Micro Flashcards
Economic model
Provide simplified portraits of the way individuals make decisions, firms behave, and the way these two interact to establish markets.
Two general methods used to verify economic models
- Direct approach seeks to establish validity of the basic assumptions the model is based on. 2. Indirect approach shows that a simplified model correctly predicts real world events.
3 general features of economic models
Ceteris Paribus assumption, economic decision makers seek to optimise something, distinction between positive and normative questions
Exogenous variables
Variables that are outside of the decision makers control. E.g households: price of goods, firms: price of inputs
Endogenous variables
Variables that are determined within a model as the result of a decision. E.g household: quantities bought, firms: output produced
Agents
Rational and optimise. They choose the best given some economic constraints. Assume they have perfect information.
Supply in competitive market
Sellers bring these to the market. The goods could be exchanged for other goods or income. If selling goods doesn’t make an improvement, rationality will prevent suppliers from producing.
Demand in competitive market
Buyers of goods have a reservation price, maximum willingness to pay.
Perfect competition
Buyers and sellers have no influence on price or quantity so R = p*q
Equilibrium in competitive market
Prices adjust until demand equals supply. The market clears
Ordinary monopoly
Seller can influence price by controlling supply, R = p(q)*q
Positive economics
Descriptive. Determines how resources are in fact allocated in an economy.
Normative economics
Judgemental. Taking a stance about what should be done.
Pareto improvement
Occurs if one persons welfare can be improved without detriments to others. If not possible, we have Pareto efficiency
Although marshallian model is useful …
It is a partial equilibrium model, looking at only one market at a time.
Production possibility frontier
Various amounts of two goods that an economy can produce using its available resources during some period. Reminds us that resources are scarce. Shows us the opportunity cost of producing more of one good as the decrease in amount of the other.
Preference
Is a binary relation over objects of choice
Satiation
The effect where the more of a good one possesses, the less one is willing to give up to get more of it. Once you pass satiation point, consuming more means enjoying it less.
Utility
A function that represents a preference.
Utility function
Translates the behaviour of the consumer into a mathematical function.
Utility rankings
Ordinal rankings. It’s not possible to compare utilities of different people.
Preferences must be..
Rational: complete and transitive and monotone
A preference is complete if:
x>y or y>x or both (x~y)
A preference relation is transitive if:
x>y and y>z then x>z
A preference relation is continuous if:
The upper contour and low contour sets are closed. The sets have no holes.
Consequence of transitivity
Different indifference curves cannot intersect
Ordinal utility
Any increasing transformation of a utility function also gives a utility function
Ordinal
Tells us that one bundle is better than another. Doesn’t tells by how much.
Indifference curve
Represents all alternative combinations of x and y for which an individual is equally well off.
why is the slope of an IC negative
to show that if the individual is forced to give up some of y, they must be compensated by an additional amount of x to remain indifferent between two bundles.
convexity definition
a set of points is said to be convex if any two points within the set can be joined by a straight line that is contained completely within the set. Implies that ICs are differentiable almost everywhere.
strict convexity
assume differentiability everywhere,
marginal rate of substitution
is the negative tradeoff between two goods - (dU/dx)/(dU/dy), px/py
perfect substitutes: U(x) = ax1 + bx2
MRS is constant (equal to a/b), convexity holds.
perfect complements: U(x) = min{ax1, bx1}
smallest numbers decide utility, MRS = 0/infinity/not defined. strict not strong monotonicity. convexity but not strict convexity. L-shaped ICs. consumption occurts at the vertices of the ICs.
Cobb Douglas: U(x) = x^a + x^b
satisfy all requirements in strongest form: convexity and monotonicity. MRS changes along Ic.
constant elasticity of substition: U(x) = (x^§ + x^§)^1/§
value of delta is not euqal to 0 but smaller than 1. if delta was larger than 1, IC would be concave. when delta = 1, this is perfect subtitutes. as delta approaches 0, the function approaches Cobb Douglas. as delta approaches negative infinity, the function approaches perfect complements.
