,p_x}=-1) (unit elastic),
(\varepsilon_{x,m}=1) (income elasticity equals 1),
cross-price elasticity is 0 if (x) doesn’t depend on (p_y) (as in this Cobb–Douglas form where spending shares separate).

2) Consumer Surplus, Welfare, and Demand Interpretation

Marshallian and Hicksian demands

Intermediate exams sometimes distinguish:

Marshallian demand (x(p,m)): demand given prices and income.
Hicksian demand (h(p,u)): demand given prices and target utility (u).

While full derivations may be optional, you should know the conceptual link:

Marshallian comes from maximising utility with a budget.
Hicksian comes from minimising expenditure for a fixed utility.

This matters because welfare changes from price changes can be measured through money-metric concepts.

Consumer surplus under linear demand

For a linear inverse demand:
[
p(q)=a-bq
]
Consumer surplus (CS) when the market price is (p^) and quantity is (q^=(a-p^)/b) is:
[
CS=\frac{1}{2}(q^)(a-p^*)
]
or equivalently area under demand above price. Many exam questions provide a demand schedule and ask you to compute CS before and after a tax.

Change in welfare from a small price change

With a general demand curve, the welfare change can be approximated using the area concept:

If price increases, CS falls.
If demand is elastic, CS changes “more sharply” with price.

Some courses prefer a decomposition using compensating variation or equivalent variation. Even if not fully tested, the direction of change should be consistent:

Price increases reduce purchasing power.
Transfers (like income compensation) can keep utility constant.

Example: A tax and deadweight loss (DWL)

Suppose the consumer faces a price that includes a specific tax (t). Let the pre-tax equilibrium price be (p) and post-tax price paid by consumers be (p_c=p+t). If supply and demand are known, you compute:

new quantity (q_t),
consumer surplus reduction,
producer surplus reduction,
tax revenue,
deadweight loss as the net loss.

Key exam intuition:

Tax revenue is the rectangle (t \times \text{quantity}).
Deadweight loss is the triangle representing lost surplus from the contraction in quantity.

3) Producer Theory: Costs, Profit Maximisation, and Supply

Production and the production function

A production function maps inputs to output:
[
q=f(L,K)
]
In exams, you often see simplified cases such as:

linear homogeneity,
diminishing marginal returns (e.g., as (L) increases holding (K) fixed),
short-run with fixed capital (K).

Cost functions: total, average, marginal

For given output (q), costs are:

Total cost (TC(q)),
Average cost (AC(q)=TC(q)/q),
Marginal cost (MC(q)=\frac{dTC}{dq}).

A standard relationship: if (TC) is differentiable,

(MC) intersects (AC) at its minimum (under standard convexity assumptions).

Profit maximisation

Profit is:
[
\pi(q)=p\cdot q – TC(q).
]
Maximise with respect to (q) (or choose inputs to produce (q) and then optimise).

FOC (for interior solutions):
[
p=MC(q).
]
Second-order conditions typically require convex costs, so (MC) increasing.

Example: quadratic cost

Let:
[
TC(q)=c q^2 + d q + F,\quad c>0.
]
Then:
[
MC(q)=2cq+d.
]
If the firm is price-taking at price (p), the profit-maximising quantity satisfies:
[
p=2cq+d \Rightarrow q^*=\frac{p-d}{2c}.
]

If (p) is too low, the firm may choose (q=0) (shutdown rule). In many intermediate courses:

The firm produces if (p \ge \min AVC(q)),
where (AVC(q)) is variable cost per unit.

4) Competitive Equilibrium and Market Clearing

Supply and equilibrium

In a competitive market:

Firms are price takers: (p=MC) (for producing firms).
Consumers take prices as given: demand derived from utility maximisation.

Equilibrium occurs where:
[
\text{Demand}(p)=\text{Supply}(p).
]

Welfare theorems (high-yield)

A classic exam conceptual piece is:

Under standard assumptions (complete markets, perfect competition), competitive equilibrium is Pareto efficient.
If preferences are convex and production sets are convex, efficiency results follow.

