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## Consumer Theory

### Preferences

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

Reflexivity - Something is at least as good as itself.

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

### Preferences can be represented as Utility functions

Ordinal Utility v.s. Cardinal Utility

Marginal Utility, Change in total utility from a change in one unit of good., Diminishing marginal utility, 2nd derivative of Utility function is negative, strictly convex., For a two variable function, we can get marginal utilities for both x and y.

Indifference curves, Can be used to map out utility functions and preferences., Slope of indifference curve = Marginal Rate of Substitution = dy/dx = -MUx/MUy, Decreasing marginal rate of substitution., Second derivative is negative., The more of x you have, the more willing you are willing to give some of it up to get one unit of y.

Cobb-Douglas Utility Function, MRS = -ay/bx

Perfect Substitutes, U(x,y) = a(x+y)

Perfect Complements, U(x.y) = min(ax,by)

### The Budget Constraint

PxX+ PyY = m, m,Px,Py are exogenous.

Solving a two variable constrained optimization problem., trying to maximise utility function., There's an implicit assumption that x,y larger or equals to zero., Turning it into a three variable unconstrained maximisation problem, Lagrange Multiplier, Substitution method, Substituting the constraint into the utility function, Graphical approach, Slope of MRS = Slope of budget line., Optimality condition, MRS is the rate at which consumers are willing to trade goods along the indifference curve., At the optimal point, the rate at which you are willing to trade the goods is equal to the rate at which the market is willing to trade the goods., We only want the highest indifference curve, and hence tangency is a necessary condition for an interior optimum.

### Income and Substitution effects

Normal goods, Demand increases when income increases

Inferior goods, Demand falls when income increases.

Change in prices, Increase in price of good X, Two effects, Income effect, Change in quantity of a good due to a fall in consumer's real income., Substitution effect, Change in quantity of a good due to a change in relative prices., Slutsky Equation, change in x = change in x due to substitution effect and change in x due to income effects., Hicks Substitution Effect, Rolling the budget line by keeping the original utility constant.

Change in Income, Fall in income

Giffen Goods, Inferior goods, When your income rises, you purchase less of them., No close substitutes, There's no substitutes consumers can conveniently switch to., Take up a large proportion of consumer's income, 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.

How to measure utility changes?, Compensating variation, How much income is needed to be taken away or given back to the consumer to compensate for the rise or fall in utility?, i.e. to go back to his original level of utility, Equivalence variation, 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?

### Market Demand

Summation of individual demands

Consumer surplus, The difference between the reservation price and the market price., Total benefit accrued to consumers from consuming the equilibrium amount D(p) at price p.

Elasticity, Responsiveness of one variable to a change in another., Price elasticity of Demand, Responsiveness of demand to a change in price of a good., Income elasticity of Demand, Responsiveness of demand to a change in consumer's income, > 0: normal good, > 1: luxury good, < 0: inferior good, Cross Elasticity of Demand, Responsiveness of demand to a change in price of another good., > 0 meaning substitutes, < 0 meaning complements/poor substitutes, Measured in percentage changes., |e| = |dq/dp * (P/Q)|, >1, Elastic, <1, inelastic, 0, perfectly inelastic, infinity, perfectly elastic, 1, unitary elastic

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

Budget and Time Constraint, n + l = T, i.e. we must allocate our time between work and leisure., Assumption: wage rate of w., extending from this using the concept of opportunity cost, every hour spent working is one hour not spent on leisure., conversely, the opportunity cost of leisure can be thought of as w., Combining the above two..., PxX + lw = Tw

Utility function, function of amount of good, x, and leisure time, l., Maximise... Lagrangian multiplier or using graphical approach., MRS(l,x) = Px/w, x = w(n/Px)

Extending the concept to construct a Labour Supply Curve, Income effect, When the real wage increases, consumers can buy more of X but also consume more of leisure., Substitution Effect, When the real wage increases, the "price" or opportunity cost of leisure goes up, so consumers will substitute away from leisure to consumption/work., When the wage increases, x goes up because both substitution and income effects are positive., l may go up if income effect dominates. (n goes down), labour supply will have a negative slope, I may go down if substitution effect dominates.(n goes up), labour effect will have a positive slope., Individual Supply Curve

### Inter-temporal Choice

Key concepts, Present Value, The value of all future receipts discounted into the present., Money at different points in time has different value. The value depends on the discount factor involved., discount factor is < 1 and is typically related to the interest rate., The higher the interest rates, the larger the discount factor., Present value basically means we try to factor in this discount factor and weigh everything in the present time period., Savings, We can save and consume less in the current period in order to increase future consumption

Budget Constraint, Expressing everything in present value, Present value of consumption = Present value of income, Suppose, I1, I2, c1, c2 are the variables., c1 = I1 - S, c2 = I2 + (1+i)S, S represents savings, It can be negative, in which case the consumer is a net borrower.

