Title: oxford / 302 microeconomics

Front: Utility function of two goods: [$]u(j,k)=j^\alpha k^\beta[/$]
What are the demands? (as a function of income, prices)
Back: [$] x_{j}^{*}=\frac{\alpha}{\alpha+\beta}\frac{m}{p_j} [/$]
[$] x_{k}^{*}=\frac{\beta}{\alpha+\beta}\frac{m}{p_k} [/$]
An interpretation of this solution is that the consumer spends a fraction [$]\frac{\alpha}{\alpha+\beta}[/$] of his income purchasing good j, and a fraction [$]\frac{\beta}{\alpha+\beta}[/$] purchasing good k.

Front: Utility function of two goods: [$]u(j,k)=\alpha ln(j) + \beta ln(k)[/$]
What are the demands? (as a function of income, prices)
Back: [$] x_{j}^{*}=\frac{\alpha}{\alpha+\beta}\frac{m}{p_j} [/$]
[$] x_{k}^{*}=\frac{\beta}{\alpha+\beta}\frac{m}{p_k} [/$]
An interpretation of this solution is that the consumer spends a fraction [$]\frac{\alpha}{\alpha+\beta}[/$] of his income purchasing good j, and a fraction [$]\frac{\beta}{\alpha+\beta}[/$] purchasing good k.

Front: definition of a competitive market
Back: agents are price-takers

Front: first welfare theorem
Back: any allocation that is the result of a walrasian equlibrium is pareto-efficient

Front: walras' law
Back: [$]\forall p, \sum_{i}{p_iz_i(\mathbf{p})}=0[/$]
the total value of excess demand is zero at any set of prices.
implication: if we find a p for which excess demand is zero for N-1 goods, then excess demand is zero for the Nth good as well.

Front: Consumer j's consumption of good i
Back: [$]x_i^j[/$]

Front: what does an edgeworth box represent?

Front: full mathematical characterisation of pareto efficiency
Back: An allocation [$]x=(x^1,...,x^M)[/$] is Pareto efficient iff:
[$$]\forall j, \max_{x^1,...,x^M} U^j(x^j)[/$$]
[latex]subject to:
\begin{align}
\forall i, \sum_{j}x_i^j =\sum_j e_i^j \tag{N equations} \\
\forall j' \neq j, u^{j'} (x^{j'}) \geq \bar{u}^{j'} \tag{M-1 equations}
\end{align}
[/latex]

Front: second welfare theorem
Back: Assume that preferences are convex and there are non-increasing returns to scale on production. Then any Pareto efficient allocation can be achieved as a Walrasian equilibrium with appropriate initial endowments.

Front: proof that at pareto efficient allocations, marginal rates of substitution are equal

Front: intuitive proof of first welfare theorem
Back: * in equilibrium, all consumers maximize utility by equating MRS between any two goods to the respective price ratio.
* in equilibrium, all consumers face the same prices
* so in equilibrium all consumers have the same MRS between any two goods
* but (see see Varian p. 607-8) this implies that the equilibrium allocation is Pareto efficient.

Front: quasi-linear utility

Front: concise proof of first welfare theorem

Front: vNM theorem
Back: A preference ordering over lotteries can be expressed as maximising a utility function iff the preference ordering satisfies:

Front: Public Goods: definition
Back: A public good is a non-rival good (zero opportunity cost of consumption).
It can be both non-excludable (`pure' public goods, like defence) or excludable (`excludable' public goods, like scrambled TV)

Front: Public good: pareto efficiency condition
Back: [latex]
The private good is x, the public good is G. Two consumers, a and b, have utilities $u_a(x_a,G)$ and $u_b(x_b,G)$. The cost of producing each good are $c_G(.)$ and $c_x(.)$. An allocation $(x,G)$ is pareto efficient iff
$$\frac{ \frac{d u^a}{d G} }  {\frac{d u^a}{d x^a}  } + \frac{   \frac{d u^b}{d G} } {\frac{d u^b}{d x^b}} = \frac{c'_G(G)}{c'_x(x^a+x^b)} $$
This is called the Samuelson rule.
[/latex]

Front: Externalities: definition
Back: An externality occurs when one economic agent's actions affect another directly, rather than through a market.

Front: Externalities: competitive equilibrium condition
Back: When each agent solves her utility maximisation problem, the agent recieving the externality treats it as a constant.

