Micro Flashcards
demand function in Cobb-Douglas economy
x for good 1=(aI)/p1; x for good 2 = (1-a)I/p2
weak vs strong Pareto efficiency
weak: ther is no feasible allocation x’ such that all agents strictly prefer x’ to x strong: there is no feasible allocation x’ such that all agents weakly prefer x’ to x, and some agents strictly prefers x’ to x generally, when we say sth is “Pareto efficient” we refer to its weak definition
weakness of the concept of PE
an allocation where one agent gets everything htere is in the economy and all other agents get nothing will be PE, assuming the agent who has everything is not satiated
non-satiation
There are diminishing returns to consuming more of a good, but you can never consume so much of it that having more incurs a disutility. Local nonsatiation is implied by monotonicity of preferences.
how to find PE allocations (in the Edgeworth box / two-person case)
fixing one agent’s utility function at a givenlevel and maximizing the other agent’s utility function s.t. this constraint MRS for both agents must be the same–>tangency condition
Pareto set
set of Pareto efficient points - these are the loci of tangencies drawn in the Edgeworth box; also known as the contract curve,since it gives the efficient set of contracts fr allocations
Walrasian equilibria vs. PE allocations
one-to-one correspeondence between the two each Walrasian equilibrium satisfies the FOC for utility maximization that the MRS between the two goods for each agent be equal to the price ratio between the two goods. Since all agents face the same price ratio at a Walrasian equilibrium–>all agents have same MRS. At any PE allocaiotn, the MRS must be equal across agents as well (bc moving from that point would make one person not optimal, i.e. worse off–>PE)
Walrasian equilibrium definition
an allocation-price pair(x,p) is a Walrasian equilibrium if the allocaiton is feasible, and each agent is making tan optimal choice from his budget set
First Theorem of Welfare Economics
if (x,p) is a Walrasian equilibrim, then x is PE
Second Theorem of Welfare Economics
every PE allocation is a Walrasian equilibrium
Pareto dominate & Pareto criterion
the allocation x’ is said to Pareto dominate x if everyone prefers x’ to x. If each individiual prefers x’ to x, it seems noncontroversia to asert tat x’ is better than x and any projects that move us from x to x’ should be undertaken. This is the Pareto criterion.
public vs. private goods & excludability vs rivalry in consumption
public vs private: The consumption of private goods only affects a single economic agent. Consider bread: You and I comsume different amounts of bread and, if I consume a particular loaf of bread, you are excluded from consuming the same loaf of bread.
Private consumption of a discrete public good
if consumer 1 buys the public good, he receives u(PG)-c utility, and if consumer 2 doesn’t want to buy the good he receives u(PG)>U(PG)-c utility. Thus, both consumers want to free ride and don’t buy the public good–>prisoner’s dilemma. The dominant strategy here is (don’t buy, don’t buy). Thus, the net result is that the good isn’t provided at all, even though it would be efficient to do so.
Voting for a discrete public good - case 1
The amount of a publci good is often determined by voting - will this generally result in an efficient provision?
Ex. 1: 3 consumers who vote to decide whether or not to provide a public good which costs $99 to privde. If a majortiy votes in favor of provision they will split the cst equally and each pay $33. The reservation prices of the three consumers are r1=90, r2=30, r3=30–>sum of reservation prices>cost of provision. But, in this case only 1 will votein favor because only he receives a net positive benefit if the good is provided–>problem with majority voting: only measures ordinal preferences for the public good, whereas the eficiency condition requires a comparison of WTP. 1 would be willing to compensate the other consumers to vote in favor, but that option might not be available.
Voting for a discrete public good (2)
different kind of voting: individuals state their WTP for the public good and the good will be provided if the sum of the stated WTP>c. If the cost shares are fixed–>no equilibrium. consider the example of three voters before: 1 is made better off if the good is provided, so he may as well announce an arbitratily large positive #. For 2&3 other way round
Voting for a discrete public good (3)
each person announces their WTP for the public good. If the sum of stated WTP≥c, the good will be provided and each person pays the amount he announced. In this case if provision of the public good is PE, then this is an equilibrium. Any set of announcements such that each agent’s announcement≤his reservation price and that sum up t the cost of the public good is an equilibrium. However, there are many other inefficient equilibria, e.g. all agents announcing 0 WTP for the public good will typically be an equilibrium.
Efficient provision of a continuous public good
amount of public good: f(g1+g2) and the utility of agent i is Ui(f(g1+g2),wi-gi)
the condition for eficiency in the case of continuous provision of the publci goodis that the sum of marginal WTP = marginal cost of provision
Private provision of a continuous public good
Supose that each agent independently decides how much he wants to contribute to the public good. If agent 1 thinks that agent 2 will contribute g2, then 1’s utility maximization problem is maxg1 u1(g1+g2,w1-g1) such that g1≥0. The constrain is a natural restriction in this case; it says that 1 can voluntarily incerase the amount of the public good, but he cannot unilateraly decrease it
reservation price
A reservation is a limit on the price of a good or a service. On the demand side, it is the highest price that a buyer is willing to pay; on the supply side, it is the lowest price at which a seller is willing to sell a good or service.
