# Year 1 Undergrad Microeconomics

##### by Zhu Hanfei 03/01/2014

# Year 1 Undergrad Microeconomics

by Zhu Hanfei# 1. Consumer Theory

## 1.1. Preferences

### 1.1.1. Complete - Have defined preferences, not allowed to have no opinion

### 1.1.2. Reflexivity - Something is at least as good as itself.

### 1.1.3. Transitivity - if A > B and B > C then A > C.

## 1.2. Preferences can be represented as Utility functions

### 1.2.1. Ordinal Utility v.s. Cardinal Utility

### 1.2.2. Marginal Utility

1.2.2.1. Change in total utility from a change in one unit of good.

1.2.2.2. Diminishing marginal utility

1.2.2.2.1. 2nd derivative of Utility function is negative, strictly convex.

1.2.2.3. For a two variable function, we can get marginal utilities for both x and y.

### 1.2.3. Indifference curves

1.2.3.1. Can be used to map out utility functions and preferences.

1.2.3.2. Slope of indifference curve = Marginal Rate of Substitution = dy/dx = -MUx/MUy

1.2.3.2.1. Decreasing marginal rate of substitution.

1.2.3.2.2. Second derivative is negative.

### 1.2.4. Cobb-Douglas Utility Function

1.2.4.1. MRS = -ay/bx

### 1.2.5. Perfect Substitutes

1.2.5.1. U(x,y) = a(x+y)

### 1.2.6. Perfect Complements

1.2.6.1. U(x.y) = min(ax,by)

## 1.3. The Budget Constraint

### 1.3.1. PxX+ PyY = m

1.3.1.1. m,Px,Py are exogenous.

### 1.3.2. Solving a two variable constrained optimization problem.

1.3.2.1. trying to maximise utility function.

1.3.2.2. There's an implicit assumption that x,y larger or equals to zero.

1.3.2.3. Turning it into a three variable unconstrained maximisation problem

1.3.2.3.1. Lagrange Multiplier

1.3.2.4. Substitution method

1.3.2.4.1. Substituting the constraint into the utility function

1.3.2.5. Graphical approach

1.3.2.5.1. Slope of MRS = Slope of budget line.

## 1.4. Income and Substitution effects

### 1.4.1. Normal goods

1.4.1.1. Demand increases when income increases

### 1.4.2. Inferior goods

1.4.2.1. Demand falls when income increases.

### 1.4.3. Change in prices

1.4.3.1. Increase in price of good X

1.4.3.1.1. Two effects

### 1.4.4. Change in Income

1.4.4.1. Fall in income

### 1.4.5. Giffen Goods

1.4.5.1. Inferior goods

1.4.5.1.1. When your income rises, you purchase less of them.

1.4.5.2. No close substitutes

1.4.5.2.1. There's no substitutes consumers can conveniently switch to.

1.4.5.3. Take up a large proportion of consumer's income

1.4.5.3.1. Income effect must be sufficiently large to offset the substitution effect. such that overall the change in demand is positive when the price of the good increases.

### 1.4.6. How to measure utility changes?

1.4.6.1. Compensating variation

1.4.6.1.1. How much income is needed to be taken away or given back to the consumer to compensate for the rise or fall in utility?

1.4.6.2. Equivalence variation

1.4.6.2.1. Suppose now the price change had not taken place, how much income has to be taken away from the consumer to create the same fall in utility?

## 1.5. Market Demand

### 1.5.1. Summation of individual demands

### 1.5.2. Consumer surplus

1.5.2.1. The difference between the reservation price and the market price.

1.5.2.2. Total benefit accrued to consumers from consuming the equilibrium amount D(p) at price p.

### 1.5.3. Elasticity

1.5.3.1. Responsiveness of one variable to a change in another.

1.5.3.1.1. Price elasticity of Demand

1.5.3.1.2. Income elasticity of Demand

1.5.3.1.3. Cross Elasticity of Demand

1.5.3.2. Measured in percentage changes.

1.5.3.3. |e| = |dq/dp * (P/Q)|

1.5.3.3.1. >1

1.5.3.3.2. <1

1.5.3.3.3. 0

1.5.3.3.4. infinity

1.5.3.3.5. 1

## 1.6. Maximising Utility from a choice between work and leisure

### 1.6.1. Budget and Time Constraint

1.6.1.1. n + l = T

1.6.1.1.1. i.e. we must allocate our time between work and leisure.

1.6.1.2. Assumption: wage rate of w.

1.6.1.2.1. extending from this using the concept of opportunity cost, every hour spent working is one hour not spent on leisure.

1.6.1.2.2. conversely, the opportunity cost of leisure can be thought of as w.

1.6.1.3. Combining the above two...

1.6.1.3.1. PxX + lw = Tw

### 1.6.2. Utility function

1.6.2.1. function of amount of good, x, and leisure time, l.

