Micro and Game Theory

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Uncompensated Demand

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Maximizing utility given prices and total income
Max U(x,y) s.t. pxX + pyY = I

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Compensated demand

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Minimize expenditure while holding utility constant, given existing prices
Min PxX + PyY s.t. U(x,y) = fixed U

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Three properties of utility functions

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1. Completeness
2. Reflexivity
3. Transitivity

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Completness

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We always know the consumer's preferences between bundles. either x > y, y > x, or they are equal.

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Reflexitivity

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Each bundle is at least as good as itself

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Transitivity

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If a > b, and b > c, then a > c

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Weak Monotonicity

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If x is at least as large as bundle y, then a rational consumer will see bundle x as at least as good as bundle y. In other words, if you manually go through the bundle or vector and bundle x has at least as much of each good as bundle y, then weak monotonicity holds.

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Strong monotonicity

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Same as weak monotonicity, but if x >= y, and x and y are not the same point, then x is strongly preferred to y.

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monotonic transformation of a utility function

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a utility function that represents the same preferences as the original utility function. In a strictly increasing function, doing something like squaring or adding three to a utility function will not impact its demonstration of preferences.

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Convexity of a utility function

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If x is >= z and y >= z, then any convex combination of x and y will be preferred to bundle z. Example, 30% of x and 70% of y will still always be greater than z

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Strict convexity

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If x >= z, y >= z, and x and y are not the same, then any combination of x and y will be strictly preferred over z

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Homogeneous function

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If you can pull out a scaler and be left with the same original function, it is homogeneous

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Homothetic preferences

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The MRS of a homogeneous function does not change as a result of a monotonic transformation

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Strictly, directly revealed preferences

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When we are given prices, income, and a consumption decision at any given time, we can calculate to see if alternate bundles were feasible and thus if we would prefer them or not.
That is, if we could've afforded an alternative bundle but still decided to go with the original one, then it is strictly directly revealed preferred over the others.

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Indirectly revealed preferences

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When given two different budget sets, the bundle selected over the bundle present in both budget sets is indirectly revealed as preferred to the bundle that was not selected over the common bundle

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Generalized Axiom of Revealed Preference (GARP)

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In a chain of bundles where each bundle is preferred to the previous, the final bundle should not be strictly preferred to the first

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Afriat's Theorem

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GARP is a sufficient and necessary condition for consumption data to be consistent with utility maximization. If GARP is violated (two bundles are revealed preferred over each other) then we cannot generate a utility function that rationalizes the data.

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Substitution effect

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Describes how consumption is impacted by changing relative income, prices. when consumers react to an increase in a good's price by consuming less of that good and more of other goods

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Income effect

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expresses the impact of increased purchasing power on consumption

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Slutsky Equation

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formula for decomposing the effects of a price change into substitution and income effects. Total effect = substitution effect + income effect

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Walrasian equilibrium

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a vector of prices, and a consumption bundle for each agent, such that (i) every agent's consumption maximizes her utility given prices, and (ii) markets clear: the total demand for each commodity just equals the aggregate endowment. The excess demand function equals zero.

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Edgeworth Box

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diagram showing all possible allocations of either two goods between two people or of two inputs between two production processes

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Finding an equilibrium price in an Edgeworth Box

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1. Find a price where each party is buying and selling goods at a rate that satisfies them and where the excess demand is equal to zero.
2. Find the price vector and relative price slope that allows the MRS of the two individuals to be equal and for the excess demand to be zero

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A Walrasian equilibrium without room for a pareto improvement yields

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MRS of each party is equal to the relative price slope

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Pareto Efficiency

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describes an allocation in which the only way to make any individual or group of individuals better off would require making at least one other person worse off

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First Welfare Theorem of Economics

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the allocation of resources in a perfectly competitive equilibrium is pareto efficient. This says that a Walrasian equilibrium is pareto efficient. Allowing the markets to achieve equilibrium is hypothetically enough. However this doesn't hold in the real world because we do not have perfect property rights over all goods that influence our utility

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When is the first welfare theorem violated?

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by the existence of externalities, monopolies, any situation where market power exists

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Walras' Law

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The invisible hand will settle prices so that excess supply in one market will balance out with the excess demand in another market. For any price vector, excess demand will equal zero because if markets clear in all other settings, then the final market must clear as well.
A law stating that in a general equilibrium model, one of the market clearing constraints is redundant
and will hold if all other constraints hold.