Front: Deadweight loss from a wedge is...
Back: Proportional to:
expenditure (p*q)
the ad-valorem wedge squared ((t/p)^2)
Increasing in:
price-elasticities of supply and demand

Front: Coase theorem
Back: assumptions:
well-defined property rights
costless bargaining
Efficiency version: the outcome of bargaining over an externality will be Pareto efficient, regardless of the initial allocation of property rights.
Invariance version: The outcome of bargaining over an externality will be Pareto efficient, and will always be the same allocation, regarless of the initial allocation of property rights.
Insight: Externalities are only externalities because there is no market for them, not because they are 'external'. Pigouvian taxation is not the only way to correct an externality.

Front: indifference curves in state-contingent income space
Back: slope of the indifference curve:

Front: first-order stochastic dominance

Front: second-order stochastic dominance

Front: certainty equivalent, risk premium

Front: lottery (definition)

Front: concavity
Back: [latex]
A function $f$ is concave on an interval $I$ iff $\forall x,y \in I$ and $\forall \alpha \in [0,1]$
$$f((1-\alpha )x+\alpha y)\geq (1-\alpha )f(x)+\alpha f(y)$$
A function is concave simpliciter iff it is concave on its domain.
\bigskip
If $f$ is twice-differentiable, then $f$ is concave if and only if $f'' \leq 0$.
[/latex]

Front: pick two different MRSs of the same sign, quickly propose a pareto imporvement
Back: did you succeed?

Front: sum of terms of a geometric series

Front: sum of terms of arithmetic series

Front: optimal pigouvian tax
Back: [latex]
Suppose we are considering a good with only negative and no positive externalities. Then
$$t = C_{social}'(q*) - C_{individual}'(q*)$$
Where q* is such that
$$MB(q*)= C_{social}'(q*)$$
[/latex]

Front: when is bertrand relevant, when is cournot relevant?
Back: If capacity and output can be easily changed, Bertrand is a better model. e.g. a firm that sells woodcutting services can just let the wood saws run all night if there is more demand.
If output and capacity are difficult to adjust, then Cournot is generally a better model. e.g. to produce a new airplane, boeing needs a new factory, 100s of workers, etc.

Front: convex preferences
Back: A set of convex-shaped indifference curves displays convex preferences. Convex preferences with their associated convex indifference mapping arise from quasi-concave utility functions. All concave functions are quasi-concave functions.

Front: insurance
Back: we can think of an insurance contract as specifying wealth levels in (two) states of the world. a contract is a point is state-contingent income space

Front: agency cost
Back: in a principal-agent problem, the risk premium associated with the compensation lottery for an agent who exerts optimal level of effort e*

Front: overview of information asymmetry

Front: signalling model
Back: * Represent in (s,w) space.
* Pooling equilibrium: no signalling, same wage for all.
* Separating equilibrium: signalling, different wages.
* The low-type is better off in the pooling equilibrium than in any separating equilibrium: The high-type may be better or worse off in the pooling equilibrium.

Front: screening model
Back: * Think of contracts offered by insurance company.
* Represent in state-contingent income space

Front: moral hazard model
Back: * Intuition: the trade-off is between risk-sharing and strength of incentives. Agent needs to be compensated for taking on some risk. There is a welfare loss for the principal.
* Contract offered must satisfy individual rationality (agent wishes to participate) and incentive compatibility (agent wishes to exert the level of effort desired by the principal)

Front: measures of risk aversion

Front: the trade-off between efficiency and market power
Back: This is known as the "Williamson trade-off", the maxim "rectangles are bigger than triangles" is sometimes applied.
Note: if the cost reducitn is sufficiently large, the monopoly price will be below the old marginal cost, so even consumer surplus increases. This is known as "drastic innovation".

Front: measuring market power

Front: measuring market concentration

Front: diversion ratio

Front: diversion ratio and merger policy

Front: duopoly (overview)

Front: taxes and quotas
Back: The (non-) equivalence of taxes and quotas (Weitzman):
• Pigovian tax or quota requires that the policymaker has full information
• If the policy maker does not know the exact loss function or firm’s cost function he/she
will not achieve the efficient outcome
• What policy instrument (tax or quota) minimises the loss?

Front: arrow's theorem
Back: Suppose we want to aggregate purely ordinal preferences over sure prospects into a group preference (also merely ordinal). And suppose we want the social preference function to satisfy:
1. U: the condition of unrestricted domain. The domain of the function which maps a
set of individual preferences into a social preference ordering is unrestricted.
2. D: non-dictatorship. A dictator is defined as someone whose choice between all
pairs of alternatives is decisive, i.e. determines the social choice regardless of
the preferences of all other members of the group.
3. P: Pareto principle. If everyone prefers a to b then a should be preferred to b in the social ordering.
4. I: independence of irrelevant alternatives. Suppose we have [a, b]. Now let the individual orderings
 change in a way which leaves unchanged each i’s preferences between a and b.
Then [a, b] should continue to hold. Thus a change only of c’s position in the
individual preference orderings would not of itself change the social ordering of
a and b.
Arrow proved the general impossibility theorem: no SWF exists which
satisfies the four conditions, U, P, I and D, and which can produce a transitive preference
ordering over social states.

