Лекции_Микроэкономика

Note that production decision does not depend on consumers preferences.

Separation theorem

If markets for intertemporal claims are perfect, individuals can separate investment decisions

(aimed at maximizing wealth) and consumption decision (dependent on consumer’s time preferences).

Present value rule

Due to separation theorem production decision can be delegated to managers. Managers that maximize the wealth of the firm will be making the correct investment decisions for all the owners individually regardless of the possibly differing time-preferences of the owners.

Net present value of investment project in two-period model: NPV I I F I . 1 r

If we have T periods and Rt I is the net income in period t , then

Discrete case. If the number of investment projects is finite and these projects are mutually exclusive, then the project with the maximum possible present value should be chosen (given that it is positive).

If projects are not mutually exclusive then all projects with positive NPVs should be adopted.

2.3 Applications of NPV rule: non-renewable (or exhaustible) resources

(is not covered under new syllabus)

Consider exhaustible resources industry (minerals or fossil fuels). The stock of the resource Q is constant. The owner should decide how much of the stock to extract and sell in each period. Consider two-period model. Assume that demand function is stable over time and P q - diminishing in q . Let extraction costs per unit of resource be constant and equal to c in any period.

Competitive industry.

In case of competitive industry firm is a price taker. Let pt stays for current price and pt 1 - for future price.

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A.Friedman ICEF-2012

Monopolist

As monopolist is a price maker, marginal profit equals to MR c , thus he will extract in both

Analytical derivation of the Hotelling rule for the monopolistic industry.

Graphical solution and comparison with competitive case (linear demand functions and zero marginal extraction costs).

Q

Graphical analysis suggests that monopolist would be more conservative: he would extract less today and more tomorrow. As a result the prices for a monopolized industry would be initially higher and become lower at later dates.’

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price

monopoly

Competitive industry

time

extraction

monopoly

Competitive industry

time

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3. CHOICE UNDER UNCERTAINTY

3.1 Gambles and contingent commodities

A state of the world is the outcome of uncertain situation.

Contingent commodity is the amount of consumption whose level depends on which state of the world occurs.

Flipping coin game.

Suppose for each dollar you bet in a flipping coin game, you win (and get your bet back) if a heads comes up and lose your bet when a tail comes up.

States of the world: state 1- tails comes up and state 2- heads comes up.

Contingent commodities: consumption if tails comes up (denote by c1 ) and consumption if heads comes up ( c2 ).

Endowment point - consumption bundle of contingent commodities that is available when you make no trades with the market.

In case of flipping coin game initially person has income of w in either state of the world.

Budget constraint for contingent commodities shows how much of each contingent commodity you can have in each state of the world.

Let us denote the bet by z and assume that bet can never be negative, then we have the following system that describes the budget constraint:

If we solve this system with respect to z , then we get the budget constraint of the form: c2 w w c1 or c1 c2 w 1 where 0 c1 w .

c 2

Note that after the state of the world is determined, the person will consume only one contingent commodity that corresponds to the state of the world that takes place.

Budget constraint does not extend to the right from endowment point as it was assumed that agent is not allowed to select the other side of the original bet. That is, we do not allow individual to make a bet in which he wins $1 if tails comes up and lose $ if heads comes

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up. If person is allowed to take both sides of the gamble, then his budget constraint will be a straight line that goes through initial endowment with a slope of .

c2

Budget constraint when both sides of the bet can be taken

Generalisation

Suppose that the terms of the gamble are such that consumption changes by x1 in the first state of the world (if tail appears) and by x2 in the second state of the world (if heads appears). Then we have the following system that describes the budget constraint:

c1 w x1 z

c2 w x2 z

If we solve this system with respect to z , then we get the budget constraint of the form:

s 1

Expected value of the gamble is the weighed sum of outcomes, where weights are equal to

S

probabilities: EV x ps xs .

s 1

EV of the gain in original flipping coin game is $1 p1 p2 $1 p1 1 p1 . If coin is symmetric then p1 0.5 and EV gamble = 0.5 1 .