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Externalities in Edgeworth Boxes

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Negative utility functions related to other individual's consumption of a good. If this is ignored, it can create a pareto inefficient allocation. With externalities, corner pareto efficient allocations may exist where the MRS of both agents are not equal but, because no other point on the indifference curves hold a better outcome, it is still Pareto efficient

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constant returns to scale

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the property whereby long-run average total cost stays the same as the quantity of output changes

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increasing returns to scale

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when long-run average total cost declines as output increases

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decreasing returns to scale

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when long-run average total cost increases as output increases

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local returns to scale

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Production functions that don't satisfy any of the global return to scale definitions. Instead, the output depends on where we fall in terms of capital and labor

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producer's problem: maximizing output

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set the MRTS for k and l = relative price level

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Producer's dual problem: minimizing cost

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Choosing a level of output and finding a cost function (the lowest possible amount of money spent to achieve this output)
Then take the derivative of the cost function to find marginal cost

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Short-Run Firm Supply Curve

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No entry or exit of firms. Locate the profit-maximizing quantity where price = marginal cost

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Long-run firm supply

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Pick the point where MC = ATC.
If ATC > MC, firms are losing money, some exit
If ATC < MC, firms are profiting, some join

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Long-run industry supply

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Under the assumption that individual firms' average cost function does not change, the industry supply curve is flat. // a curve that shows the relationship between price and quantity supplied by the industry once firms adjust fully to any change in market demand.
But, if more firms entering the market changes individual cost functions by making labor and capital more expensive, then the long-run industry supply curve can change over time

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Consumer surplus

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The area above the equilibrium price. Total surplus can be calculated as the sum of the difference between each individual's reservation price for the purchase and the amount they actually pay // the amount a buyer is willing to pay for a good minus the amount the buyer actually pays for it

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Producer surplus

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The inverse of consumer surplus. How much each producer made between the production requirement to the equilibrium price. // the amount a seller is paid for a good minus the seller's cost of providing it

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Producer surplus in long-run

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the existence of a producer surplus in the long-run represents some kind of rent increase throughout the economy. Whoever owns the inputs of that good is earning more

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nominal income

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income not adjusted for purchasing power

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Exchange rate method

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Calculate nominal income by multiplying consumption by prices then deflating the income with the exchange rate as a fraction.
Can be problematic as it can imply that a certain country consumes more when this is actually not the case

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Big Mac Index

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Tool for calculating purchasing power parity that compares prices of a Big Mac throughout the world. Done by calculating nominal income then dividing it by the local price of the big mac.

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Geary-Khamis method

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Summation of price of good J in country K, multiplied by country K's share of the total global consumption of good K, multiplied by a currency conversion factor for country K

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Requirements of a good approach for comparing wealth of nations

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1. weak continuity
2. Dependence on prices
3. Weak ranking restriction
4. Independence of irrelevant countries

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Weak continuity

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small changes shouldn't flip the rankings

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Dependence on prices

answer

system cannot rank all entities the same way, it must consider prices

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Weak ranking restriction

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if one country consumes more of every good than another, our function should assign a higher number to the higher consumption country

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Independence of irrelevant countries

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a change in Pakistan shouldn't affect the ordering of France and Germany

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Nash Equilibrium

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a situation in which economic actors interacting with one another each choose their best strategy given the strategies that all the other actors have chosen

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Iterative elimination of dominated strategies

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...

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Pure Strategy Nash Equilibrium

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Nash equilibrium in which all players choose a fixed strategy

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Mixed Strategy Nash Equilibrium

answer

a profile of mixed strategies such that no actor could do better by switching strategies, given the other player's strategy

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Best Response (Game Theory)

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the strategy (or strategies) which produces the most favorable outcome for a player, taking other players' strategies as given

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Subgame Perfect Nash Equilibrium

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In sequential games, a sequence of decisions made by the various players such that no player has an incentive to deviate.

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Cournot competition

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A type of market competition in which the firms simultaneously and independently choose how much to produce and the market price is set so that the quantity demanded of each firm's product equals the quantity produced

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question

Uncompensated Demand

answer

Maximizing utility given prices and total income
Max U(x,y) s.t. pxX + pyY = I

question

Compensated demand

answer

Minimize expenditure while holding utility constant, given existing prices
Min PxX + PyY s.t. U(x,y) = fixed U

question

Three properties of utility functions

answer

1. Completeness
2. Reflexivity
3. Transitivity

question

Completness

answer

We always know the consumer's preferences between bundles. either x > y, y > x, or they are equal.

question

Reflexitivity

answer

Each bundle is at least as good as itself

question

Transitivity

answer

If a > b, and b > c, then a > c

question

Weak Monotonicity

answer

question

Strong monotonicity

answer

Same as weak monotonicity, but if x >= y, and x and y are not the same point, then x is strongly preferred to y.

