Microeconomics

Introduction

Microeconomics is the study of how individuals and firms make decisions in a world of scarcity. Every agent has limited resources. Consumers have a limited budget. Producers have limited time, capital and technology. Every choice is therefore a trade-off.

The notes describe microeconomics as a series of constrained optimization exercises. Consumers and firms try to make themselves as well off as possible, given their constraints. Microeconomics deals with individual decision units and the exchange between them. Macroeconomics deals with aggregates such as economic growth, booms and busts, and unemployment.

Opportunity cost is the value of the next best alternative that is given up. Every action and every inaction has a cost, because the agent could have done something else instead. Nothing is free; the notes connect this idea to the nickname “dismal science”. In the course, price and opportunity cost are treated as the same thing.

The course assumes that people are motivated by self-interest and act with rationality. Deviations from this model, studied by behavioural economics, exist, but the basic model is learned first.

Markets

A Market is a collection of buyers and sellers who, through actual or potential interaction, determine the price of highly interchangeable products. On the buying side, consumers buy goods and firms buy labour and other inputs. On the selling side, consumers sell labour, resource owners sell inputs, and firms sell goods. A price is the rate at which money is swapped for a good.

The extent of a market ties it to one group of closely related products sold within a geographic boundary. Whether fast-food restaurants in Rome and Milan form one market depends on whether buyers substitute between them. The notes distinguish three types of market structure by the number of sellers:

Competitive market: many firms sell to many consumers.
Monopoly: one firm sells to many consumers.
Oligopoly: a few firms sell to many consumers.

The course answers three questions about markets:

How are the market price and quantity determined? (See market equilibrium.)
Is the market outcome efficient, that is, does it maximise the benefits for all agents? (See economic efficiency.)
When and how should the government intervene, for example with taxes, subsidies or regulation?

Models

An economic model is a description of the relation between two or more economic variables. Models are not laws. They rest on simplifying assumptions and are never fully true, although they are usually close to the truth. The design trade-off is tractability, meaning few steps, against realism. Economics errs on the side of tractability: all models are wrong, but some are useful.

The scientific method of economics runs in a loop: observation, then theory in the form of a model, then extra implications, then a test with new observations, then refinement. A quantitative model gives numbers, for example a 10% rise in costs leading to a 5% rise in price. A qualitative model gives only directions, for example higher costs leading to a higher price.

% Δ costs = + 10 % \Rightarrow % Δ P = + 5 %

The MIT course asks for every model at three levels: intuitive (it can be explained to a non-economist), graphical and mathematical. The mathematics is the least important level but the easiest to test.

Positive analysis studies the way things are. Normative analysis studies the way things should be. Normative analysis requires positive analysis first. A free-market outcome may still be undesirable for three standard reasons: market failure such as fraud and imperfect information, equity or fairness, and behavioural mistakes.

In a capitalist economy, firms and individuals decide what to produce and consume, subject to rules set by the government. In a command economy, the government makes the production and consumption decisions. Adam Smith’s invisible hand states that consumers and firms serving their own interest end up doing what is best for society. Here “best” means maximum economic surplus: the largest amount of goods that people value gets produced.

Supply and Demand

Adam Smith’s water–diamond paradox motivates the topic. Water is essential and free. Diamonds are frivolous and expensive. Prices cannot be explained by demand alone, which measures how much people want a good, or by supply alone, which measures how much of it there is. Both are needed, like the two blades of a pair of scissors: the supply and demand scissors.

Demand

The demand curve shows, at each price, how much buyers want to buy, holding fixed all other factors that affect demand. It has a negative slope. As the price goes up, the quantity demanded goes down, because buying is less attractive at a high price. By convention, price $P$ is on the vertical axis and quantity $Q$ on the horizontal axis. Historically price was drawn as if it were the dependent variable. In the model, price is the independent variable and quantity the dependent one.

A change in quantity demanded occurs when the price changes. It is a movement along the curve.
A change in demand occurs when anything else changes. The whole curve shifts.

The determinants of demand, or shifters, are consumer tastes and trends, income, population, the prices of related goods, and taxes and regulation. Related goods and income classify goods as follows:

Substitutes: when the price of the other good goes up, demand for this good goes up. Example: iPhone and Samsung phones.
Complements: when the price of the other good goes up, demand for this good goes down. Example: petrol and cars.
Normal good: when income goes up, quantity demanded goes up.
Inferior good: when income goes up, quantity demanded goes down.

The demand function writes quantity demanded as a function of price and of the other factors:

Q^{d} = D (P, other factors)

In the linear case, demand is $Q d = A - BP$ . Solving for price gives inverse demand. Its vertical intercept $A/B$ is the choke price, its horizontal intercept is $A$ , and its slope is $-1/B$ .

Q^{d} = A - B P

P = \frac{A}{B} - \frac{1}{B} Q^{d}

The signs in a demand function show how goods are related. The notes read the following demand for corn, where $M$ is income:

Q_{corn} = 5 - 2 P_{corn} + 4 P_{potatoes} - 0.25 P_{butter} + 0.0003 M

$-2P corn$ : the own-price coefficient. It is always negative.
$+4P potatoes$ : a plus sign on the price of another good. Potatoes are a substitute for corn.
$-0.25P butter$ : a minus sign on the price of another good. Butter is a complement to corn.
$+0.0003M$ : a plus sign on income. Corn is a normal good.

Supply

The supply curve shows, at each price, how much sellers want to sell, holding fixed all other factors that affect supply. It has a positive slope. A higher price means more money per unit, so firms want to produce more. The curve does not start at the origin: below some price, the good costs more to make than it sells for (see costs).

A change in quantity supplied occurs when the price changes. It is a movement along the curve.
A change in supply occurs when anything else changes. The curve shifts. Good news for sellers, such as cheaper inputs, better technology or subsidies, shifts supply to the right. Bad news, such as higher input costs or taxes, shifts it to the left.

The determinants of supply are technology, input costs (labour, capital and raw materials), taxes and subsidies, and the number of firms. In the linear case, supply is $Q s = CP - D$ . The inverse supply function has slope $1/C$ .

Q^{s} = C P - D

P = \frac{D}{C} + \frac{1}{C} Q^{s}

The notes summarise the shifters of the demand curve and the supply curve:

Demand shifters: income $M$ ; a rise in the price of a substitute (+); a rise in the price of a complement (−); trends, tastes, population and taxes.
Supply shifters: better technology; input costs; subsidies (+) and taxes (−); the number of firms.

Market equilibrium

Market equilibrium is the price at which quantity demanded equals quantity supplied. Graphically, it is where the demand curve and the supply curve cross. It is the one point where both consumers and producers are happy to make the transaction, so there is no pressure to move. To find it, set $Q d (P) = Q s (P)$ , solve for the equilibrium price $P*$ , and substitute back to get the equilibrium quantity $Q*$ .

Q^{d} (P^{*}) = Q^{s} (P^{*})

Market demand and market supply are the horizontal sums of the individual curves (horizontal summation): at each price, the individual quantities are added.

Q^{D} = \sum_{i} q_{i}^{d}

Q^{S} = \sum_{j} q_{j}^{s}

Comparative statics

Comparative statics changes one thing and then compares the old equilibrium with the new one. It is “comparative” because it compares two equilibria. It is “static” because it does not study the path between them.