1.6.2.1.1. Maximise... Lagrangian multiplier or using graphical approach.

1.6.2.1.2. MRS(l,x) = Px/w

1.6.2.1.3. x = w(n/Px)

### 1.6.3. Extending the concept to construct a Labour Supply Curve

1.6.3.1. Income effect

1.6.3.1.1. When the real wage increases, consumers can buy more of X but also consume more of leisure.

1.6.3.2. Substitution Effect

1.6.3.2.1. When the real wage increases, the "price" or opportunity cost of leisure goes up, so consumers will substitute away from leisure to consumption/work.

1.6.3.3. When the wage increases, x goes up because both substitution and income effects are positive.

1.6.3.3.1. l may go up if income effect dominates. (n goes down)

1.6.3.3.2. I may go down if substitution effect dominates.(n goes up)

1.6.3.3.3. Individual Supply Curve

## 1.7. Inter-temporal Choice

### 1.7.1. Key concepts

1.7.1.1. Present Value

1.7.1.1.1. The value of all future receipts discounted into the present.

1.7.1.1.2. Money at different points in time has different value. The value depends on the discount factor involved.

1.7.1.1.3. Present value basically means we try to factor in this discount factor and weigh everything in the present time period.

1.7.1.2. Savings

1.7.1.2.1. We can save and consume less in the current period in order to increase future consumption

### 1.7.2. Budget Constraint

1.7.2.1. Expressing everything in present value

1.7.2.1.1. Present value of consumption = Present value of income

1.7.2.2. Suppose, I1, I2, c1, c2 are the variables.

1.7.2.2.1. c1 = I1 - S

1.7.2.2.2. c2 = I2 + (1+i)S

1.7.2.3. S represents savings

1.7.2.3.1. It can be negative, in which case the consumer is a net borrower.

### 1.7.3. Marginal Rate of Time preference

1.7.3.1. MRTP = Py/Px, Py = 1/1+i

1.7.3.2. Solve for c1,c2

1.7.3.2.1. If c1 < I1, net saver

1.7.3.2.2. if c1 > I1, net borrower

# 2. Producer Theory

## 2.1. Firms aim to maximise profits Profits = Revenue - Costs

### 2.1.1. Revenue = xPx

2.1.1.1. How does firms produce x? Not every x is feasible.

2.1.1.2. Price might be related to the quantity produced (i.e. P(x)

### 2.1.2. Costs

2.1.2.1. The production function defines the production process, how inputs are transformed into outputs.

2.1.2.1.1. f(x,y)

2.1.2.1.2. Cobb Douglas

2.1.2.1.3. Isoquants

2.1.2.1.4. Short run v.s. Long run

### 2.1.3. The firm's problem

2.1.3.1. Profit maximization

2.1.3.1.1. max (revenue - costs) by choosing x,y.

2.1.3.1.2. Exogenous factors: price of inputs x, price of inputs y, price of goods sold.

2.1.3.1.3. Equivalent to choosing combination of (x,y) because price is not determined by the firm.

2.1.3.2. A reframe: Cost Minimization

2.1.3.2.1. Assume firms produce a fixed quantity zbar.

2.1.3.2.2. And then optimize the output, knowing the combination of inputs that will minimize the cost from first part.

2.1.3.2.3. Fixed Costs & Sunk Costs

# 3. Markets

## 3.1. Strategic Model

### 3.1.1. Players take actions to influence prices and quantities that take into account others' behaviour; each players thinks he can influence prices by changing his or her behaviour

3.1.1.1. Players can trade amongst themselves.

3.1.1.2. A pareto efficient outcome is a set of feasible allocation that has no other set of feasible allocation pareto dominating over it.

3.1.1.2.1. In other words, it is the state of affairs that cannot be improved without making someone in the allocation worse off. i.e. there is at least one person who strictly prefers the current state of allocation, a, all other states, b.

3.1.1.2.2. Might not be desirable in the sense that it might be grossly inequitable.

## 3.2. Non-Strategic Model

### 3.2.1. Agent takes what the market price as given. Unable to influence prices. Competition behaviour

### 3.2.2. Not modelled by game theory

3.2.2.1. Not entirely true is it?

### 3.2.3. Market price given, each seller/buyer decides whether to transact at the given market price.

3.2.3.1. If Willing buyers > Willing sellers, prices will increase

3.2.3.1.1. Excess demand

3.2.3.2. Conversely, if willing buyers < willing sellers, prices will fall.

3.2.3.2.1. Excess Supply

3.2.3.3. The theory thus assumes that the price has somehow adjusted so this is true.

3.2.3.3.1. At an equilibrium, any willing buyer will have found a willing seller.

### 3.2.4. Given a set of prices, Px,Py, each player has his initial endowment and a budget line on which they can trade from their initial endowment.