But exam questions also test limitations:

with externalities, monopoly power, or information asymmetry, market outcomes may be inefficient.

You should be able to explain why:

If an externality exists, private marginal costs differ from social marginal costs.
If the firm has market power, price exceeds marginal cost, implying under-consumption relative to the efficient outcome.

Numerical equilibrium example (common structure)

Suppose:

Inverse demand: (p=100-2q).
Firm marginal cost: (MC(q)=q+10).
With perfect competition, set (p=MC):
[
100-2q=q+10 \Rightarrow 90=3q \Rightarrow q^=30.
]
Then:
[
p^=100-2(30)=40.
]
You can then compute surplus areas if you’re asked:
Consumer surplus:
[
CS=\frac12 \cdot q^* \cdot (p_{\text{max}}-p^*)
]
where (p_{\text{max}}) is intercept when (q=0), here (100).
Producer surplus depends on the supply curve or cost line.

5) Market Power: Monopoly and Welfare Effects

Monopoly behaviour

A monopolist chooses quantity to maximise profit where:
[
\pi(q)=p(q)\cdot q – TC(q).
]
Instead of (p=MC), monopoly satisfies:
[
MR(q)=MC(q),
]
where marginal revenue (MR) is less than price for downward-sloping demand.

For linear demand:
[
p(q)=a-bq,
]
revenue (TR=pq=(a-bq)q=a q-b q^2), so:
[
MR(q)=a-2bq.
]

Pricing and deadweight loss

With monopoly:

Price is higher than competitive price.
Quantity is lower than competitive quantity.
Deadweight loss arises as consumers forgo mutually beneficial trades between the monopoly quantity and the competitive quantity.

A high-yield exam diagram interpretation:

Triangle of DWL between (MC) and demand curve (marginal benefit) over the reduced quantity range.

Example with linear demand and constant marginal cost

Let:

(p=100-2q),
(MC=30) (constant),
ignore fixed costs.

Competitive: (p=MC\Rightarrow 100-2q=30\Rightarrow q=35,\ p=30).

Monopoly: (MR=MC).
Here (MR=100-4q). Set (100-4q=30\Rightarrow 4q=70\Rightarrow q_M=17.5).
Then (p_M=100-2(17.5)=65).

So monopoly reduces quantity from (35) to (17.5), increasing price from (30) to (65). If asked, compute DWL as:

area between demand and (MC) from (17.5) to (35):
[
DWL=\frac12 (35-17.5)\cdot ( \text{marginal benefit at } q=17.5 – MC )
]
But the exact numeric depends on which triangle boundaries are computed; in exams, you must match the diagram’s intercepts and slopes.

6) Elasticities and Policy Incidence

Elasticity determines who bears a tax

When a per-unit tax is imposed in a competitive market, equilibrium shifts. The key result: tax incidence depends on relative elasticities, not on statutory responsibility.

If demand is more inelastic than supply, consumers bear a larger share.
If supply is more inelastic than demand, producers bear more.

Exam-ready incidence intuition

Consider the extreme:

If demand is perfectly inelastic, quantity doesn’t change; the entire tax burden falls on consumers through higher consumer price.
If supply is perfectly inelastic, the entire tax burden falls on consumers? Actually, with perfectly inelastic supply, the consumer price changes more because quantity can’t adjust on the supply side; incidence splits accordingly. The safe method is: compute equilibrium with tax and compare price changes.

In computational exams, you’re often given linear demand and supply:

Inverse demand: (p= a-bq),
Inverse supply: (p=c+dq),
With tax (t): consumers pay (p_c), producers receive (p_p), with (p_c=p_p+t).
Solve:
[
a-bq = p_c,\quad p_p=c+dq,\quad p_c=p_p+t.
]
Then:
[
a-bq = c+dq+t \Rightarrow q = \frac{a-c-t}{b+d}.
]
Compute (p_c) and (p_p) and then incidence shares.