When one curve shifts, the direction of change is clear. A shift of demand moves price and quantity in the same direction. A shift of supply moves them in opposite directions.

Demand rises: price up, quantity up.
Demand falls: price down, quantity down.
Supply rises: price down, quantity up.
Supply falls: price up, quantity down.

When both curves shift, one variable can always be signed. The other depends on which shift is bigger, so it is ambiguous.

Demand up, supply up: price ambiguous, quantity up.
Demand down, supply down: price ambiguous, quantity down.
Demand up, supply down: price up, quantity ambiguous.
Demand down, supply up: price down, quantity ambiguous.

Elasticity

The slope of a curve shows how sensitive quantity is to price, but slope depends on units: kilos or tonnes, euros or cents. Elasticity is a pure number. It is the percentage change in one variable for a one-percent change in another, so it does not depend on units. The elasticity of $X$ with respect to $Y$ measures the responsiveness of $X$ to small changes in $Y$ . The concept applies to demand, supply and income.

Price elasticity of demand

The price elasticity of demand $ε D P$ is the percentage change in quantity demanded divided by the percentage change in price. For a discrete change, the percentage changes use the old values $P 0$ and $Q 0$ . For a small change, the ratio becomes a partial derivative times the ratio of initial price to initial quantity.

ε_{P}^{D} = \frac{% Δ Q^{d}}{% Δ P} = \frac{\frac{Q_{1} - Q_{0}}{Q_{0}}}{\frac{P_{1} - P_{0}}{P_{0}}}

ε_{P}^{D} = \frac{\partial Q^{d}}{\partial P} \cdot \frac{P_{0}}{Q_{0}}

With linear demand $Q d = a - bP$ , the derivative is $\partialQ d /\partialP = -b$ , so the elasticity at the initial point is:

ε_{P}^{D} = - b \frac{P_{0}}{Q_{0}}

Demand slopes down, so $ε D P \leq 0$ . The Bocconi notes write the categories with the sign; the MIT wording often uses the absolute value. Each category describes what a 1% price rise does:

Elastic demand: $ε < -1$ , that is $|ε| > 1$ . Quantity falls by more than 1%.
Inelastic demand: $-1 < ε < 0$ . Quantity falls by less than 1%.
Unit elastic demand: $ε = -1$ . Quantity falls by exactly 1%.
Perfectly inelastic demand: $ε = 0$ . The curve is vertical and quantity does not change.
Perfectly elastic demand: $ε = -\infty$ . The curve is horizontal and consumers stop buying.

Substitutability determines elasticity. The more substitutes a good has, the more elastic its demand. Luxuries, goods sold in competitive markets with many close substitutes, and goods that take a large share of the budget have elastic demand. Necessities with no plausible substitute, such as insulin for a diabetic, are close to perfectly inelastic. Goods with perfect substitutes, such as two brands of gum, are close to perfectly elastic: a price one cent higher loses the whole market.

Elasticity changes along a linear demand curve. Since $ε = -bP/Q$ , demand is elastic at a high price, where $Q$ is small, and inelastic at a low price, where $Q$ is large. It is unit elastic at the midpoint. A linear demand curve is therefore not a constant-elasticity curve.

Elasticity and total expenditure

Total expenditure $TE$ is price times quantity. It equals the total revenue of sellers before taxes. In percentage terms, the change in total expenditure is approximately the sum of the percentage changes in price and quantity. The sign of the change in $TE$ therefore depends on which percentage change is bigger, that is, on the price elasticity of demand.

TE = P \cdot Q

% Δ TE \approx % Δ P + % Δ Q

Elastic demand ( $|ε| > 1$ ): a price rise lowers $TE$ and a price fall raises it, because quantity moves more than price in percentage terms.
Inelastic demand ( $|ε| < 1$ ): a price rise raises $TE$ and a price fall lowers it, because quantity moves less than price.
Unit elastic demand: $TE$ does not change, because the two effects cancel.

Total expenditure reaches its maximum at the point where demand is unit elastic. On a linear demand curve this point is the midpoint.

Other elasticities

The income elasticity of demand $ε D M$ measures how quantity demanded responds to income $M$ . The cross-price elasticity of demand measures how the demand for good $x$ responds to the price of good $y$ . Both follow the pattern of the price elasticity: a derivative times a ratio of initial values.

ε_{M}^{D} = \frac{% Δ Q^{d}}{% Δ M} = \frac{\partial Q^{d}}{\partial M} \cdot \frac{M_{0}}{Q_{0}}

ε_{p_{y}}^{D_{x}} = \frac{% Δ Q_{x}^{d}}{% Δ p_{y}} = \frac{\partial Q_{x}^{d}}{\partial p_{y}} \cdot \frac{P_{y, 0}}{Q_{x, 0}}

$ε D M > 0$ : a normal good. Within normal goods, the MIT wording separates a luxury good ( $ε D M > 1$ ), whose budget share rises with income, from a necessity good ( $0 < ε D M < 1$ ), whose spending rises while its budget share falls.
$ε D M < 0$ : an inferior good.
Positive cross-price elasticity: the goods are substitutes.
Negative cross-price elasticity: the goods are complements.

The price elasticity of supply $ε S P$ is non-negative, because supply slopes up. Supply is elastic if the value is above 1, inelastic if below 1, and unit elastic if equal to 1. Perfectly elastic supply is horizontal, with elasticity $\infty$ . Perfectly inelastic supply is vertical, with elasticity $0$ : the seller offers a fixed quantity whatever the price.

ε_{P}^{S} = \frac{% Δ Q^{s}}{% Δ P} = \frac{\partial Q^{s}}{\partial P} \cdot \frac{P_{0}}{Q_{0}^{s}} \geq 0

Worked example

Take demand $Q d = 100 - 4P$ at the initial price $P 0 = 10$ . The initial quantity is 60. The slope $\partialQ d /\partialP$ is −4, so the price elasticity is about −0.67 and demand is inelastic at this point. The expenditure rule then predicts that a price rise raises total expenditure. At a price of 11, quantity is 56, and total expenditure rises from 600 to 616.

Q_{0} = 100 - 4 \cdot 10 = 60

ε_{P}^{D} = - 4 \cdot \frac{10}{60} \approx - 0.67

{TE}_{0} = 10 \times 60 = 600

{TE}_{1} = 11 \times 56 = 616

Preferences and Utility Functions

The demand curve comes from utility maximization, which has two components. Preferences describe what people want; prices and income do not enter them. The budget constraint describes what people can afford. The consumer maximises happiness, given preferences, subject to the budget constraint. The notes call this topic the “you won the lottery” lecture: it covers only preferences, with no constraint yet.

The consumer’s problem in four steps

The Bocconi course frames consumer choice as four steps. Each step builds on the one before.

Preferences: rank all the alternatives. A higher place in the ranking gets a higher number.
Budget constraint: given prices and income $M$ , find the affordable bundles.
Choice: pick the highest-ranked bundle among the affordable ones.
Demand: change one price, hold the rest fixed, repeat step 3, and trace the demand curve.

A consumption bundle is a list of quantities of goods. The notation $A ≻ B$ states a strict preference: $A$ is preferred to $B$ . The notation $A \sim B$ states indifference between the two bundles.