question

monotonic transformation of a utility function

answer

question

Convexity of a utility function

answer

If x is >= z and y >= z, then any convex combination of x and y will be preferred to bundle z. Example, 30% of x and 70% of y will still always be greater than z

question

Strict convexity

answer

If x >= z, y >= z, and x and y are not the same, then any combination of x and y will be strictly preferred over z

question

Homogeneous function

answer

If you can pull out a scaler and be left with the same original function, it is homogeneous

question

Homothetic preferences

answer

The MRS of a homogeneous function does not change as a result of a monotonic transformation

question

Strictly, directly revealed preferences

answer

question

Indirectly revealed preferences

answer

When given two different budget sets, the bundle selected over the bundle present in both budget sets is indirectly revealed as preferred to the bundle that was not selected over the common bundle

question

Generalized Axiom of Revealed Preference (GARP)

answer

In a chain of bundles where each bundle is preferred to the previous, the final bundle should not be strictly preferred to the first

question

Afriat's Theorem

answer

question

Substitution effect

answer

Describes how consumption is impacted by changing relative income, prices. when consumers react to an increase in a good's price by consuming less of that good and more of other goods

question

Income effect

answer

expresses the impact of increased purchasing power on consumption

question

Slutsky Equation

answer

formula for decomposing the effects of a price change into substitution and income effects. Total effect = substitution effect + income effect

question

Walrasian equilibrium

answer

question

Edgeworth Box

answer

diagram showing all possible allocations of either two goods between two people or of two inputs between two production processes

question

Finding an equilibrium price in an Edgeworth Box

answer

question

A Walrasian equilibrium without room for a pareto improvement yields

answer

MRS of each party is equal to the relative price slope

question

Pareto Efficiency

answer

describes an allocation in which the only way to make any individual or group of individuals better off would require making at least one other person worse off

question

First Welfare Theorem of Economics

answer

question

When is the first welfare theorem violated?

answer

by the existence of externalities, monopolies, any situation where market power exists

question

Walras' Law

answer

question

Externalities in Edgeworth Boxes

answer

question

constant returns to scale

answer

the property whereby long-run average total cost stays the same as the quantity of output changes

question

increasing returns to scale

answer

when long-run average total cost declines as output increases

question

decreasing returns to scale

answer

when long-run average total cost increases as output increases

question

local returns to scale

answer

Production functions that don't satisfy any of the global return to scale definitions. Instead, the output depends on where we fall in terms of capital and labor

question

producer's problem: maximizing output

answer

set the MRTS for k and l = relative price level

question

Producer's dual problem: minimizing cost

answer

Choosing a level of output and finding a cost function (the lowest possible amount of money spent to achieve this output)
Then take the derivative of the cost function to find marginal cost

question

Short-Run Firm Supply Curve

answer

No entry or exit of firms. Locate the profit-maximizing quantity where price = marginal cost

question

Long-run firm supply

answer

Pick the point where MC = ATC.
If ATC > MC, firms are losing money, some exit
If ATC < MC, firms are profiting, some join

question

Long-run industry supply

answer

question

Consumer surplus

answer

question

Producer surplus

answer

question

Producer surplus in long-run

answer

the existence of a producer surplus in the long-run represents some kind of rent increase throughout the economy. Whoever owns the inputs of that good is earning more

question

nominal income

answer

income not adjusted for purchasing power

question

Exchange rate method

answer

question

Big Mac Index

answer

Tool for calculating purchasing power parity that compares prices of a Big Mac throughout the world. Done by calculating nominal income then dividing it by the local price of the big mac.

question

Geary-Khamis method

answer

Summation of price of good J in country K, multiplied by country K's share of the total global consumption of good K, multiplied by a currency conversion factor for country K

question

Requirements of a good approach for comparing wealth of nations

answer

1. weak continuity
2. Dependence on prices
3. Weak ranking restriction
4. Independence of irrelevant countries

question

Weak continuity

answer

small changes shouldn't flip the rankings

question

Dependence on prices

answer

system cannot rank all entities the same way, it must consider prices

question

Weak ranking restriction

answer

if one country consumes more of every good than another, our function should assign a higher number to the higher consumption country

question

Independence of irrelevant countries

answer

a change in Pakistan shouldn't affect the ordering of France and Germany

question

Nash Equilibrium

answer

a situation in which economic actors interacting with one another each choose their best strategy given the strategies that all the other actors have chosen

question

Iterative elimination of dominated strategies

answer

...

question

Pure Strategy Nash Equilibrium

answer

Nash equilibrium in which all players choose a fixed strategy

question

Mixed Strategy Nash Equilibrium

answer

a profile of mixed strategies such that no actor could do better by switching strategies, given the other player's strategy

question

Best Response (Game Theory)

answer

the strategy (or strategies) which produces the most favorable outcome for a player, taking other players' strategies as given

question

Subgame Perfect Nash Equilibrium

answer

In sequential games, a sequence of decisions made by the various players such that no player has an incentive to deviate.

question

Cournot competition

answer

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