Marginal Rate of Time preference, MRTP = Py/Px, Py = 1/1+i, Solve for c1,c2, If c1 < I1, net saver, a change in interest rates, Income effects, You become poorer today because the returns on your savings go down, this leads to less present consumption, Substitution Effects, The opportunity cost of consumption today decreases, so you will increase present consumption., Overall effects unclear., if c1 > I1, net borrower, A change in interest rates, Income effect, A fall in interest rates means you become richer, leading you to consumer more today., Substitution effect, The opportunity cost of consumption today goes down., Both work in the same direction

## Producer Theory

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

Revenue = xPx, How does firms produce x? Not every x is feasible., Price might be related to the quantity produced (i.e. P(x)

Costs, The production function defines the production process, how inputs are transformed into outputs., f(x,y), Partial Differentiation yields the marginal product of each resource., MPx tells us how much output increases when we increase a little bit of x while keeping y fixed., Diminishing Marginal Product., However, marginal product is INCREASING in the other variable., i.e. the cross partial derivative Fxy, is positive., Cobb Douglas, Returns to scale, Constant returns to scale, If we multiply both inputs by factor k, then output will rise by factor k., If output doubles, costs doubles., Increasing returns to scale, If we multiply both inputs by factor k, output will increase by more than k., If output doubles, costs less than double. C(2z) < 2c(z), Decreasing returns to scale, If we multiply both inputs by factor k, output will increase by less than k., If output doubles, costs more than double. c(2z) > 2c(z)., A function may exhibit different types of returns to scale at different domains., i.e. a low level of production may have increasing returns to scale, but high level may result in decreasing returns., Concept is closely linked to economies of scale and MES. A market with increasing returns to scale will have relatively few firms, conversely, a market will decreasing returns to scale setting in early, will have relatively many firms., Isoquants, Analogous to the indifference curves of the utility function, Marginal rate of technical substitution (MRTS), Analogous to the MRS of the indifference curves., Rate at which you can exchange one resource for another and still produce the same amount of output., Assumption that MRTS is decreasing, i.e. diminishing marginal rate of technical substitution, Implies that the greater amount of x used, the more x the firm needs to use for every unit of y reduced in the production process., Linked to the idea of a fixed and variable factor of production, Gives all the combinations of two resources that allows you to produce a given level of output., Short run v.s. Long run, SR: There are some fixed factor of production, We can write the costs function as c(z,xbar), LR: All factors are variable, We can write the cost function as c(z,x(z)), where we choose x(z) to minimize the costs given z., In the SR, since x= xbar is fixed, the firm may not be producing at the level of x which minimises cost of producing a given z., Hence, SR costs > LR costs except when xbar = x*(z), Diagram, Similarly, for marginal costs, the LRMC and SRMC only coincide when xbar = x*(z).