A = (x_{A}, y_{A})

Preference assumptions

Models rely on simplifying assumptions. The question is not whether the assumptions are true but whether they are sensible, that is, roughly consistent with reality. The Bocconi course states three principles:

Ranking Principle: the agent can rank all bundles according to his preferences. It combines completeness and transitivity.
Choice Principle: among the affordable bundles, the agent picks the one ranked highest. This principle describes behaviour, not preferences.
More-is-better, or non-satiation: more is always better than less.

The Ranking Principle has two parts:

Completeness: for any two bundles, the agent can say $A ≻ B$ , $B ≻ A$ or $A \sim B$ . The agent may say “I don’t care”, which is indifference. The agent may never say “I don’t know”.
Transitivity: if $A ≻ B$ and $B ≻ C$ , then $A ≻ C$ .

A ≻ B and B ≻ C \Rightarrow A ≻ C

The Bocconi notes add that more-is-better is not a rationality principle. It is only a regularity that people usually follow. It rests on free disposal: the agent can throw away unwanted units at no cost, so an extra unit can never hurt. The principle ranks a bundle with more of both goods above another. It cannot rank a bundle with more of one good and less of the other. It fails for goods that must be combined in fixed doses, such as coffee and sugar.

The MIT course states three assumptions on preferences instead: completeness, transitivity and non-satiation. Completeness and transitivity together form the Ranking Principle, and non-satiation is more-is-better. The Choice Principle has no MIT counterpart, because MIT folds it into maximising utility subject to the budget. Non-satiation is where the economics comes in: it gives the model its power. It does not say that the next unit gives as much happiness as the last one; that is the question of diminishing marginal utility. It only says that the agent always wants more.

Indifference curves

An indifference curve (IC) is the set of all consumption combinations among which the agent is indifferent. It is a graphical map of preferences. In the notes’ example, bundle $A$ has 2 pizzas and 1 cookie, $B$ has 1 pizza and 2 cookies, and $C$ has 2 pizzas and 2 cookies. The agent is indifferent between $A$ and $B$ , which lie on the same IC. $C$ is preferred by more-is-better and lies on a higher IC. Everything above and to the right of an IC is better. Everything below and to the left is worse.

The family of indifference curves is the set of all ICs of one agent over bundles of $x$ and $y$ . Each of its four properties follows from one assumption:

The agent prefers higher ICs. This follows from more-is-better.
ICs slope downward. An upward slope would mean that more of both goods leaves the agent indifferent, which violates more-is-better. ICs are also thin: a thick curve would contain two bundles, one with more of both goods.
ICs never cross. A crossing would make one bundle indifferent to two bundles that are not indifferent to each other. This violates transitivity.
Exactly one IC passes through every bundle. This follows from completeness: the agent always knows how he feels about a bundle.

The Bocconi notes classify the shapes of indifference curves by the preferences behind them:

Convex ICs describe convex preferences, the standard case. People prefer a mix to extremes, for example variety in food. The curves bow toward the origin.
Concave ICs describe people who prefer extremes, for example one very good bag rather than two average bags. The course only asks to recognise this case.
If one good has no value, the ICs are vertical (good $y$ has no value) or horizontal (good $x$ has no value).
Perfect complements, such as right and left shoes, give L-shaped ICs, with $U = min(x, y)$ .
Perfect substitutes, where only the total quantity matters, give straight-line ICs, with $U = ax + by$ .
A bad lowers wellbeing when consumed, for example pollution or work. With a bad on one axis, ICs slope upward: more of the good is needed to compensate for more of the bad.

U = \min (x, y)

U = a x + b y

Rate of substitution and MRS

The rate of substitution between two bundles $A$ and $B$ on the same indifference curve is the number of units of $y$ that the agent is willing to give up to get some extra $x$ and stay indifferent. A high rate of substitution means that the agent values $x$ highly relative to $y$ . It is the subjective value of $x$ in terms of $y$ . In the notes’ example, the move from $A = (2, 3)$ to $B = (3, 2)$ gives a rate of 1.

- \frac{Δ y}{Δ x}

- (\frac{- 1}{1}) = 1

The marginal rate of substitution $MRS x,y$ is the rate at which the agent is willing to give up $y$ to get a small extra amount of $x$ , staying on the same IC. It is the absolute value of the slope of the IC at a point. It also equals the ratio of the marginal utilities.

{MRS}_{x, y} = - \frac{d y}{d x} |_{U = \bar{U}} = \frac{{MU}_{x}}{{MU}_{y}}

The Bocconi notes write the MRS as a positive number, the absolute slope. The MIT notes write it as the slope itself, $-MU x /MU y$ . The object is the same, so the sign convention of each question matters.

Along a convex IC the MRS falls as $x$ increases. This is a diminishing marginal rate of substitution. An agent with a lot of pizza and few cookies gives up a lot of pizza for one cookie. An agent with a lot of cookies gives up very little. Convexity of ICs, which reflects convex preferences, is the same thing as diminishing MRS. Between agents, a steeper IC at a bundle means a higher MRS: that agent values $x$ more relative to $y$ . If $MRS Ann > MRS Mark$ , Ann likes $x$ more.

Utility functions

A utility function is a mathematical representation of preferences. It assigns a number $U(x, y)$ to every bundle, aggregating and weighting its items so that bundles can be ranked. It must give the same number to all bundles on the same IC and a higher number to bundles on a higher IC. A function that does both preserves the ranking.

Utility is ordinal, not cardinal: the numbers only rank bundles. This is the idea of ordinal utility, as opposed to cardinal utility. “Utils” do not exist. It is not possible to say that one agent’s utility is 20% higher than another’s, and utility cannot be added across people; demand curves are added instead. Any increasing monotonic transformation of $U$ represents the same preferences.

To get an IC from a utility function, fix $U = Ū$ and solve for $y$ . With $U(M, B) = M + 2B$ , the IC is a straight line, which means that the goods are perfect substitutes.

U (M, B) = M + 2 B

B = \frac{\bar{U} - M}{2}

Marginal utility $MU x$ is the additional utility from one more unit of $x$ , holding the quantity of every other good constant. It is the utility of the next unit, given the units already consumed. Economics studies the marginal decision: “do you want the next cookie?” is easier to answer than “how many cookies do you want?”.

{MU}_{x} = \frac{\partial U}{\partial x}

Diminishing marginal utility means that $MU$ falls as consumption of a good rises. Each unit still adds happiness, by more-is-better, but less than the one before. With $U = \sqrt(PC)$ and $P = 2$ pizzas, the first cookie adds 1.41, the second 0.59 and the third 0.45. Marginal utility stays positive; it diminishes.

U = \sqrt{P C}

The link between the MRS and marginal utilities follows from the fact that utility does not change along an IC. The total differential of utility is zero, and rearranging gives the MRS:

{MU}_{x} d x + {MU}_{y} d y = 0

- \frac{d y}{d x} = \frac{{MU}_{x}}{{MU}_{y}}

Diminishing marginal utility in each good makes the MRS fall along the curve: as $x$ grows, $MU x$ falls and $MU y$ rises.