Front: pooling equilibrium of signalling game

Front: separating equilibrium of signalling game

Front: screening game setup

Front: screening game pooling equilibrium

Front: screening game separating equilibrium

Front: moral hazard game set up

Front: moral hazard game benchmarks

Front: moral hazard game with a risk-averse manager

Front: attitudes to risk and indifference curves

Front: insurance with two agents

Front: horizontal vs vertical differentiation
Back: Vertical differentiation: Goods vary in quality. If the goods sell at
the same price, everybody chooses the higher quality variety. e.g. Medical care.
Horizontal differentiation: Goods vary on dimensions other than quality. If the
goods sell at the same price, consumers differ in their ranking.
e.g. Breakfast cereals

Front: approximate risk premium
Back: [latex]For a lottery with mean $w$ ("initial wealth"), and variance $\sigma^2$ around $w$, the risk premium is $$r\approx\frac{1}{2}A(w)\sigma^2$$  $$ r\approx-\frac{1}{2}\frac{u''(w)}{u'(w)}\sigma^2$$. This is known as the Arrox-Pratt approximation.[/latex]
Other fields (not shown on cards):
notes: see http://assets.press.princeton.edu/chapters/s7945.pdf

Front: Walrasian equilibrium of an exchange economy
Other fields (not shown on cards):
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\begin{document}
\section{Notation}
\begin{itemize}
\item $N$ goods, $M$ consumers
\item $x_i^j$ is consumption of good $i$ by consumer $j$
\item $x^j = (x_1^j,...,x_N^j)$ is a consumption bundle for $j$
\item an allocation is matrix $x=(x^1,...,x^M)$
\item $e_i^j$ is $j$'s endowment of good $i$
\item $p=(p_1,...,p_N)$ is a price vector
\item ${x_i^j}^*(p)$ is demand for good $i$ by consumer $j$
\end{itemize}
\section{Definition of demand}
\begin{itemize}
\item Consumers solve
$$
{x^j}^* = \arg\max_{x^j} u^j(x^j) \qquad \text{subject to} \\
\sum_i x_i^j p_i \leq \sum_i e_i^j p_i
$$
This assumption tells us that the consumers
\begin{itemize}
\item care only about their own consumption ($u^j(x^j$))
\item are price-takers (they optimise taking $p_i$ as given)
\end{itemize}
\item Firms are price-takers and maximise profits i.e. they solve
$${y^f}^* = \arg\max_{y_f} \sum_i p_i y_i^f \qquad \text{subject to } y_f \in Y^f
$$
\end{itemize}
\section{Definition of excess demand and Walrasian equilibrium}
\begin{itemize}
\item Excess demand for good $i$ is $z_i(\mathbf{p}) = \sum_j\mathbf{x}_i^{j*}(\mathbf{p}) - \sum_je_i^j$.
\item A Walrasian equilibrium is a price vector $p^*$ for which $z_i(p^*)=0$ for all $i$
\end{itemize}
\section{Theorem}
If the excess demand function $z_i(p)$ is continuous in $p$ [sufficient condition: convex preferences and non-increasing returns to scale], then a Walrasian equilibrium exists.
\section{Adding a production side}
\begin{itemize}
\item $N$ goods, $M$ consumers, $K$ firms
\item Firm $f$ is a technology $Y^f \subseteq \real^N$. An element $\mathbf{y}^f \in Y^f$ is a production plan (outputs measures positively, inputs measured negatively). For example $Y^f = \setbuild{(-y_1,y_2)}{y_2 \leq f(y_1), y_1 \geq 0}​$. (Consumers own firms, and receive shares of their profit which enter income.)
\item Firms are price-takers and maximise profits i.e. they solve
$${y^f}^* = \arg\max_{y_f} \sum_i p_i y_i^f \qquad \text{subject to } y_f \in Y^f
$$
\item Redefine consumer optimisation problem to reflect the fact that they earn income from their share in firms
\item Redefine excess demand for good $i$ as $z_i(\mathbf{p}) = \sum_j\mathbf{x}_i^{j*}(\mathbf{p}) - \sum_fy_i^{f*}(\mathbf{p}) - \sum_je_i^j$.
\end{itemize}
\end{document}

Front: types of goods