Consumer and producer surplus changes

After computing (p_c, p_p, q), you can compute:

CS drop: rectangle + triangle depending on which price changes occur.
PS drop similarly.
Tax revenue: (t\times q_t).
DWL: total CS + PS loss minus tax revenue.

This is a standard structure in intermediate exams; practising these steps is usually enough to secure marks.

University of Johannesburg — ECO2003F Intermediate Microeconomics Course Notes

University of Johannesburg course assessments frequently test the same core microeconomics tools, but with more emphasis on interpreting results from models rather than only computing them. Students often face questions that require: (i) choosing the correct equilibrium concept (competitive vs monopoly), (ii) reasoning about the sign of elasticities, (iii) understanding the difference between efficient and market outcomes under distortions, and (iv) handling corner solutions (e.g., non-negativity constraints leading to a “bang-bang” demand).

1) Utility, Indifference Curves, and Choice under Budget Constraints

Indifference curves and marginal rate of substitution (MRS)

A core diagram concept is:

Indifference curves represent bundles that yield the same utility.
MRS is the slope:
[
MRS_{xy}=\frac{MU_x}{MU_y}.
]
At optimum with an interior solution:
[
MRS_{xy}=\frac{p_x}{p_y}.
]
This is the “tangency condition” and a common exam mark allocation.

Corner solutions: when equalising marginal utilities fails

With constraints like (x\ge 0, y\ge 0), the optimum might be at a corner. This happens for:

perfect substitutes,
Leontief preferences,
or when FOCs suggest negative consumption.

Exam tip: check whether the derived interior solution violates non-negativity. If it does, re-solve under boundary conditions.

Perfect substitutes example

Let:
[
u(x,y)=x+2y.
]
The consumer chooses the good with higher utility per rand.
Compute unit “value per price”:

If price ratio makes (x) more attractive than (y), spend all on (x), so (y=0).
Otherwise spend all on (y), so (x=0).

This kind of question appears in multiple-choice or short-answer sections because it assesses conceptual understanding of preference geometry.

2) From Preferences to Demand: Substitution and Income Effects

Normal vs inferior goods

In intermediate courses, you are often asked to classify goods based on income elasticity or based on Slutsky decomposition.

A good (x) is:

normal if income increases raises demand ((\frac{\partial x}{\partial m}>0)),
inferior if the opposite.

For Cobb–Douglas with (x^*=\alpha m/p_x), (x) is normal.

Slutsky decomposition

When price (p_x) changes, the total effect splits:

Substitution effect (holding utility constant),
Income effect (due to changed purchasing power).

Even if full derivations are not required, you should know the direction:

For normal goods, substitution effect from a price increase reduces demand.
Income effect could offset, but typically makes inferior goods behave differently.

Example: Quasilinear preferences (simplification)

If:
[
u(x,y)=\sqrt{x}+y,
]
then (y) is like money, and the Hicksian demand often has simpler structure. Quasilinear models make consumer surplus equal to area under demand, reinforcing welfare calculation methods.

3) Duality: Expenditure Minimisation and Welfare

Expenditure function and indirect utility

The expenditure function (e(p,u)) gives the minimum spending needed to reach utility (u). The indirect utility function (v(p,m)) gives the maximum utility achievable.

Welfare changes such as:

compensating variation,
equivalent variation,
depend on comparing (e(p,u)) across price regimes.

Exam questions may ask: “Is CV larger than EV?” The correct answer depends on whether the change increases or decreases prices and the curvature properties (convexity of preferences). When uncertain, rely on the monotonic direction: increased prices reduce welfare, so both EV and CV reflect the required compensation to restore original utility.

Computation strategy

If given a specific functional form (like Cobb–Douglas), welfare calculations become more tractable:

Use known expenditure shares,
convert to money-metric changes,
compute areas if demand is linear.

4) Firm Theory: Competitive Supply and Shutdown

Deriving competitive supply from cost minimisation

A firm chooses (q) given price (p) to maximise profit:
[
\max_{q\ge 0} \left(pq – TC(q)\right).
]
In competitive equilibrium, firms supply where:
[
p=MC(q)
]
provided production is optimal and profitable enough relative to shutdown.