The firm's problem, Profit maximization, max (revenue - costs) by choosing x,y., In the SR: one of the input is fixed., Essentially we are solving a single variable maximisation problem., Algebraic method, differentiation, Graphical method, Suppose we fix the profit, then we can draw isoprofit lines with z-y axes, The outmost line that is tangent with the production function will be the optimal amount of y., Exogenous factors: price of inputs x, price of inputs y, price of goods sold., Equivalent to choosing combination of (x,y) because price is not determined by the firm., A reframe: Cost Minimization, Assume firms produce a fixed quantity zbar., Find out what is the cost minimising combination of inputs for that quantity., Again we are exogenously given: Px, Py and zbar., Algebraically, We can express inputs as a function of Px, Py and zbar., Graphically, we can define iso-cost curves, where costs are fixed along the curve - based on the cost function expressed in x,y., This can be combined with the production function which constrains the amount of output that can be produced, and we can thus find the minimum cost at which the output can be produced using the combination of inputs, x,y., And then optimize the output, knowing the combination of inputs that will minimize the cost from first part., Choosing z depends on various factors., We can define Average costs and Marginal Costs, Think Bowl shaped LRAC and different segments for increasing and decreasing returns to scale explanation, AC (z) = F/z + Cv(z)/z, Marginal Costs are closely related to AC, MC(1) = AC(1), If AC is rising, MC must be > AC., If AC is falling, MC must be < AC., hence, when they intersect, min pt of AC must be equal MC., AVC is the integral of MC summed over output z., We can define cost function as a function of fixed costs F + variable costs., Depends on the type of Market, Perfect Competition, Large number of firms, Each firm has no control over prices, price taker, p is given., At the optimal level of production, MC=MR, when MC > MR, firms do better to produce less, When MC < MR, firms do better to produce more., From the supply curve, we can examine at the different output produced at each price., Supply curve is the upward sloping portion of the MC curve that is above AVC., we know p = MC(z) for optimality, solve for second derivative of profit max, we get MC'(z) > 0, i.e. upward sloping part of MC., Shut-down condition is revenue < variable costs. price < average variable costs. from 1) we know that P = MC, hence MC < AVC is the shutdown condition for a PC firm., Producer Surplus, Difference between seller's reservation price and actual price they got paid for., (P-AVC(z))* z, Also known as Economic Rent., LR Supply, MC(z)= MC(z, x(z)), This will coincide with the SRMC when x*(z) = xbar., Generally more price elastic, since more factors can be varied., Shutdown condition is when MC < AC., i.e. the supply curve is when LRMC > LRAC., Market Supply, Analogous to Market Demand, Horizontal Summation of firms' supply, the interaction of market demand and market supply curves will determine the market price., equilibrium price p*, At this price, firms can either be making normal, subnormal or supernormal profits in the SR., Supernormal, Normal Profits, Subnormal profit, In the LR, firms move to their long run costs curves., Firms making losses will now exit the market., Firms can also enter the market, Free entry and exit in the LR. No fixed costs/sunk costs., No barriers to entry in the PC market., Hence, in the LR, if all firms have same cost structures, derive z* where average costs is minimised and let p* = C(z*)/z* any price below p* firms will make a loss. p* will be the equilibrium price., Only NORMAL profits possible in the LR, hence P =AC(z*) must be satisfied., Entry and exit will take place until all firms make normal profits., As a new firm enters, the industry supply curve will FLATTEN., Eventually, as more and more firms enter, the LIMIT of the curve is a flat supply curve at P=P*, Homogenous products., Market flow:, 1. Industry Supply and Demand interact to determine prices., 2. Given prices, firms will decide how much to produce based on MR=P=MC, since firms are price takers., If Q that satisfies above results in P > AVC(Q) in the SR and P > AC(Q) in the LR, the firm will produce, if not the firm will leave the market/ not produce., It follows that if Q is such that P>AC(Q), at profit maximising point, it follows that the LRAC which is the summation of all the individual firm's AC curves is not at its minimum point, hence the availability of supernormal profits will cause more firms to enter the market and eventually prices will fall until P=AC(Q) at both the industry level and at the individual firm level, at which point, no firm will have the incentive to enter because they will make negative profits from entering., Monopoly, Only 1 firm, firm controls price and how much it wants to sell., Monopolist now faces a different problem, Max profits = p(z)z - c(z), MR=MC, Each additional unit lowers prices of all previous units, but adds to the total revenue., using Elasticity concept to express MR, MR=MC P>MC, Deadweight loss, 1st Degree price discrimination can result in no deadweight loss., Natural Monopoly, High Sunk costs/Fixed costs, Power Grids, railway lines, Low Marginal Costs, Hence forcing firms to sell at a low price through regulation can result in losses., Fixed Costs & Sunk Costs, Fixed costs are costs independent of the level of production, Sunk costs are the costs incurred that cannot be recovered., Suppose you bought a floorspace and pay 1000 a month. But you can sublet it for 800., Then 200 is the sunk costs, The 1000 is fixed costs, Sunk costs affect entry and exit.