Cobb–Douglas utility

The Cobb–Douglas utility function is the one met most often in the course. It represents convex preferences. Its marginal utility in each good and its MRS are:

U (x, y) = x^{a} y^{b}

{MU}_{x} = a x^{a - 1} y^{b}

{MU}_{y} = b x^{a} y^{b - 1}

{MRS}_{x, y} = \frac{a}{b} \cdot \frac{y}{x}

For example, with $U = XY 4$ the exponents are $a = 1$ and $b = 4$ , so:

MRS = \frac{1}{4} \frac{Y}{X}

Constraints, Choices and Demand

This topic completes the consumer’s problem. It adds the budget constraint to preferences, finds the optimal choice, and derives the demand curve. It then splits the response to a price change into a substitution effect and an income effect.

The budget constraint

For most of the course, spending equals income: there is no saving and no borrowing. A bundle $(x, y)$ is affordable if it costs no more than income $M$ . The budget constraint and the budget line are:

p_{x} x + p_{y} y \leq M

p_{x} x + p_{y} y = M

y = \frac{M}{p_{y}} - \frac{p_{x}}{p_{y}} x

The intercepts are $M/p x$ , where all income goes on $x$ , and $M/p y$ . The slope is $-p x /p y$ , the negative of the price ratio. The budget set is the triangle of all bundles with $p x x + p y y \leq M$ . The budget line holds the bundles that cost exactly $M$ . The MIT course calls the set of choices available at given income and prices the opportunity set.

The slope is the rate at which the market lets the consumer trade $y$ for $x$ : every extra unit of $x$ costs $p x /p y$ units of $y$ . MIT names it the marginal rate of transformation (MRT). It is the opportunity cost of $x$ in terms of $y$ . The MRS is what the consumer wants; the MRT is what the market allows.

MRT = - \frac{p_{x}}{p_{y}}

The comparative statics of the budget line are as follows:

Income changes: a rise in $M$ shifts the line out in parallel; a fall shifts it in. The slope does not change: the market controls the slope, and income controls the level.
$p x$ changes: the $y$ -intercept stays fixed and the line rotates (pivots) around it. A rise in $p x$ rotates it inward and makes it steeper; a fall rotates it outward and makes it flatter.
$p y$ changes: the $x$ -intercept stays fixed and the line rotates around it.

Example: with $p x = 4$ , $p y = 2$ and $M = 200$ , the budget line has slope $-4/2 = -2$ and intercepts $x = 50$ and $y = 100$ .

4 x + 2 y = 200

Optimal choice

The Choice Principle picks the preferred bundle among the affordable ones. In the notes’ example, the ranking is $C ≻ B ≻ D ≻ A ≻ E$ and only $B$ , $A$ and $E$ are affordable. The choice is $B$ .

C ≻ B ≻ D ≻ A ≻ E

Graphically, more-is-better puts the chosen bundle on the budget line, so it costs all of $M$ . The consumer wants the highest indifference curve he can reach. That IC is at a tangency with the budget line: the slope of the IC equals the slope of the budget line. With convex ICs, the interior solution solves two equations in two unknowns $(x*, y*)$ : the tangency condition below and the budget line $p x x + p y y = M$ .

{MRS}_{x, y} = \frac{p_{x}}{p_{y}} \Leftrightarrow \frac{{MU}_{x}}{p_{x}} = \frac{{MU}_{y}}{p_{y}}

MIT calls the tangency condition the fundamental equation of consumer choice: the MRS equals the MRT. The marginal benefit, what the consumer wants, equals the marginal cost, what the constraint allows. The rearranged form is the bang-for-the-buck equation: the marginal utility per euro must be equal across goods.

MRS = MRT

If $MU x /p x > MU y /p y$ , or equivalently $MRS > p x /p y$ , the next euro gives more utility when spent on $x$ , so the consumer buys more $x$ and less $y$ . If $MRS < p x /p y$ , the consumer buys less $x$ and more $y$ . At the optimum the consumer is indifferent about where the last euro goes.

The notes’ choice diagram shows three ICs. The tangency bundle $D$ lies on the highest IC that touches the budget line. Bundle $A$ is affordable but lies on a lower IC; there $MRS > p x /p y$ , so the consumer moves toward $D$ . The highest IC is not affordable.

Worked example

Take $U = XY 4$ , $M = 300$ , $p x = 0.4$ and $p y = 1$ . The Cobb–Douglas MRS is $Y/(4X)$ . Setting it equal to the price ratio gives $Y$ in terms of $X$ . Substituting into the budget line gives the optimal bundle.

\frac{1}{4} \frac{Y}{X} = \frac{0.4}{1} \Rightarrow Y = 1.6 X

0.4 X + 1.6 X = 300 \Rightarrow X^{*} = 150

Y^{*} = 1.6 \cdot 150 = 240

The Cobb–Douglas shortcut confirms the result: the consumer spends one fifth of income on $X$ .

X^{*} = \frac{1}{5} \cdot \frac{300}{0.4} = 150

Cobb–Douglas demand

With $U = x a y b$ , the consumer spends the fixed share $a/(a + b)$ of income on $x$ and $b/(a + b)$ on $y$ . Demand for each good depends only on its own price and income. This is a feature of the Cobb–Douglas utility function, not a general rule.

x^{*} = \frac{a}{a + b} \cdot \frac{M}{p_{x}}

y^{*} = \frac{b}{a + b} \cdot \frac{M}{p_{y}}

Corner solutions

A corner solution arises when there is no tangency. With perfect substitutes $U = ax + by$ , the ICs are straight. The consumer spends everything on the good with the higher marginal utility per euro, comparing $a/p x$ with $b/p y$ . If the two are equal, any bundle on the budget line is optimal.

\frac{{MU}_{x}}{p_{x}} = \frac{a}{p_{x}}

\frac{{MU}_{y}}{p_{y}} = \frac{b}{p_{y}}

With perfect complements $U = min(x, y)$ , there is no tangency either. The consumer chooses the kink of the IC and uses the budget line to solve:

x = y

From choice to the demand curve

Hold $p y$ and $M$ fixed and change $p x$ . The budget line rotates, the tangency moves, and the chosen $x*$ changes. Solving the two optimality conditions with $p x$ left as a symbol gives the individual demand function for $x$ . Plotting the price–quantity pairs derives the demand curve from tastes and the budget alone; nothing else lies underneath it.

x^{*} = f (p_{x})

Individual demand gives the number of units one consumer wants at every price, holding everything else fixed. Market demand adds the individual quantities at each price. The other good also reacts to $p x$ :

y^{*} = g (p_{x})

If $y$ rises when $p x$ rises, for example $y* = 10p x$ , then $x$ and $y$ are substitutes.
If $y$ falls when $p x$ rises, for example $y* = 10/p x$ , they are complements.
If $y$ does not move, for example $y* = 40$ in the Cobb–Douglas case, the goods are unrelated.

Normally a rise in $p x$ lowers $x$ . The exception is the Giffen good, a strongly inferior good whose demand rises when its price rises.

Income changes and Engel curves

Hold $p x$ and $p y$ fixed and change $M$ . The budget line shifts in parallel and the optimum moves. Plotting income against the quantity chosen gives the Engel curve, an MIT term. Its slope in percentage terms is the income elasticity $ε D M$ .

Normal good: a rise in $M$ raises $x$ . The Engel curve slopes upward and $ε D M > 0$ . This is the usual case.
Inferior good: a rise in $M$ lowers $x$ , and $ε D M < 0$ . Potatoes and fast food are examples: they are cheap and filling, and consumers drop them when they can afford steak.
Two goods cannot both be inferior. If income rises and the consumer buys less of both, he ends up inside the budget line, which violates more-is-better.