Shephards Lemma
dE/dpi = h*i
Marshallian demand solves
maximisation problem
Hicksean demand solves
minimisation problem
Roy’s identity
x*i = - Vpi / Vy. Consumers demand given by ratio of marginal indirect utilities.
corner solution
an individual’s preferences may be such that they obtain maximum utility by choosing no amount of one of the goods.
marshallian demand for cobb douglas
x1=(ay)/p1, x2=(by)/p2
direct utility
utility across all possible consumption bundles
indirect utility
represents the maximum utility a consumer can attain given the prices of goods and the consumer’s income. Substituting marshallian demand into utility function gives indirect utility. V(p,w) = max U(x) subject to p*x <= w
properties of indirect utility
continuous in prices, utility and income, small changes have small effects. assume differentiability. homogenous of degree 0, strictly increasing in income and decreasing in prices.
properties of the expenditure function
continuous in prices and target utility. homogenous of degree 1 in prices. expenditure function and indirect utility are inverse functions of one another. expenditure is strictly increasing and unbounded in target utility. non-decreasing and concave in prices
homogenous of degree 0
if you change prices and income, the outcome doesnt change
homogenous of degree 1
if you double prices, outcome doubles
normal good
dx/dm >= 0
inferior good
dx/dm < 0
income effect
the change in demand for a good caused by a change in consumer’s purchasing power or real income.
substitution effect
consumers replace one good with another due to price change
increase in demand
outward shift of demand curve
increase in quantity demanded
movement along the curve
compensated demand function can be found from
differentiating expenditure function with respect to prices
inferior good def
when income increases, consumption decreases
income elasiticty def
measures the sensitivity of changes in consumption relative to changes in income.
income elasticity formula
(dxi/dm)(m/xi)
own price elasiticty
changes in demand relative to changes in price: (dxi/dpi)(pi/xi)
cross price elasiticty
(dxi/dpj)(pj/xi)
inelastic demand
-1 < e < 0
elastic demand
e < -1
slutsky equation in elasticity form
e = ẽ - se, where s=(px/m) is income share
whether hicksean and marshallian price elasticities differ much depends on…
the importance of income effects in the overall demand.
the slutsky elasticity equation shows that hicksean and marshallian own-price elasticities will be similar if….
either of the two conditions hold: 1) the share of income devotes to x is small, 2) the income elasticity of demand of x is small. These both reduce the importance of the income effect
engel’s law
percentage of income allocated for food purchases decreases as a household’s income rises, while the percentage spent on other things increases.
cournot aggregation
budget share s1 of good 1 is small when good 1 has many substitutes (ϵi1>0), and big when it has many complements (ϵi1<0)
euler theorem
the net sum of all price elasticities together with the income elasticity for a good must sum to zero
consumer surplus
monetary measure of the utility gains and losses that individuals experience when price changes. Area below the compensated demand curve and above market price.
compensating variation
compensation required to reach same utility level after a price change: CV = E(p1x, py, U) - E(p0x, py, U)
1st proof that the slope of compensated demand curve is negative
based on quasi-concavity of utility functions, any IC must exhibit diminishing MRS, any change in price will induce a quantity change in opposite direction when moving along IC
2nd proof slope of compensated demand curve is negative
derives from shephards lemma. because the expenditure function is concave in prices, the compensated demand function (the derivative of expenditure) must have a negative slope
theory of revealed preferences
two bundles, A and B. At some price and income level, the individual can afford both bundles but chooses A, A has been revealed preferred to B. Principle of rationality says under any different price-income arrangement, B can never be revealed preferred to A. If B is chosen at another price-income arrangement, the individual could not afford A.
primal relationship among demand concepts
maximise U(x,y) s.t. I = (p_x)x + (p_y)y –> indirect utility function U*=V(p_x, p_y, I) –> Roys identity –> marshallian demand x(p_x, p_y, I) = (dV/dp_x)/(dV/dI)
marshallian demand in relation to indirect utility function
x(p_x, p_y, I) = (dV/dp_x) / (dV/dI)
dual relationship among demand concepts
minimise E(x,y) s.t. Ū=U(x,y) –> expenditure function E*=E(p_x, p_y, Ū) –> shephards lemma –> compensated demand x(p_x, p_y, U) = dE/dp_x
two goods are said to be gross substitutes is
dxi/dpj > 0
two goods are said to be gross complements if
dxi/dpj < 0
consumption smoothing
saving some income so you can rely on savings if you lose job. income is smooth over time
weak axiom of revealed preferences
if both x and y are available in 2 different price situations and x is chosen in the first, consistent behaviour is revealed if y is not chosen in the next instant
if x is directly revealed preferred to y, and y is directly revealed preferred to z, then…
x is indirectly revealed preferred to z
Strong axiom of revealed preferences SARP
if x is indirectly or directly revealed preferred to y then y cannot be indirectly or directly revealed preferred to x.
preference sign with a star
preference is based on choice which is based on the revealed preference
will a rational and continuous preference always generate a choice function that satisfied WARP?
yes.