A gamble with zero expected monetary gain is called actuarially fair gamble.

If expected monetary gain is different from zero, then such a gamble is said to be unfair.

Gamble with positive expected gain is said to be favourable; gamble with negative expected gain is called unfaivourable.

The fair odds line is a budget constraint reflecting the opportunities presented by an actuarially fair gamble.

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Going back to our example of symmetric flipping coin we can illustrate the fair odds line as a

Budget constraint when both sides of the bet can be taken

Note that expected value of consumption remains constant along fair odds line:

EV c c1 p1 c2 p2 w x1 z p1 w x2 z p2 w z x1 p1 x2 p2 w.

Preferences

Three types of attitude toward risk can be distinguished.

A person is said to be risk averse if he prefers a certain prospect with a particular expected value to an uncertain prospect with the same expected value.

A person is said to be risk neutral if he is indifferent between a certain prospect with a particular expected value and an uncertain prospect with the same expected value.

A person is said to be risk loving if he prefers an uncertain prospect with a particular expected value to a certain prospect with the same expected value.

In order to illustrate certain prospects we will draw certainty line – the locus of all possible certain consumption levels (i.e. the line c2 c1 ).

Indifference curves for risk neutral agent are given by straight lines parallel to fair odds line. Reason: any uncertain prospect for risk neutral agent is equivalent to certain bundle with the same EV. Note that along FOL expected value of consumption is constant, thus all these points lie on the same indifference curve. As more is better, agent becomes better off while moving along certainty line further from the origin.

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U1

Fair odds line

c1

To illustrate indifference curves for a risk averse agent, let us take two points on fair odds line: certainty point A and some uncertain prospect B. As both bundles have the same expected value of consumption but A is certain, then by definition risk averse agent would prefer certain bundle A to any uncertain prospect like B that gives the same EV of consumption. It means that all points on FOL would give lower utility than A. In other words, A belongs to indifference curve that lies further from the origin. As a result ICs cannot be bowed outward as in diagram (1). Otherwise point B would bring higher utility than A, which contradicts to risk aversion. ICs cannot cross fair odds line at certainty line as in diagram (2). Otherwise risky prospect (D) would be equivalent to the certain one (A) with the same EV.

Thus the IC of risk averse agent satisfies the following properties:

absolute value of the slope (MRS) at certainty points is equal to the ratio of probabilities (absolute value of the slope of FOL);

ICs are bowed in.

U1

Fair odds line

c1

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Exercise. Show that indifference curves of risk lover are bowed out and at certainty points have slope that is the same as the slope of FOL.

Optimal bet

The case of risk averse agent.

By definition, a risk-averse agent will never participate in fair game (i.e. will make zero bet) as his initial endowment lies on certainty line and is preferred to any risky prospect that belongs to FOL.

Fair odds line = Budget line

c1

Fair game

If game is favourable (this is the case if 1), then risk averse agent will take some risk and optimal bet would be positive as risk is compensated by positive expected gain.

Budget constraint

c1

Unfair favourable game

If game is unfavourable (this is the case if 1), then risk averse agent will make zero bet as expected consumption at any point on budget line is less than at initial endowment and, in addition, the endowment point is certain.

Budget line

Fair odds line

c1

Unfair unfaivourable game

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3.2 Expected utility

In presence of uncertainty utility depends on the quantities of contingent commodities and corresponding probabilities. In principle, probabilities can enter utility function in quite complex ways. Under some additional requirements on preferences utility function takes

S

linear in probabilities form: U c1 ,c2 , ,cS ; p1 , p2 , , pS ps u cs . A utility function that

s 1

takes this form is called a von Neumann-Morgenstern utility function or expected utility function (EUF).

EUF and attitude toward risk

Risk-averse person: U c1 ,c2 ; p,1 p pu c1 1 p u c2 u pc1 1 p c2 for any

u c2

u pc1 1 p c2 pu c1 1 p u c2

u c1

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