Preferences and prices are given, but income has sources. It comes from work (labour supply), from lending and borrowing (choices involving time) and from an initial endowment such as an inheritance.

Substitution effect and income effect

A price change does two things at once. The decomposition returns in labour supply and in saving, where the two effects fight each other.

The substitution effect is the change in quantity when the price changes, holding utility constant: the consumer stays on the old IC and faces the new price ratio. Graphically, an imaginary budget line with the new slope is drawn tangent to the old IC. The move from the old optimum to this point is the substitution effect. It is always negative: a higher price always lowers the compensated demand for the good, because the tangency of a steeper line with the same IC lies further left.
The income effect is the change in quantity caused by the change in purchasing power, holding prices at their new level. A price rise makes the consumer effectively poorer: the opportunity set shrinks although income does not. For a normal good the income effect of a price rise is negative; for an inferior good it is positive.

The substitution effect and the income effect combine as follows:

Normal good, price up: substitution −, income −, total − (less is bought).
Normal good, price down: substitution +, income +, total + (more is bought).
Inferior good, price up: substitution −, income +, total ambiguous.
Inferior good, price down: substitution +, income −, total ambiguous.

In the notes’ diagram, $p x$ rises and the budget line moves from $BC 1$ to $BC 2$ . The substitution effect moves the consumer from $a$ to $b$ along the old IC. The income effect moves him from $b$ to $c$ , a parallel shift in to $BC 2$ . Here $x$ is a normal good, so both effects cut $x$ .

A Giffen good is an inferior good whose income effect is so large that it beats the substitution effect. Its demand curve slopes upward. Giffen goods are very rare; rice for the poorest households in China is the one clean example. The notes compare them to gryphons: imaginary for consumption goods. The same fight between the two effects is common for labour supply and saving.

Labour Supply

The supply of labour gives, for every wage, the number of hours a worker is willing to work. It uses the same consumer-choice model with new names. Instead of choosing between two goods, the consumer chooses how hard to work.

The trick is to model the good, not the bad. Work is a bad and consumption is a good, and the model cannot handle a good against a bad. So the bad is flipped into a good: the model uses leisure $N$ and recovers labour $L$ as total time $T$ minus leisure. Leisure and consumption are both normal goods, so everything from preferences and choice applies.

L = T - N

Setup

The Bocconi course sets up the labour supply model in three parts, following the consumer’s problem:

Preferences: convex Cobb–Douglas ICs over leisure $N$ on the horizontal axis and a consumption good $C$ on the vertical axis. Both are normal goods. $T$ is total hours available per day, net of sleep, and $L = T - N$ is hours of labour.
Budget constraint: $p C$ is the price of consumption, often normalised to 1 as the numeraire, so that $C$ is measured in euros. The wage $w$ is the price of leisure. $E$ is the initial endowment, or non-labour income, such as an inheritance or transfers.
Choice: the tangency of an IC with the budget line.

Income from work plus the endowment pays for consumption. Moving the value of leisure to the left-hand side gives the full income form: spending on consumption plus spending on leisure equals full income.

p_{C} C = E + w L = E + w (T - N)

p_{C} C + w N = E + w T

The budget line has slope $-w/p C$ . Its endpoints are the no-work point, $N = T$ with $C = E/p C$ , and the work-all-hours point, $N = 0$ with $C = (E + wT)/p C$ . The interior solution sets the MRS between leisure and consumption equal to $w/p C$ and uses the budget constraint to find $(N*, C*)$ . Optimal labour follows from optimal leisure.

{MRS}_{N, C} = \frac{w}{p_{C}}

L^{*} = T - N^{*}

The wage is the price of leisure. Every hour not worked is an hour of wage not earned, so the price of leisure is its opportunity cost. By taking leisure, the worker spends the money he could be earning.

A corner solution occurs if the tangency gives $N* > T$ , since there are not enough hours in the day. The best the agent can do is not to work, and consumption is funded by the endowment alone:

N^{*} = T

C^{*} = \frac{E}{p_{C}}

From choice to the labour supply curve

Fix $p C$ , $E$ and $T$ . Leave $w$ as a symbol and solve the two conditions. The result, $N*$ as a function of $w$ , is the demand for leisure. Total time minus that demand is the labour supply function. The method mirrors the derivation of the demand curve.

N^{*} = f (w)

L^{*} = T - f (w)

When the wage rises from $w 1$ to $w 2$ , the budget line pivots around the no-work point $(T, E/p C)$ . The worker still has at most $T$ hours of leisure, but each hour worked buys more consumption.

A wage rise

For an ordinary consumption good, a price rise cuts demand for two reasons that work together: the good is more expensive (substitution) and the consumer is poorer (income). For leisure the two effects push in opposite directions, because a higher price of leisure makes the worker richer, not poorer.

Substitution effect: a higher $w$ makes leisure more expensive, so the worker takes less leisure and works more. Sitting on the sofa feels worse when the worker could earn more. The Bocconi notes call this effect B.
Income effect: a higher $w$ raises purchasing power, so the worker wants more of every normal good, including leisure. He takes more leisure and works less. The Bocconi notes call this effect A. A lottery winner does not work harder.

The effects of a wage rise on leisure and labour are:

Substitution effect: leisure down, labour up.
Income effect: leisure up, labour down.
Net effect: ambiguous for both leisure and labour.

The Bocconi rule settles which effect wins. At low wages, below a threshold $w̄$ , the substitution effect dominates (B > A) and labour supply slopes upward. At high wages, above $w̄$ , the income effect dominates (A > B) and labour supply bends back. The result is a backward-bending labour supply curve. This is not a Giffen curiosity. It needs no inferior good, only that leisure is normal and labour is its mirror image.

The MIT notes give a target income intuition. A student saving for a 200-euro bike at 10 euros an hour works 20 hours. If the wage rises to 20 euros, 10 hours are enough. The higher wage made the student work less. No assumption was violated: the income effect simply won.

L = \frac{200}{10} = 20

L = \frac{200}{20} = 10

An empirical note in the MIT material: in the 1970s married women had a large labour supply elasticity, between 0.5 and 1. They had a real alternative use of time and almost no income effect, since a wage rise does not make a non-worker richer. Men had an elasticity near zero, or slightly backward bending. Today the female elasticity has fallen to about 0.2.

Choices Involving Time

The syllabus calls this topic “Money Supply” (Bernheim and Whinston, sections 10.1–10.2). It uses the same consumer-choice model. The two goods are consumption today and consumption tomorrow. Saving is a bad, like labour, so the model again uses the complementary good, consumption in each period, and recovers saving as income today minus consumption today.

s = m_{0} - c_{0}

Interest

The principal is the amount of money borrowed or lent. Interest is the price of the loan. The interest rate $r$ is interest divided by principal. Lending $M 0$ today at rate $r$ returns $M 1$ tomorrow.

M_{1} = M_{0} + r M_{0} = M_{0} (1 + r)

Over $t$ periods, compounding means earning interest on interest. Future value and present value convert money between dates:

FV = PV {(1 + r)}^{t}

PV = \frac{FV}{{(1 + r)}^{t}}

A euro tomorrow is worth less than a euro today, because a euro today can be invested and become $1 + r$ euros tomorrow. Present value is the value of a future payment in today’s euros. Future payments are divided by $(1 + r) t$ ; this is discounting. Payments at different dates cannot be added until they are discounted to the same date, just as a pound of apples cannot be added to a pound of gold.