Isoquants
gives all input bundles that produce exactly y units of output
Properties:
- An Isoquant Slopes Downward from Left to Right: This implies that the Isoquant is a negatively sloped curve. This is because when the quantify of factor K (capital) is increased, the quantity of L (labor) must be reduced so as to keep the same level of output.
- An Isoquant that Lies Above and to the Right of Another Represents a Higher Output Level: It means a higher isoquant represents higher level of output.
- Isoquants Cannot Cut Each Other:
If two isoquant are drawn to intersect each other, then it is a negation of the property that higher Isoquant represents higher level of output to a lower Isoquant. The intersection at point E shows that the same factor combination can produce 100 units as well as 200 units. But this is quite absurd. How can the same level of factor combination produce two different levels of output, when the technique of production remains unchanged.
- Isoquants are Convex to the Origin: This property implies that the marginal significance of one factor in terms of another factor diminishes along an ISO product curve. In other words, the isoquants are convex to the origin due to diminishing marginal rate of substitution.
monotone preferences
In economics, an agent’s preferences are said to be weakly monotonic if, given a consumption bundle, the agent prefers all consumption bundles that have more of all goods.
Much of consumer theory relies on a weaker assumption, local nonsatiation.
IC curves
gives all input bundles that produce exactly y units of output
Properties:
- Negatively-sloped: The indifference curves must slope down from left to right. This means that an indifference curve is negatively sloped. It slopes downward because as the consumer increases the consumption of X commodity, he has to give up certain units of Y commodity in order to maintain the same level of satisfaction.
- Convex to the origin: This is equivalent to saying that as the consumer substitutes commodity X for commodity Y, the marginal rate of substitution diminishes of X for Y along an indifference curve. As the consumer moves from A to B to C to D, the willingness to substitute good X for good Y diminishes. This means that as the amount of good X is increased by equal amounts, that of good Y diminishes by smaller amounts.
- Indifference Curve Cannot Intersect Each Other: Given the definition of indifference curve and the assumptions behind it, the indifference curves cannot intersect each other. It is because at the point of tangency, the higher curve will give as much as of the two commodities as is given by the lower indifference curve. This is absurd and impossible.
- Don’t touch the axes: One of the basic assumptions of indifference curves is that the consumer purchases combinations of different commodities. He is not supposed to purchase only one commodity. In that case indifference curve will touch one axis. This violates the basic assumption of indifference curves.
Leontief utilities
Leontief utility functions represent complementary goods. For example:
Suppose x is the number of left shoes and y the number of right shoes. A consumer can only use pairs of shoes. Hence, his utility is min(x,y)
Properties:
- A consumer with a Leontief utility function has the following properties:
- The preferences are weakly monotone but not strongly monotone: having a larger quantity of a single good does not increase utility, but having a larger quantity of all goods does.
- The preferences are weakly convex, but not strictly convex: a mix of two equivalent bundles may be either equivalent to or better than the original bundles.
MRTS
The rate at which one factor can be substituted for another while holding the level of output constant.
Shifts in the BL
The price line is determined by the income of the consumer and the prices of goods in the market. If there is a change in the income of the consumer or in the prices of goods, the price line shifts in response to a exchange in these two factors.
- Income changes: When there is change in the income of the consumer, the prices of goods remaining the same, the price line shifts from the original position. It shifts upward or to the right hand side in a parallel position with the rise in income. A fall in the level of income, product prices remaining unchanged, the price line shifts left side from the original position. With a higher income, the consumer can purchase more of both goods than before but the cost of one good in terms of the other remains the same.
- Price changes. Now let us consider that there is a change in the price of one good. The income of the consumer and price of other good is held constant. When there is a fall in the price of one good say commodity A, the consumer purchases more of that good than before. A price change causes the budget line to rotate about a point.
Why is consumer EQ where MRS=price ratio?
- Budget Line Should be Tangent to the Indifference Curve: The consumer cannot be in equilibrium at any other point on ICs. For instance, point R and S lie on lower IC1 but yield less satisfaction. As regards point U on IC3, the consumer no doubt gets higher satisfaction but that is outside the budget line and hence not achievable to the consumer. The consumer’s equilibrium position is only at point C where the price line is tangent to the highest attainable indifference curve IC2 from below.
- Slope of the Price Line to be Equal to the Slope of Indifference Curve: Geometrically, at tangency point C, the consumer’s substitution ratio is equal to price ratio Px / Py. It implies that at point C, what the consumer is willing to pay i.e., his personal exchange rate between X and Y (MRSxy) is equal to what he actually pays i.e., the market exchange rate.
CRTS
When an increase in inputs (capital and labour) cause the same proportional increase in output - f(tx)=tf(x).
In economic theory we often assume that a firm’s production function is homogeneous of degree 1 (if all inputs are multiplied by t then output is multiplied by t). A production function with this property is said to have “constant returns to scale”.