3.2.4.1. Simple utility maximisation problem

3.2.4.2. A competitive equilibrium is defined as a set of prices (Px,Py), that when each agent acts as a price taker and chooses his best bundle at those prices, then both markets will clear.

3.2.4.2.1. At equilibrium, it must be such that each agent chooses the same point on the edgeworth box.

3.2.4.2.2. A competitive equilibrium must be pareto efficient

3.2.4.2.3. Walras Law

3.2.4.2.4. Second fundamental theorem of welfare economics

# 4. Game Theory

## 4.1. Strategic Games

### 4.1.1. Players

### 4.1.2. Strategies

### 4.1.3. Payoffs/Preference orderings

## 4.2. Nash Equilibrium

### 4.2.1. A profile of actions, each of which is an optimum choice given the actions of other players.

4.2.1.1. Assumption that beliefs about other players are correct.

4.2.1.2. If everyone share this belief, then it is optimum to play this way.

4.2.1.2.1. Stable social norm: if everyone expects others to behave this way, then they should act in a certain way.

4.2.1.3. Can be derived from the intersection of the best response functions.

4.2.1.3.1. Given a profile of actions by other players, there will be a set of actions that are optimum for player i, that are called best responses to actions of all other players.

### 4.2.2. Strict Nash Equilibrium versus Weak Nash Equilibrium

4.2.2.1. A nash equilibrium is strict if choosing the nash equilibrium strategy set produces outcome that is strictly better than other strategy sets for given the profile of all other players.

### 4.2.3. Mixed equilibrium

4.2.3.1. To find the probability of playing strategy set (a,b) with (p,1-p) for player 1, we look at player 2's payoffs. Suppose player 2 is mixing, then he must be indifferent between his strategy sets.

4.2.3.1.1. To make player 2 indifferent between his strategy sets, we can compute the mixed strategy of player 1.

## 4.3. Prisoner Dilemma

### 4.3.1. Nash equilibrium is pareto inefficient

### 4.3.2. However, strategy DC is dominant.

## 4.4. Dominance

### 4.4.1. Games may be solved by iterated deletion of dominated strategies.

4.4.1.1. dominance solvable games.

### 4.4.2. A strategy Ai is strictly dominated for player i, if there exists another strategy A'i such that utility of choosing A'i given action profile of other players is always higher than utility of choosing Ai.

### 4.4.3. Weak dominance

4.4.3.1. A strategy is weakly dominated if there exists another strategy a'i for which utility of playing a'i given action profile of other players is larger or equal to utility of playing ai, with strict inequality for some Ai.

## 4.5. Externalities

### 4.5.1. Individuals do not take externality into account in their decisions - outcomes tend to pareto inefficient

### 4.5.2. Tragedy of the Commons

4.5.2.1. having well defined property rights will internalise the externalities.

### 4.5.3. Congestion Charging

4.5.3.1. Governments can own the roads and put a price on driving.

4.5.3.2. This method of pricing the externality is called Pigou Taxation

## 4.6. Public Goods and the Free-rider problem

### 4.6.1. Non excludable

4.6.1.1. Cannot prevent non-payers from consuming the good

### 4.6.2. Non rivalrous

4.6.2.1. Each additional user does not deplete the benefits of the good

## 4.7. Cournot Game

### 4.7.1. Firm choose quantity.

4.7.1.1. p(q)q - c(q)

## 4.8. Bertrand Game

### 4.8.1. Firms choose price.

4.8.1.1. q(p)p - c(p)

4.8.1.2. Look at the different possibilities, examine the best response functions

4.8.1.3. Take note of the monopoly price p=(a-c)/2

## 4.9. Hotelling game

### 4.9.1. Tendency to choose the same product, convergence of behaviour

## 4.10. Dynamic Games

### 4.10.1. Backward induction

### 4.10.2. Sub-game perfect equilibrium

4.10.2.1. For games with imperfect information.

4.10.2.2. A subgame perfect equilibrium is a strategy set for each player that must induce a nash equilibrium in each of the sub-games.

### 4.10.3. Stackelberg Game

4.10.3.1. First Mover advantage

4.10.3.2. Credibility

### 4.10.4. Ultimatum Game

### 4.10.5. Hold-up Game

### 4.10.6. Finitely Repeated games

4.10.6.1. The final play of the game at T forms a subgame.

4.10.6.1.1. In the subgame, they must play Nash, as defined by subgame perfect N.E.

### 4.10.7. Infinitely Repeated Game

4.10.7.1. Nash Reversion Strategy

4.10.7.1.1. Always play DC until someone plays C

4.10.7.1.2. If you were to deviate, you should deviate at period 1 to maximise the gain, because of discounting.

4.10.7.2. A subgame in this game is defined as

4.10.7.2.1. any period t, including the history of the game with pay-offs from t-1 and extending into the future t+1 is the rest of the subgame.