Two consequences follow. First, the earlier one saves, the more compounding helps: at 7%, saving 3,000 a year for the first 15 of 48 working years beats saving the same amount for the last 33 years. Second, a flat payment $f$ received forever, a perpetuity, has a present value of about $f/r$ .

PV \approx \frac{f}{r}

The MIT notes distinguish two rates. The nominal interest rate $i$ is measured in euros. The real interest rate $r$ is measured in goods and equals the nominal rate minus inflation $π$ . Only the real rate matters for choices. Unless a question says otherwise, inflation is zero and $i = r$ .

r = i - π

Intertemporal choice

The model of intertemporal choice has two periods, $t = 0$ (today) and $t = 1$ (next year). There are two goods, $c 0$ and $c 1$ , with prices $p 0$ and $p 1$ , and two incomes, $m 0$ and $m 1$ . The agent can borrow or lend any amount at the interest rate $r$ . The intertemporal budget constraint states that the present value of consumption equals the present value of income:

p_{0} c_{0} + \frac{p_{1} c_{1}}{1 + r} = m_{0} + \frac{m_{1}}{1 + r}

In the $(c 0, c 1)$ plane the budget line has the slope below. With $p 0 = p 1 = 1$ , consumption tomorrow equals income tomorrow plus saving with interest, the slope is $-(1 + r)$ , and saving is $s = m 0 - c 0$ .

slope = - \frac{p_{0} (1 + r)}{p_{1}}

c_{1} = m_{1} + (1 + r) (m_{0} - c_{0})

The interior solution sets the MRS between consumption today and consumption tomorrow equal to the absolute slope of the budget line. With prices equal to 1, the condition is $MRS = 1 + r$ .

{MRS}_{c_{0}, c_{1}} = \frac{p_{0} (1 + r)}{p_{1}}

The slope has a clear reading. $p 0$ is the price of the good this year, and $p 1 /(1 + r)$ is the price of the good next year seen from today. The price of consuming today is $1 + r$ : every unit eaten today costs $1 + r$ units tomorrow. The interest rate is the opportunity cost of consumption today, exactly as the wage is the opportunity cost of leisure.

The endowment point consumes exactly the income of each period. It is always affordable, whatever $r$ , so the budget line pivots around it when $r$ changes.

(\frac{m_{0}}{p_{0}}, \frac{m_{1}}{p_{1}})

If $c 0 < m 0 /p 0$ , the agent is a saver (a lender) and moves up and to the left of the endowment.
If $c 0 > m 0 /p 0$ , the agent is a borrower and moves down and to the right.

In the notes’ diagram, the budget line runs from $m 1 + (1 + r)m 0$ on the vertical axis to $m 0 + m 1 /(1 + r)$ on the horizontal axis, through the endowment point. The chosen bundle lies left of the endowment, so the agent is a saver and saves $s = m 0 - c 0 *$ .

A rise in the interest rate

A higher $r$ makes the budget line steeper, pivoting around the endowment point. As always, the change splits into a substitution effect and an income effect:

Substitution effect: consumption today becomes more expensive, so $c 0$ falls and saving rises. The sign is the same for everybody.
Income effect for a saver: each euro saved now pays more, so the saver is richer and wants more of everything, including $c 0$ . Saving falls. The two effects fight, and the net effect on saving is ambiguous. With a savings target, a higher rate even allows the saver to save less.
Income effect for a borrower: the loan costs more, so the borrower is poorer and $c 0$ falls. Both effects cut $c 0$ , so a borrower unambiguously borrows less.

A saver stays a saver after a rise in $r$ : the old bundle is still affordable, and the endowment side of the line is now worse. A borrower may switch to saving. In summary, the net effect on $c 0$ is ambiguous for a saver (substitution −, income +) and negative for a borrower (substitution −, income −). The same fight between the two effects appears in labour supply.

Worked example

Take $U = c 0 c 1$ , $m 0 = 100$ , $m 1 = 0$ , $p 0 = p 1 = 1$ and $r = 0.10$ . The intertemporal budget constraint and the tangency condition give the optimum:

c_{0} + \frac{c_{1}}{1.1} = 100

MRS = \frac{c_{1}}{c_{0}} = 1.1 \Rightarrow c_{1} = 1.1 c_{0}

c_{0} + c_{0} = 100 \Rightarrow c_{0}^{*} = 50

c_{1}^{*} = 1.1 \cdot 50 = 55

s = 100 - 50 = 50

The Cobb–Douglas shortcut gives the same result: half of the present value of income, 100, goes to each period. If $r$ rises to 0.20, $c 0 *$ stays at 50, because for this utility function the income and substitution effects exactly cancel. Consumption tomorrow rises:

c_{1}^{*} = 1.2 \cdot 50 = 60

Technology and Production

The second half of the constrained optimization story asks where the supply curve comes from. The firm does not maximise production. It maximises profit, total revenue minus total cost. Profit maximization implies cost minimisation: whatever quantity the firm sells, it wants to produce it as cheaply as possible. Topics 8 and 9 build the cost function. Profit maximisation itself comes after the first partial.

Π (Q) = P Q - TC (Q)

The Bocconi course sets out a four-step roadmap from technology to supply:

The production function gives the inputs needed for each output.
For each output, the firm picks the input mix that costs least. This gives total cost $TC(Q)$ .
For a given price $P̄$ , the firm picks $Q$ to maximise $Π(Q) = P̄Q - TC(Q)$ .
Repeating step 3 for every price traces the supply curve.

The production function

The production function assigns to every input combination the maximum output the firm can produce with its technology. It converts inputs into output as a utility function converts goods into happiness, with one difference: output is a real, measurable quantity. Lower-case $q$ denotes one firm and upper-case $Q$ the market.

Q = F (L, K)

Inputs are also called factors of production. Labour $L$ is measured in workers or hours. Capital $K$ covers machines, buildings and land: everything workers use to make things. There is one output.

A variable input can be adjusted over the period considered, such as hours of work. A fixed input cannot be adjusted over that period, such as plant size.

The short run is a period over which one or more inputs is fixed. In the course, capital is fixed at $K̄$ and labour is variable.
The long run is a period over which all inputs are variable. There is no $K̄$ .

Q = F (L, \bar{K})

The length of the long run depends on the production process: days for a food stall, decades for a power plant. The exact length is not examined.

Average and marginal product

The average product is output per unit of an input. The marginal product is the additional output from the next unit of an input, holding the other input fixed. $MP L$ is the production analogue of marginal utility.

{AP}_{L} = \frac{Q}{L} = \frac{F (L, K)}{L}

{AP}_{K} = \frac{Q}{K}

{MP}_{L} = \frac{\partial F}{\partial L}

{MP}_{K} = \frac{\partial F}{\partial K}

For example, with the production function $Q = KL$ , $AP L = K$ for given $K$ , $AP K = L$ for given $L$ , and $MP L = K$ .

