Micro

Question 1

demand function in Cobb-Douglas economy

Answer 1

x for good 1=(aI)/p1; x for good 2 = (1-a)I/p2

Question 2

weak vs strong Pareto efficiency

Answer 2

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

Question 3

weakness of the concept of PE

Answer 3

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

Question 4

non-satiation

Answer 4

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.

Question 5

how to find PE allocations (in the Edgeworth box / two-person case)

Answer 5

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

Question 6

Pareto set

Answer 6

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

Question 7

Walrasian equilibria vs. PE allocations

Answer 7

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)

Question 8

Walrasian equilibrium definition

Answer 8

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

Question 9

First Theorem of Welfare Economics

Answer 9

if (x,p) is a Walrasian equilibrim, then x is PE

Question 10

Second Theorem of Welfare Economics

Answer 10

every PE allocation is a Walrasian equilibrium

Question 11

Pareto dominate & Pareto criterion

Answer 11

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.

Question 12

public vs. private goods & excludability vs rivalry in consumption

Answer 12

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.

Question 13

Private consumption of a discrete public good

Answer 13

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.

Question 14

Voting for a discrete public good - case 1

Answer 14

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.

Question 15

Voting for a discrete public good (2)

Answer 15

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

Question 16

Voting for a discrete public good (3)

Answer 16

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.

Question 17

Efficient provision of a continuous public good

Answer 17

amount of public good: f(g₁+g₂) and the utility of agent i is U_i(f(g₁+g₂),w_i-g_i)

the condition for eficiency in the case of continuous provision of the publci goodis that the sum of marginal WTP = marginal cost of provision

Question 18

Private provision of a continuous public good

Answer 18

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

Question 19

reservation price

Answer 19

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.

Question 20

Isoquants

Answer 20

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.

Question 21

monotone preferences

Answer 21

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.

Question 22

IC curves

Answer 22

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.

Question 23

Leontief utilities

Answer 23

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.

Question 24

MRTS

Answer 24

The rate at which one factor can be substituted for another while holding the level of output constant.

Question 25

Shifts in the BL

Answer 25

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.

Question 26

Why is consumer EQ where MRS=price ratio?

Answer 26

• 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.

Question 27

CRTS

Answer 27

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”.

Question 28

DRTS

Answer 28

Increasing inputs leads to a proportionally smaller increase in output - f(tx)

Question 29

IRTS

Answer 29

An increase in inputs leads to bigger proportional increase in output - f(tx)>tf(x)

Question 30

Homogeneous functions

Answer 30

A function f(x) is homogeneous of degree k if f(tx)=t^kf(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)

Question 31

positive monotonic transformation

Answer 31

if g is a strictly incerasing function, that is a function for which x>y implies that g(x)>g(y)

Question 32

homothetic function

Answer 32

• 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

Question 33

Profit maximization and its implications

Answer 33

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

Question 34

What happens to factor demand and output supply of a profix-maximizing firm if all prices are doubled?

Answer 34

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.

Question 35

Hotelling’s lemma

Answer 35

a firm’s supply function is the derivative of the firm’s profit function w.r.t. price

Question 36

Preference axioms (without them a utility function cannot exist)

Answer 36

• 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

Question 37

duality

Answer 37

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

Question 38

Roy’s identity

Answer 38

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

Question 39

Shepard’s lemma

Answer 39

h(p,u) = de(p,u) / dp

Question 40

individually rational allocations

Answer 40

where U(x)≥U(initial endowment)

Question 41

PE on Edgeworth box edges

Answer 41

left & bottom: MRS(b)>MRS(a)

right and top: MRS(a)>MRS(b)