Shutdown rule

If costs include fixed costs (F), shutdown depends on variable cost:

Produce if (p \ge \min AVC(q)),
Shut down if (p < \min AVC(q)).

Shutdown questions test whether you can compute (AVC) and find its minimum, which often requires calculus or using given quadratic forms.

Example: shutdown with quadratic variable costs

Suppose variable cost:
[
VC(q)=q^2+4q.
]
Then:
[
AVC(q)=\frac{VC(q)}{q}=q+4\quad \text{for } q>0.
]
The minimum of (AVC) occurs as (q\to 0^+) (approaching 4). So shutdown occurs if (p<4). This shows how sometimes the minimum lies at the boundary rather than interior.

5) Monopoly, Market Power, and Pricing under Costs

Monopoly with constant marginal cost

In the linear-demand, constant-MC case above, you can compute monopoly quantity and price quickly using MR=MC.

However, intermediate exams often also include:

increasing marginal costs (MC(q)),
or given cost functions like (TC(q)=\frac{1}{2}q^2+10q+F).

In those cases, you:

compute (MC(q)),
set (MR(q)=MC(q)),
compute (p(q)),
compute profit:
[
\pi = p q – TC(q).
]

Deadweight loss comparison

A typical exam short-answer:

“Explain why monopoly generates DWL even if it maximises profit.”
Your response:
efficient allocation equates demand (marginal benefit) with social marginal cost,
monopoly equates MR with MC, but MR is below price by the amount of marginal effect of reduced quantity on revenue,
so quantity is lower than efficient quantity, creating DWL.

Example: compute profit and compare to perfect competition

If in perfect competition (p=MC), profit can be zero in long run if entry drives it. But with monopoly, profit is positive. Yet the positive profit does not imply efficiency; deadweight loss can still be large.

6) Regulation and Antitrust Instruments

Price regulation

Two common regulatory benchmarks:

Marginal-cost pricing (set (p=MC)): can replicate competitive output but may lead to losses if there are fixed costs.
Average-cost pricing (set (p=AC)): ensures break-even (zero economic profit) but distorts quantity relative to efficiency.

In exams, you might be asked:

Which policy yields higher welfare?
Answer depends on whether fixed costs are significant and on whether the regulator can set and commit to price accurately.

Taxes vs regulation

Consider:

a per-unit tax shifts supply up (for competitive markets) and can reproduce an externality-correcting wedge if set equal to marginal external damage.
a subsidy can correct positive externalities.

You should be able to state:

In externality settings, the “Pigouvian” policy equals the marginal external cost/benefit.

Even if externalities appear in a later section, regulation can be asked as an extension.

Cape Peninsula University of Technology (CPUT) — ECO2003F Intermediate Microeconomics Course Notes

CPUT modules often emphasise applications, especially interpreting microeconomic results in real-world settings relevant to South Africa: pricing of utilities, market power concerns, student budget constraints, and policy instruments like excise taxes or VAT-like price changes. While the math stays consistent, the exam sometimes tests your ability to connect formal outcomes to policy meaning.

1) Market Demand, Empirical Elasticities, and Interpretation

Elasticity signs and what they mean

A table of typical elasticity sign interpretations:

Elasticity	If value is…	Interpretation
(\varepsilon_{x,p_x})	negative	normal downward-sloping demand
(\varepsilon_{x,p_y})	positive	goods are substitutes
(\varepsilon_{x,p_y})	negative	goods are complements
(\varepsilon_{x,m})	positive	good is normal
(\varepsilon_{x,m})	negative	good is inferior

This is frequently assessed in conceptual questions.

Estimating and using elasticity with log-linear models

Sometimes exams use a demand model like:
[
\ln q = A + B \ln p
]
Then elasticity equals:
[
\varepsilon=-B
]
for own-price elasticity (depending on sign conventions). Being able to quickly interpret (B) is valuable.