index numbers are
averages
quantity index with weights w1, w1; baseline x1B, x2B; choice at time x1T, x2T
= (w1x1T + w2x2T) / (w1x1B + w2x2B)
endowments
time, assets
Paasche index with weights p1T, p2T
= (p1x1T + p2x2T) / (p1x1B + p2x2B)
if net demand > 0
net buyer
if net demand < 0
net seller
changes in endowment affect
only the budget constraint, equivalent to income changes
changes in prices affect
jointly affect income and price
price offer curve
The curve containing all the utility- maximising bundles
slutsky equation with endowment income effect
dxi(p,m)/dpj = (dxi(p,U)/dpj) + (wj - xj(p,m))*(dxi(p,m)/dm)
what does strict convexity ensure
there is only one tangent line at IC
how to get whole market demand
add up all the demands at every price
competitive market with endowments
consumers are endowed with goods and choose optimally. consumers have no influence on how trade happens
pure exchange
feasible (re)allocations
pareto efficient allocation
a feasible allocation such that there is no way to improve any of the consumers without harming someone
contract curve
the set of all pareto efficient allocations
tangency condition to get the contract curve
in a two goods case, two MRS are equal at one particular allocation in feasible set.
market clearing
when supply = demand
blocking coalition
a group of consumers that object to a proposed feasible allocation because they lose out. they could leave the economy and do better
barter equilibrium
a feasible allocation that cannot be blocked by a coalition and is pareto efficeint.
equation for the minimum utility required before blocking allocations
Ux(wx)
core of the exchange economy
the set of all barter equilibria - all feasible and unblocked allocations
allocation in the core
there is no excess demand or supply
if excess equation is positive
excess demand
if excess equation is negative
excess supply
walras’ law
at all prices the value of excess demand is 0.
theorem of walrasian equilibrium
in a market where consumer behaviour accords with their preferences, then excess demand is continuous in prices, homogenous of degree 0, and Walras’ law holds
walrasian equilibrium price vector
z(p*)=0
1st Welfare Theorem
the WEA x(p*) is an allocation in the core, it is pareto efficient - under certain assumptions, any competitive equilibrium in an exchange economy is Pareto efficient. This means that at equilibrium, no other allocation can improve the welfare of one consumer without making another worse off
2nd welfare theorem
there exists prices p** such that for an appropriate redistribution of intial endowments, allocation is WEA at prices p**.
production function
indicates the maximum amount of output that can be generated by the efficient combination of inputs
production function assumptions
strictly increasing, continuous and differentiable, strictly quasi concave
isoquants
combinations of inputs that lead to the same output level
isoquants - perfect substitutes
straight diagonal lines
isoquants - perfect complements
L shaped
isoquants - cobb douglas
curves
isoquants - CES
curves that go more into 0,0
marginal product
measures the additional effect on production of one factor input by keeping others constant - is positive as f is increasing. df/dy
change in marginal product will be
negative due to diminishing returns
factor productivity
average product of a factor input, f(y)/y. can be seen as a measure of efficiency when other factors fixed
Marginal rate of technical substitution
measures the substitutability of two factors, MPy,i / MPy,j
elasticity of substitution
unit free measure of the degree of substitutability between factors
short run
at least one factor input is fixed
long run
firm can change all factor inputs
f(sy) = sf(y) = 1
constant returns to scale
f(sy) = sf(y) > 1
increasing returns to scale
f(sy) = sf(y) < 1
decreasing returns to scale
when q(t) = f(t,y) = A(t)f(y) what is changes over time
dq/dt = (dA/dt)f(y) + A(df(y)/dt)
cost function
C(w,q) = min w*y such that f(y) >= q
cost function for two inputs
C(w1, w2, q) = min w1k + w2l
isoquants
combinations of input 1 and input 2 that deliver the same level of output, MRTS = w1/w2
properties of the cost function
C is zero for lowest production level; continuous in input prices and target output; homogenous of degree 1 in prices; cost is strictly increasing; non-decreasing in prices; concave in prices
average cost
TC(q) / q = C(w,q) / q
marginal cost
MC(q) = dC(w,q)/dq
SR average cost
C(q) / q + F/q = AVC + AFC
SR marginal cost
dC(q)/dq + dF/dq
LR decision of firm
whether to enter the market or not
if AVC is decreasing
MC < AVC
if AVC is increasing
MC > AVC
if AC is decreasing
MC < AC
if AC is increasing
MC > AC
SR costs vs. LR costs
SR costs are at least as high as LR costs
SR decision of firm
whether to supply to the market or not
LR equilibrium
all the profit will be distributed among the acting firms until there are no profits
profit function
pi (p,w) = max pq - wy
1st order condition of profit max
p = MC(q*)
2nd order condition of profit max
d^2C(q) / dq^q >= 0 or MC(q) is increasing