The Bocconi notes explain the link between average and marginal with workers. Five workers produce 4 units each on average, so $Q = 20$ . A sixth worker alone adds 10. The new average product is 5:

\frac{20 + 10}{6} = 5

The average rises when the marginal unit is above the average and falls when it is below. The $MP L$ curve crosses the $AP L$ curve at the maximum of $AP L$ . The same arithmetic links marginal cost and average cost.

Assumptions on technology

The Bocconi course places three assumptions on technology:

(A1) Free disposal: the firm can dispose of unwanted inputs at no cost, so more of an input never lowers output.
(A2) Productive Inputs Principle: increasing the amounts of all inputs strictly increases the output the firm can produce. It mirrors more-is-better for consumers.
(A3) Law of diminishing marginal returns: holding other inputs fixed, the marginal product of an input eventually declines as more of it is used. With one shovel and six workers, the sixth digs less than the fifth, because there is only so much capital to work with. The law describes a range of production, not a universal truth, but every firm reaches that range.

Isoquants and the MRTS

An isoquant is the set of all input combinations $(L, K)$ that produce the same output efficiently. It is the firm’s indifference curve, but tangible. For $Q = KL$ , the bundles $(L = 1, K = 2)$ and $(L = 2, K = 1)$ both lie on the isoquant $Q = 2$ . The family of isoquants contains the isoquants for all output levels. Its properties follow the same logic as those of ICs:

Isoquants are thin.
Isoquants do not slope upward.
Isoquants of the same technology do not cross.
Isoquants further from the origin represent more output.

The marginal rate of technical substitution $MRTS L,K$ is the rate at which the firm can replace capital with labour, giving up $K$ and adding $L$ , while keeping output unchanged, for a small change in $L$ . It is the absolute value of the slope of the isoquant. It varies along the isoquant because of diminishing marginal products. With many machines and one worker, one extra worker replaces many machines. With many workers, an extra worker replaces very little capital.

Output does not change along an isoquant, so the total differential of output is zero. Rearranging gives the MRTS as a ratio of marginal products:

{MP}_{L} d L + {MP}_{K} d K = 0

{MRTS}_{L, K} = - \frac{d K}{d L} |_{Q = \bar{Q}} = \frac{{MP}_{L}}{{MP}_{K}}

The formula has the same structure as the MRS, $MRS = MU x /MU y$ . The MIT notes write the slope with the minus sign, $-MP L /MP K$ .

Two extreme technologies bound the usual case, as with indifference curves:

Perfect substitutes, $Q = aL + bK$ : straight isoquants with a constant $MRTS = a/b$ . Example: a robot and a worker who do the same job.
Perfect complements, or Leontief technology, $Q = min(aL, bK)$ : L-shaped isoquants. Examples: one driver per truck, a left shoe per right shoe. The MRTS jumps from $\infty$ to 0 at the corner.

Q = a L + b K

Q = \min (a L, b K)

Returns to scale

Returns to scale describe what happens to output when the firm increases all inputs in the same proportion. The question is not labour against capital but scaling the whole operation. The test compares $F(tL, tK)$ with $tF(L, K)$ for $t > 1$ , usually $t = 2$ :

Constant returns to scale (CRS): doubling all inputs exactly doubles output.
Increasing returns to scale (IRS): doubling all inputs more than doubles output.
Decreasing returns to scale (DRS): doubling all inputs less than doubles output.

F (2 L, 2 K) = 2 F (L, K)

F (2 L, 2 K) > 2 F (L, K)

F (2 L, 2 K) < 2 F (L, K)

Graphically, with CRS the isoquant for $Q = 200$ is twice as far from the origin as the one for $Q = 100$ . With IRS it is closer than that; with DRS it is farther.

The Cobb–Douglas production function has a shortcut. Scaling both inputs by $t$ scales output by $t$ raised to the sum of the exponents. The technology has CRS if $α + β = 1$ , IRS if $α + β > 1$ and DRS if $α + β < 1$ . For example, $Q = \sqrt(LK)$ has CRS and $Q = KL$ has IRS.

Q = A L^{α} K^{β}

F (t L, t K) = t^{α + β} F (L, K)

Increasing returns come from specialisation of tasks as the firm grows, as in mass production. Decreasing returns come from limited managerial capacity: coordination gets harder and slacking gets harder to monitor. Returns to scale cannot increase forever. Such a firm would keep getting more productive as it grew and would end up owning the whole economy.

Diminishing marginal returns and decreasing returns to scale are different ideas. Diminishing marginal returns concern one input rising with the other fixed, an idea of the short run. Decreasing returns to scale concern all inputs rising together, an idea of the long run. A CRS technology can, and normally does, have diminishing marginal returns to each input.

Over time the production function itself shifts. The multiplier $A$ is total factor productivity: how much more output comes from the same inputs. The standard of living depends on productivity: more capital or a faster-growing $A$ . Malthus missed this when he predicted starvation from the fixed supply of land.

Q = A (t) F (L, K)

Costs

The goal is a cost function $TC(Q)$ that shows how the cost of production varies with output. Labour costs $w$ per unit, the wage. Capital costs $r$ per unit, the rental rate: the firm does not buy a machine or a worker but rents them for the period. Using $(L, K)$ costs $wL + rK$ . Cost minimization finds the cheapest input bundle that can produce $Q$ :

TC (Q) = \min_{L, K} {w L + r K : F (L, K) \geq Q}

The problem has two steps. First, fix $Q$ and pick the cheapest bundle $(L*, K*)$ on that isoquant. Second, repeat for every $Q$ to get the input demands $L*(Q)$ and $K*(Q)$ and total cost:

TC (Q) = w L^{*} (Q) + r K^{*} (Q)

Short-run cost

In the short run capital is fixed at $K̄$ , so there is nothing to choose. Producing $Q$ requires the labour that solves $F(L, K̄) = Q$ , that is, the inverse function of the production function:

L = f^{- 1} (Q)

TC (Q) = w f^{- 1} (Q) + r \bar{K} = VC (Q) + FC

Variable cost $VC(Q)$ covers all costs that depend on $Q$ ; here it is the labour bill. Fixed cost $FC$ is the constant term, paid even at $Q = 0$ :

FC = r \bar{K}

Worked examples

In the Bocconi example, $Q = 4L 2$ , $w = 15$ , $r = 25$ and $K̄ = 4$ . For $Q = 1$ , labour is one half and total cost is 107.5. In general, labour is half the square root of $Q$ , which gives the cost function:

1 = 4 L^{2} \Rightarrow L = \frac{1}{2}

TC (1) = 15 \cdot \frac{1}{2} + 25 \cdot 4 = 107.5

L = \frac{1}{2} \sqrt{Q}

TC (Q) = 7.5 \sqrt{Q} + 100

The MIT running example uses $Q = \sqrt(LK̄)$ , $w = 5$ , $r = 10$ and $K̄ = 1$ . Then $L = Q 2$ . The fixed cost is 10 and the variable cost is $5Q 2$ .

Q = \sqrt{L \bar{K}}

L = Q^{2}

TC (Q) = 10 + 5 Q^{2}

Average and marginal cost

Average cost is the cost per unit over the whole range produced. It splits into average variable cost and average fixed cost. Marginal cost is the cost of producing the next unit. It is the single most important cost concept, because it drives the supply curve. With a non-linear technology, marginals and averages differ, so they must be kept separate.