2) Utility/Consumption with Budget Constraints in Context

Example scenario: transport and food budgets

A typical South African-student-style example: a student consumes transport (T) and food (F) with income (m). Let prices be (p_T, p_F). Demand depends on preferences and constraints.

If you assume Cobb–Douglas preferences:
[
u(T,F)=T^{\alpha}F^{1-\alpha},
]
then:
[
T^=\alpha \frac{m}{p_T},\quad F^=(1-\alpha)\frac{m}{p_F}.
]
This supports policy relevance:

If transport costs rise (increasing (p_T)), demanded transport declines proportionally if (\alpha) is fixed.
Food demand changes too, depending on spending shares.

Exams may ask “Which good shows a larger percentage reduction?” For Cobb–Douglas, percentage reduction in (T) is the same as percentage increase in (p_T) because elasticity is -1.

3) Welfare with Taxes: From Formal Areas to Real Policy

Specific tax and welfare components

A structured approach for any tax welfare question:

Compute pre-tax equilibrium (q_0, p_0).
Introduce tax (t): consumers pay (p_c), producers receive (p_p=p_c-t).
Solve new equilibrium (q_t).
Compute surplus changes:
- CS loss depends on area between demand and consumer price.
- PS loss depends on area between producer price and supply (or MC).
- Tax revenue: (t q_t).
Deadweight loss:
[
DWL = (\Delta CS + \Delta PS) – \text{Tax revenue}
]
noting that (\Delta CS) and (\Delta PS) are negative changes.

This method prevents common algebra mistakes.

Example: linear demand and supply with numeric parameters

Let:

Inverse demand: (p=60-2q).
Inverse supply (marginal cost): (p=10+q).

Without tax:
[
60-2q = 10+q \Rightarrow 50=3q \Rightarrow q_0=\frac{50}{3}\approx 16.67.
]
Price:
[
p_0 = 10+q_0 = 10+\frac{50}{3}=\frac{80}{3}\approx 26.67.
]

With tax (t=6) (specific tax):
Consumer price (p_c=p_p+6). Set:
[
60-2q = (10+q)+6 \Rightarrow 60-2q=16+q \Rightarrow 44=3q \Rightarrow q_t=\frac{44}{3}\approx 14.67.
]
Producer price:
[
p_p = 10+q_t = 10+\frac{44}{3}=\frac{74}{3}\approx 24.67.
]
Consumer price:
[
p_c = p_p+6 = \frac{74}{3}+6 = \frac{92}{3}\approx 30.67.
]

Now welfare:

CS before: triangle with base (q_0) and height (p_{\text{max}}-p_0), where (p_{\text{max}}=60).
[
CS_0=\frac12 q_0(60-p_0)=\frac12\cdot\frac{50}{3}\cdot\left(60-\frac{80}{3}\right)
=\frac{25}{3}\cdot\frac{100}{3}=\frac{2500}{9}\approx 277.78.
]
CS after: similarly,
[
CS_t=\frac12 q_t(60-p_c)=\frac12\cdot\frac{44}{3}\cdot\left(60-\frac{92}{3}\right)
=\frac{22}{3}\cdot\frac{88}{3}=\frac{1936}{9}\approx 215.11.
]
So:
[
\Delta CS = CS_t – CS_0 = \frac{1936}{9}-\frac{2500}{9}=-\frac{564}{9}=-62.67\ (\text{approx}).
]
PS before: triangle with base (q_0) and height (p_0-p_{\text{min supply}}), where supply intercept is 10.
[
PS_0=\frac12 q_0(p_0-10)=\frac12\cdot\frac{50}{3}\cdot\left(\frac{80}{3}-10\right)
=\frac{25}{3}\cdot\left(\frac{80-30}{3}\right)=\frac{25}{3}\cdot\frac{50}{3}=\frac{1250}{9}\approx 138.89.
]
PS after:
[
PS_t=\frac12 q_t(p_p-10)=\frac12\cdot\frac{44}{3}\cdot\left(\frac{74}{3}-10\right)
=\frac{22}{3}\cdot\left(\frac{74-30}{3}\right)=\frac{22}{3}\cdot\frac{44}{3}=\frac{968}{9}\approx 107.56.
]
So:
[
\Delta PS = \frac{968}{9}-\frac{1250}{9}=-\frac{282}{9}=-31.33\ (\text{approx}).
]
Tax revenue:
[
TR=t q_t = 6\cdot \frac{44}{3} = 88.
]