## Markets

### Strategic Model

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, Players can trade amongst themselves., A pareto efficient outcome is a set of feasible allocation that has no other set of feasible allocation pareto dominating over it., 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., When both MRS are equal to each other, this condition is satisfied, as both parties will not be willing to trade away from their indifference curves to a lower one., The line joining the set of pareto efficient points is called the contract curve., Might not be desirable in the sense that it might be grossly inequitable.

### Non-Strategic Model

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

Not modelled by game theory, Not entirely true is it?

Market price given, each seller/buyer decides whether to transact at the given market price., If Willing buyers > Willing sellers, prices will increase, Excess demand, Conversely, if willing buyers < willing sellers, prices will fall., Excess Supply, The theory thus assumes that the price has somehow adjusted so this is true., At an equilibrium, any willing buyer will have found a willing seller.

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., Simple utility maximisation problem, 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., At equilibrium, it must be such that each agent chooses the same point on the edgeworth box., An equilibrium will always exists if prices keep changing in a continuous manner., There can be more than one equilibrium., Stable v.s. Unstable equilibria, A competitive equilibrium must be pareto efficient, First fundamental theorem of welfare economics, Must be on the contract curve., Walras Law, The value of aggregate excess demand, summed over all goods must be equal to zero., Second fundamental theorem of welfare economics, The best way to deal with inequity is not interfere with free decentralised markets, but to reallocate endowments using lump sum taxes and let the market decide on prices and consumption bundles., The theorem says, as long as preferences are convex, for any Pareto efficient allocation, one can find initial endowments such that this allocation is a competitive equilibrium allocation

## Game Theory

### Strategic Games

Players

Strategies

Payoffs/Preference orderings

### Nash Equilibrium

A profile of actions, each of which is an optimum choice given the actions of other players., Assumption that beliefs about other players are correct., If everyone share this belief, then it is optimum to play this way., Stable social norm: if everyone expects others to behave this way, then they should act in a certain way., Can be derived from the intersection of the best response functions., 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., Not exactly a function because it may be Many:One

Strict Nash Equilibrium versus Weak Nash Equilibrium, 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.

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

### Prisoner Dilemma

Nash equilibrium is pareto inefficient

However, strategy DC is dominant.

### Dominance

Games may be solved by iterated deletion of dominated strategies., dominance solvable games.

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.

Weak dominance, 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.

### Externalities

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

Tragedy of the Commons, having well defined property rights will internalise the externalities.

Congestion Charging, Governments can own the roads and put a price on driving., This method of pricing the externality is called Pigou Taxation

### Public Goods and the Free-rider problem

Non excludable, Cannot prevent non-payers from consuming the good

Non rivalrous, Each additional user does not deplete the benefits of the good

### Cournot Game

Firm choose quantity., p(q)q - c(q)

### Bertrand Game

Firms choose price., q(p)p - c(p), Look at the different possibilities, examine the best response functions, Take note of the monopoly price p=(a-c)/2

### Hotelling game

Tendency to choose the same product, convergence of behaviour

### Dynamic Games

Backward induction

Sub-game perfect equilibrium, For games with imperfect information., A subgame perfect equilibrium is a strategy set for each player that must induce a nash equilibrium in each of the sub-games.

Stackelberg Game, First Mover advantage, Credibility

Ultimatum Game

Hold-up Game

Finitely Repeated games, The final play of the game at T forms a subgame., In the subgame, they must play Nash, as defined by subgame perfect N.E., apply payoffs from last stage and rollback to the previous stage at T-1, essentially it's another one-shot game, since next period payoffs are not affected by this period's choices., The same arguments for all T-i, Equilibrium strategy: play n in all period regardless of what has happened so far.

Infinitely Repeated Game, Nash Reversion Strategy, Always play DC until someone plays C, If you were to deviate, you should deviate at period 1 to maximise the gain, because of discounting., A subgame in this game is defined as, 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., Now test the nash reversion strategies, 1. If no one has deviated., Play DC, this is the same as the decision at the start of the game., Consider the payoffs of deviating at this stage v.s. not deviating., Clearly better to deviate at the first stage to maximise the payoff from deviation (minimize discounting), Depends on the discount factor which depends on patience or how much one values the future., 2. If someone has deviated, play C., Clearly nash, because if other player plays C, you are better off playing C.