AC = \frac{TC}{Q}

AVC = \frac{VC}{Q}

AFC = \frac{FC}{Q}

AC = AVC + AFC

MC = \frac{d TC}{d Q} = \frac{d VC}{d Q}

In discrete form, marginal cost is the difference in cost between $Q$ and $Q - 1$ units. Variable cost therefore builds up unit by unit:

MC (Q) = TC (Q) - TC (Q - 1) = VC (Q) - VC (Q - 1)

VC (2) = VC (1) + MC (2)

The curves have typical shapes. $AFC = FC/Q$ falls forever, because the fixed cost is spread over more units and goes to zero as $Q$ grows. $AVC$ eventually rises because of diminishing marginal returns. Their sum $AC$ is U-shaped: it falls at first as the fixed cost is paid off and rises later as the marginal product of labour falls.

Marginal cost is linked to $MP L$ . The next unit is cheap when the worker is productive and expensive when he is not. Diminishing $MP L$ is why $MC$ rises. In the same way, $AVC$ is the wage divided by the average product of labour.

MC = \frac{d VC}{d Q} = w \frac{d L}{d Q} = \frac{w}{{MP}_{L}}

AVC = \frac{w}{{AP}_{L}}

$MC$ crosses $AVC$ and $AC$ at their minimum points. The reason is arithmetic, as with average and marginal product. When the marginal unit costs less than the average, it pulls the average down. When it costs more, it pulls the average up. So the average is at its minimum exactly where it equals the marginal. In the MIT example, the curves cross at $Q = \sqrt2 \approx 1.41$ , where $AC$ is lowest.

AC = \frac{10}{Q} + 5 Q

MC = 10 Q

\frac{10}{Q} + 5 Q = 10 Q \Rightarrow Q = \sqrt{2} \approx 1.41

The efficient scale of production $Q e$ is the quantity at which $AC$ is at its minimum, where $MC = AC$ . Economies of scale exist when $AC$ falls as $Q$ rises. Diseconomies of scale exist when $AC$ rises as $Q$ rises. In the long run these mirror increasing and decreasing returns to scale.

Q_{2} > Q_{1} \Rightarrow AC (Q_{2}) < AC (Q_{1})

The notes’ cost diagram for $TC = 10 + 5Q 2$ shows $MC$ cutting $AC$ at its minimum, the efficient scale. $AFC$ falls forever and $AVC = 5Q$ rises. Because $AVC$ is linear here, $MC$ cuts it at the origin; with a U-shaped $AVC$ the crossing is at its minimum.

Long-run cost

In the long run $K$ is variable and the firm chooses the cheapest input mix. The tools are the isoquant, which shows what the technology allows, and the isocost.

An isocost line is the set of input combinations that cost the same total $C$ . It is the firm’s budget line, with slope $-w/r$ . Unlike the consumer, the firm has no given budget. The analysis draws a family of isocosts and looks for the lowest one that still reaches the isoquant.

w L + r K = C

K = \frac{C}{r} - \frac{w}{r} L

At the tangency the MRTS equals the input price ratio, and output must equal $Q$ . This is the bang-for-the-buck rule again: the last euro spent on workers must add as much output as the last euro spent on machines. If $MP L /w > MP K /r$ , the firm hires more labour and less capital.

{MRTS}_{L, K} = \frac{{MP}_{L}}{{MP}_{K}} = \frac{w}{r} \Leftrightarrow \frac{{MP}_{L}}{w} = \frac{{MP}_{K}}{r}

F (L, K) = Q

In the notes’ diagram, isocost $C 1$ cannot reach the isoquant, and $C 3$ reaches it but wastes money. The least-cost bundle is where the isoquant is tangent to the lowest reachable isocost, $C 2$ . The slope of every isocost is $-w/r$ .

Worked example

The MIT example uses $Q = \sqrt(LK)$ , $w = 5$ and $r = 10$ . The marginal products give an MRTS of $K/L$ . Tangency sets $K/L$ equal to $w/r = 1/2$ . Workers cost half as much as machines and, with this technology, are equally productive at the margin, so the firm uses twice as many workers as machines.

{MP}_{L} = \frac{1}{2} \sqrt{\frac{K}{L}}

{MP}_{K} = \frac{1}{2} \sqrt{\frac{L}{K}}

MRTS = \frac{K}{L}

\frac{K}{L} = \frac{w}{r} = \frac{1}{2} \Rightarrow K = \frac{L}{2}

Substituting into the production function gives the input demands and the long-run cost function:

Q = \sqrt{L \cdot \frac{L}{2}} \Rightarrow L^{*} = \sqrt{2} Q

K^{*} = \frac{\sqrt{2}}{2} Q

TC (Q) = w L^{*} + r K^{*} = 5 \sqrt{2} Q + 5 \sqrt{2} Q = 10 \sqrt{2} Q \approx 14.1 Q

What is general is $TC = wL* + rK*$ and the tangency rule. What is specific is everything that follows from this production function.

Leontief cost minimisation

With Leontief technology $Q = min(aL, bK)$ there is no tangency. This case is common in past exams. The firm produces at the corner of the isoquant, where $aL = bK = Q$ :

a L = b K = Q

L^{*} = \frac{Q}{a}

K^{*} = \frac{Q}{b}

TC (Q) = (\frac{w}{a} + \frac{r}{b}) Q

Input prices and the expansion path

If $w$ rises, the isocost line gets steeper. To produce the same $Q̄$ , the firm slides along the isoquant toward more capital and less labour. This is the substitution effect on the production side, always toward the cheaper input. It is the theory behind the claim that a higher minimum wage leads to workers being replaced by machines.

The expansion path is the set of least-cost bundles as $Q$ grows. Reading off the cost of each bundle gives the long-run cost function $TC(Q)$ . With Cobb–Douglas technology the optimal $K/L$ ratio is constant, so the path is a straight line from the origin.

Long run versus short run

In the long run there are no fixed costs, so total cost equals variable cost. Long-run costs are never higher than short-run costs: with one more input to optimise over, the firm can always do at least as well. The long-run average cost curve is the lower envelope of the short-run $AC$ curves, one for each plant size. A small output calls for a small plant and a large output for a large plant. A firm that builds the wrong plant, as Tesla did in 2017, is stuck on the wrong short-run curve until it can adjust capital.

TC (Q) = VC (Q)

Inputs: in the short run $K = K̄$ is fixed and $L$ is variable; in the long run both are variable.
Costs: in the short run $TC = VC(Q) + FC$ ; in the long run $TC = VC(Q)$ and $FC = 0$ .
Finding $TC(Q)$ : in the short run, invert $F(L, K̄) = Q$ ; in the long run, use the tangency $MRTS = w/r$ together with $F(L, K) = Q$ .
Shape of $AC$ : U-shaped in the short run ( $AFC$ down, $AVC$ up); in the long run it follows returns to scale.

After the partial, the course turns to profit maximization. The firm maximises $Π = PQ - TC(Q)$ by setting marginal cost equal to price in perfect competition. It produces only if the price covers the minimum of $AVC$ in the short run and the minimum of $AC$ in the long run. The resulting $Q*(P)$ is the supply curve the course started with.

MC = P

P \geq \min AVC

P \geq \min AC

Sources

Microeconomics - First Partial Notes.pdf

[cite-note-30325-first-partial-notes] Microeconomics - First Partial Notes.pdf