Total welfare change:
[
\Delta CS + \Delta PS + TR = (-62.67) + (-31.33) + 88 = -6.00.
]
Thus deadweight loss:
[
DWL = 6.00.
]

This computation is the kind of end-to-end workflow that examiners reward: correct equilibrium, correct price paid/received, correct areas.

4) Externalities and Policy Calibration (Common Intermediate Extension)

Even where externalities appear briefly, exams often include at least one question linking costs to welfare.

Negative externality

With negative externality:

private marginal cost (MC_p),
social marginal cost (MC_s = MC_p + MEC) (marginal external cost).

Competitive equilibrium equates (p=MC_p), but efficient outcome equates:
[
p = MC_s.
]
A Pigouvian tax (t) equal to marginal external damage at the efficient quantity can shift the equilibrium towards efficiency.

Positive externality

With positive externality:

(MB_s = MB_p + MEB),
and a subsidy can increase output to the socially efficient level.

Exam reasoning template

State private vs social marginal conditions.
Identify inefficiency direction:
- negative externality → overconsumption,
- positive externality → underconsumption.
Match policy:
- negative externality → tax,
- positive externality → subsidy.
Explain welfare effect qualitatively if numeric data absent.

5) Practical South African Context: Interpreting Market Power and Regulation

Utilities and pricing

Many real markets resemble imperfect competition: regulated utilities, network industries, or monopolistic service provision. Intermediate micro helps evaluate:

when prices exceed marginal costs,
when regulation needs to correct welfare loss,
when subsidies may be equity-driven.

Tuition, fees, and student choice (micro interpretation)

Even without modelling education as a firm, you can interpret a student’s choice of course load (goods are courses/time commitments) through:

budget constraints (time, money),
opportunity costs,
and trade-offs (substitution between leisure and study hours).

This is conceptual; the mathematical demand model still teaches how budget and prices reallocate consumption.

North-West University — ECO2003F Intermediate Microeconomics Course Notes

North-West University assessments frequently reward disciplined derivations and strong diagram discipline. Students often lose marks for sign errors, mixing up inverse demand vs demand, or forgetting to check assumptions (interior vs corner solutions; marginal vs average relationships). These notes therefore focus on exam-proof methods: a consistent “logic chain” from assumptions to results.

1) Graph Skills for Exam Answers

Building and using standard diagrams

You should be able to draw quickly and label:

Demand curve (downward sloping).
Supply curve (upward sloping).
Tax wedge:
- shift between consumer price and producer price.
Monopoly outcome:
- MR below demand,
- MC intersects MR at monopoly quantity,
- monopoly price on demand at that quantity.
Deadweight loss:
- between demand and MC across the quantity reduction.

Examiners expect correct shading and area identification.

Common diagram pitfalls

Using demand slope where MR should be.
Confusing intercepts when computing CS and PS.
Forgetting that tax creates two prices: one for consumers and one received by producers.

2) Step-by-Step Solution Method for Equilibrium with Taxes

General linear model framework

Assume:

Inverse demand: (p = a – bq), (a,b>0).
Inverse supply: (p = c + dq), (c,d>0).
With a per-unit tax (t):
[
p_c = a-bq,\quad p_p = c+dq,\quad p_c = p_p + t.
]
So:
[
a – bq = c + dq + t
\Rightarrow a – c – t = (b+d)q
\Rightarrow q_t = \frac{a-c-t}{b+d}.
]
Then:
[
p_p = c + d q_t,\quad p_c = p_p + t.
]

Surplus and DWL with areas

After computing (q_t), the DWL can be computed as:
[
DWL = \frac12 \cdot (t) \cdot (\Delta q),
]
where (\Delta q = q_0 – q_t), if demand and supply are linear and the wedge is measured per unit. In more general settings, you compute via area triangles carefully. For linear models, this shortcut is often consistent with exam marks.