in SR if firm decides to produce nothing then…
they will still have fixed costs so are making a loss
monopoly profit max equation
MC = MR
monopoly in practice
a monopoly cannot exploit all consumers because it cannot differentiate between consumers so it will set one price and those consumers who wish to buy above that price will be served, the others will not. revenue depends on quantity supplied with determines the price
crusoe production - at prices (p,w) : pi(p,w) = px2 - wl, iso-profits =
x2 = pi/p + (w/p)l
budget line in general equilibrium with production
= pi/p + (w/p)l
marginal rate of production transformation (MRPT) in a PPF
= - MCx1 / MCx2
benefits of general equilibrium
preferences and technology exogenous; under perfect information and price taking assumption; prices endogenous and so is WEA
limits of general equilibrium
externalities; lack of information; market power; public goods; efficiency not equal to optimal distribution
an allocation is feasible if
it respects the initial endowments of both consumers
An allocation is unblocked if
no two consumers can mutually improve their utility through trade from this allocation
walrasian equilibrium
an allocation where all markets clear (demand equals supply) and marginal rates of substitution for all consumers are equal to the price ratio
general formulation of the First Welfare Theorem for an exchange economy
Consumers: n consumers with continuous and quasi-concave utility functions Ui(xi), where xi = (xi1, xi2) represents the consumption bundle of goods 1 and 2 for consumer i.
Endowments: Each consumer has an initial endowment ei = (e1i, e2i) of both goods.
Production Efficiency: There are no production possibilities, meaning the economy solely focuses on exchange of existing endowments.
Competitive Equilibrium: There exists a price vector p = (p1, p2) for the two goods and consumption bundles xi for each consumer such that:
Market Clearing: Aggregate demand for each good equals aggregate supply, z(p*)=0
Individual Optimization: Each consumer maximizes their utility Ui(xi) given their budget constraint p * xi <= p * ei.
Theorem: Under the above assumptions, the competitive equilibrium (p, x) is Pareto efficient.
general formulation of the Second Welfare Theorem for an exchange
economy
Consumers: n consumers with continuous and quasi-concave utility functions Ui(xi), where xi = (xi1, xi2) represents the consumption bundle of goods 1 and 2 for consumer i.
Endowments: Each consumer has an initial endowment ei = (e1i, e2i) of both goods.
Pareto Efficiency: There exists an efficient allocation (x, y) for all consumers, where no other allocation can make someone better off without harming another.
Redistribution: There exists a feasible redistribution of endowments e’i = (e’1i, e’2i) for each consumer such that the sum of redistributed endowments remains the same as the original total: ∑ni=1 e’1i = ∑ni=1 ei1 and ∑ni=1 e’2i = ∑ni=1 ei2.
Theorem: Under the above assumptions, there exists a price vector p = (p1, p2) and consumption bundles xi for each consumer such that:
Competitive Equilibrium: The allocation (x, y) is a competitive equilibrium for the economy with redistributed endowments
a utility function of 3 goods that is quasi linear and homogenous of degree 1
U(x1, x2, x3) = x1 + sqrt(x2x3)
can a utility function of 2 goods be quasi linear and homogenous of degree 1
no
expenditure function def
represents the minimum cost required to achieve a certain level of utility. E(p,u) = min p*x subject to U(X) >= u
the duality result holds under what assumptions and why
holds under the assumptions of rationality, continuity, and strict convexity of preferences. These assumptions ensure a unique solution to the consumer’s optimization problem and a well-defined relationship between the indirect utility and expenditure functions
duality result
V(p,w)=u <-> E(p,u)=w. the indirect utility evaluated at utility level u is equal to income w if and only if the expenditure function is equal to income.
income effect def
describes how an increase in income or purchasing power can change the quantity of goods consumers demand
substitution effect def
the decrease in sales for a product that can be attributed to consumers switching to cheaper alternatives when its price rises
income and substitution effect on normal goods
both work in the same direction; a decrease in the relative price of the good will increase quantity demanded both because the good is now cheaper than substitute goods, and because the lower price means that consumers have a greater total purchasing power and can increase their overall consumption
income and substitution effect on normal goods
work in opposite directions
income and substitution effect on Giffen goods
positive income effect and negative substitution effect
example of preferences that are represented by a continuous utility function that allows for fat indifference curves
perfect complements: U(x) = min{ax1, bx1}
marshallian demand derived from
utility maximisation problem
hicksean demand derived from
expenditure minimisation problem