DRTS
Increasing inputs leads to a proportionally smaller increase in output - f(tx)
IRTS
An increase in inputs leads to bigger proportional increase in output - f(tx)>tf(x)
Homogeneous functions
A function f(x) is homogeneous of degree k if f(tx)=tkf(x). The two most improtant degrees in economics are the 0th and 1st degree. A 0-degree homogeneous function is one for which f(tx)=f(x), a 1st degree f(tx)=tf(x)
positive monotonic transformation
if g is a strictly incerasing function, that is a function for which x>y implies that g(x)>g(y)
homothetic function
- In consumer theory, a consumer’s preferences are called homothetic if they can be represented (i.e. monotonic transformation) by a utility function which is homogeneous of degree 1. This is true if and only if the marginal rates of substitution between any two
goods is homogeneous of degree 0 in x. With only two goods, that implies that the marginal rate of substitution is a function of the ratio of the two quantities.
- For example, in an economy with two goods x and y, homothetic preferences can be represented by a utility function U that has the following property: for every a>0: u(a*x,a*y)=a*u(x,y), i.e. CTRS
- In a model where competitive consumers optimize homothetic utility functions subject to a budget constraint, the ratios of goods demanded by consumers will depend only on relative prices, not on income or scale. This translates to a linear expansion path in income: the slope of indifference curves is constant along rays beginning at the origin. This is to say, the Engel curve for each good is linear.
- Utility functions having constant elasticity of substitution (CES) are homothetic. Linear utilities, Leontief utilities and Cobb–Douglas utilities are special cases of CES functions and thus are also homothetic.
- On the other hand, quasilinear utilities are, in general, not homothetic. E.g, the function cannot be represented as a homogeneous function, e.g. u(x,y)=squareroot(x)+y
Profit maximization and its implications
profit is maximized where MR=MC. This has two important implications:
- One decision the firm makes is to choose its level of output. The fundamental condition for profit maximization tells s that eht level of output should be chosen so that the production of one more unit of output should produce a MR=MC of production.
- Another decision of the firm is to determine how much of a specific factor - say labor - to hire. The fundamental condition for profit maximization tells us that the firm should hire an amount of laor such that the MR from employing one more unit of labor = MC of hiring that additional unit of labor
If we break up revenue and cost, we have how much a firm sells of various outputs, the price of each output, how much a firm uses of each inut and the prices of input. Setting the prices and quantities is bound by constraints:
- Technology constraint, i.e. feasibility of production plan
- market constraints, i.e. affect of actions of other agents on the firm (e.g. if competition charges lower prices etc). Simplest form of market behavior: price taking, i.e. price is an exogenous variable
Thus, the firm is only concerned with profit-max levels of outputs and inputs
What happens to factor demand and output supply of a profix-maximizing firm if all prices are doubled?
They don’t change–>if demand and supply functions change if prices are multiplied by t, then the firm is not maximizing its profits. In other words, the factor demand functions must be homogeneous of degree 0.
Hotelling’s lemma
a firm’s supply function is the derivative of the firm’s profit function w.r.t. price
Preference axioms (without them a utility function cannot exist)
- General: A set of assumptions that characterize rational preferences. If preferences are complete, reflexive, transitive, continuous, and
strongly monotonic, there exists a continuous utility function.
- Axiom of transitivity: preferences have to be consistent. If x is at least as good as (i.e. weakly preferred to) y and y is at least as good as z, then x is at least as good as z. Symbolically, this can be stated as: if A≥B and B≥C then A≥C.
- Axiom of order (Completeness): people are able to make decisions between options. Given any two options x and y then either x is at least as good as y or y is at least as good as x. For all x and y we have x≥y or y≥x or both.
- reflexivity: x is at least as good as x
- Continuity: Preferences will change with small changes in bundles.
- strong monotonicity:
- weak: if x≥y, then x≥(weakly preferred to) y
- strong: if x≥y and x/=y, then x>(strictly preferred to)y
duality
Definition: the principle that optimization problems may be viewed from either of two perspectives, the primal problem or the dual problem. This means you can maximize the primal problem (utility) or minimize the dual problem (expenditure).
Marchallian demand function: 𝐱(𝐩,𝑚) - max 𝑢 𝐱 s.t. 𝐩𝐱≤𝑚
Hicksian demand function: 𝐡(𝐩, 𝑢) - min 𝐩𝐱 s.t. 𝑢 𝐱 ≥ 𝑢
Indirect utility function: 𝑢(𝐩,𝑚) = 𝑢(𝐱(𝐩,𝑚))
Expenditure function: 𝑒(𝐩, 𝑢) = 𝐩𝐡(𝐩, 𝑢)
u(p,m) and e(p,u) are the inverse of each other
Roy’s identity
demand (x(p,m)) = -(du(p,m)/dp) / (du(p,m)/dm)
demand = -derivative of u w.r.t. p / derivative of u w.r.t. m
Shepard’s lemma
h(p,u) = de(p,u) / dp
individually rational allocations
where U(x)≥U(initial endowment)
PE on Edgeworth box edges
left & bottom: MRS(b)>MRS(a)
right and top: MRS(a)>MRS(b)