3) Monopoly Computation Template

Linear demand and general cost

Suppose:

demand: (p=a-bq),
cost: (TC(q)) known.

Steps:

Compute (TR(q) = p(q)q = (a-bq)q).
Compute (MR(q)=\frac{dTR}{dq}).
Compute (MC(q)=\frac{dTC}{dq}).
Solve (MR(q)=MC(q)) for (q_M).
Compute (p_M = a-bq_M).
Compute profit (\pi = p_M q_M – TC(q_M)).

Check reasonableness

Ensure:

(q_M\ge 0),
monopoly quantity less than competitive quantity (if demand downward sloping and MC upward/constant). If your result violates that, check arithmetic.

4) Welfare with Monopoly vs Competitive

Efficient benchmark

Efficient quantity (in a standard model with no externalities and with marginal benefit equal to demand) in a competitive environment typically satisfies:
[
p = MC(q),
]
while monopoly satisfies:
[
MR(q)=MC(q).
]
Because (MR < p), monopoly picks lower quantity.

Deadweight loss logic

DWL is the loss from trades between:

(q_M) (monopoly output) and
(q_C) (competitive output).

In linear models:

You can compute DWL as a triangle using base ((q_C-q_M)) and height ((\text{demand price at }q_M – MC)) or equivalently the difference between marginal benefit and marginal cost at the cutoff.

5) Two-Part Tariffs and Nonlinear Pricing (Often Appears)

Even in intermediate courses, nonlinear pricing may be used as an application of marginal concepts and consumer participation.

A two-part tariff charges:

an entry fee (A),
plus a per-unit price (p).

If the per-unit price equals marginal cost, and the entry fee extracts consumer surplus up to the point where consumer utility equals reservation utility (often zero), the firm can:

increase revenue,
potentially achieve output close to efficient levels while capturing surplus.

In exams, you may be asked:

choose (A) such that consumer’s net utility is non-negative,
compute firm profit given (q) under per-unit pricing.

6) Practice Problems Embedded in Exam Form

Below are “exam-like” exercises that consolidate the skills across sections.

Problem 1: Derive Marshallian demand (Cobb–Douglas)

Consumer:
[
u(x,y)=x^{\alpha}y^{1-\alpha},\quad 0<\alpha<1,
]
income (m), prices (p_x,p_y).
Task: derive (x^) and (y^).

Solution outline:

Set Lagrangian (\mathcal{L}=x^{\alpha}y^{1-\alpha}+\lambda(m-p_x x-p_y y)).
FOCs:
[
\alpha x^{\alpha-1}y^{1-\alpha}=\lambda p_x,\quad (1-\alpha)x^\alpha y^{-\alpha}=\lambda p_y.
]
Divide FOCs:
[
\frac{\alpha}{1-\alpha}\cdot \frac{y}{x}=\frac{p_x}{p_y}
\Rightarrow \frac{y}{x}=\frac{1-\alpha}{\alpha}\cdot \frac{p_x}{p_y}.
]
Use budget (p_x x + p_y y=m) to solve for (x,y). The resulting shares are:
[
x^=\alpha \frac{m}{p_x},\quad y^=(1-\alpha)\frac{m}{p_y}.
]

Problem 2: Consumer surplus change from price increase

Demand: (p=50-2q). Initial price (p_0=30), tax or increase leads to (p_1=35).
Find CS before and after.

Step:

Find quantities:
(q_0=(50-30)/2=10).
(q_1=(50-35)/2=7.5).
CS uses triangle area:
[
CS=\frac12 q (p_{\max}-p),
]
where (p_{\max}=50).
Compute:
[
CS_0=\frac12\cdot 10\cdot(50-30)=5\cdot 20=100,
]
[
CS_1=\frac12\cdot 7.5\cdot(50-35)=3.75\cdot 15=56.25.
]
Change:
[
\Delta CS=-43.75.
]

Problem 3: Tax incidence via elasticities (conceptual)

Suppose demand is more inelastic than supply.
Task: who bears more of a per-unit tax?
Answer: consumers bear the larger share because quantity falls less when demand is inelastic, so the consumer price rises more relative to the producer price.

Problem 4: Competitive equilibrium with given functions

Inverse demand: (p=120-3q). MC: (MC(q)=20+q).
Find competitive (q^) and (p^).

Set (p=MC):
[
120-3q=20+q \Rightarrow 100=4q \Rightarrow q^=25,\quad p^=120-3(25)=45.
]

Cross-Institution Master Checklist (Applies to ECO2003F Exams)

Even though these notes reference particular institutions, the exam skills are transferable. Use the checklist below as a final revision tool.

Consumer theory essentials

Can you state and use budget constraints correctly?
Can you compute MRS = price ratio at optimum?
Can you derive Cobb–Douglas demand (spending shares)?
Can you classify goods (normal/inferior) from income effects or elasticity signs?
Can you compute consumer surplus under linear demand?

Producer theory essentials

Can you compute (AC) and (MC) from a given (TC)?
Can you apply shutdown rule using (AVC) minimum?
Can you compute profit and identify when profit is zero vs positive?

Market equilibrium and policy essentials

Can you solve competitive equilibrium (D(p)=S(p)) or (p=MC)?
Can you solve equilibrium under a per-unit tax with two prices (p_c) and (p_p)?
Can you compute CS, PS, tax revenue, and DWL using correct areas?
Do you explain tax incidence using elasticities?

Market power essentials

Can you compute MR from linear demand?
Can you solve (MR=MC) and find monopoly price?
Can you explain DWL and why monopoly output is too low (relative to efficiency)?

Final Exam Strategy: How to Score Marks Efficiently

Intermediate microeconomics exams often include a mix of short conceptual questions and long calculation problems. The marks are usually allocated for method correctness, not just final numbers. A disciplined approach increases marks and reduces careless errors.

1) Start by identifying the model type

Before any algebra:

Is the market competitive (price taker) or monopoly (MR logic)?
Is the question about demand or welfare (surplus areas)?
Is there a tax/regulation (policy wedge: tax creates two prices)?

2) Write the key conditions first

Examples:

Competitive firm: (p=MC) (and shutdown check).
Monopoly: (MR=MC).
Consumer optimum: (MRS=\frac{p_x}{p_y}).

3) Then compute step-by-step

Solve for quantity.
Then price(s).
Then surplus areas.
Then interpret (welfare loss, incidence, efficiency comparison).

4) Interpret signs and magnitudes

Price up → demand down (should be true for normal demand).
Tax increases consumer price and decreases producer price.
Monopoly: price higher, quantity lower.

5) Always label clearly

In diagrams and written answers, label:

intercepts,
slopes,
consumer vs producer prices,
and which triangle is which (CS vs PS vs DWL).

This clarity is often the difference between partial and full credit when examiners can see you understand the structure even if a number is slightly off.

Summary of core outcomes to remember

Cobb–Douglas demand: constant spending shares: (x^=\alpha m/p_x), (y^=(1-\alpha)m/p_y).
Competitive firm output: (p=MC(q)), with production only if (p \ge \min AVC).
Monopoly output: (MR(q)=MC(q)), with (MR<p) for downward sloping demand.
Tax welfare accounting:
- tax revenue = rectangle (tq),
- DWL = triangles created by reduced quantity,
- incidence depends on relative elasticities.
Efficiency benchmark:
- without distortions, competitive equilibrium tends to be efficient,
- with externalities, efficiency requires aligning private and social marginal conditions (taxes/subsidies).

These tools form the backbone of ECO2003F-style intermediate microeconomics questions across South African universities and TVET-connected pathways.