Lec 2Utility Maximization

Utility is a purely theoretical construct defined as:- If an individual prefers bundle x-(x1,x2) to another bundle y-(y1,y2) then an

individual is said to get “a higher level of utility” from bundle x than bundle y- If an individual is indifferent between a bundle x and another bundle y, then

an individual is said to get “the same level of utility” from bundle x and y

The only property of a utility assignment that is important is how it orders the bundles of goodsThis is a theory of ordinal utility

A utility function U is just a mathematical function that assigns a numeric value to each possible bundle such that:

- If an individual strictly prefers bundle x to bundle y, then U(x1,x2)>U(y1,y2)- If an individual is indifferent between a bundle x and y, U(x1,x1)=U(y1,y2)

All bundles in an indifference curve have the same utility levelIf U(x1,x2) is a utility function, then any positive monotonic transformation of it is a utility function that represents the same preferences

Examples of Utility Fcns

There is no unique utility fcn representation of a preference relation

Perfect substitutes- Linear utility functions:U(x1,x2)= ax1 + bx2, with a> and b> 0

Perfect complements- Liontief utility functionsU(x1,x2) = min{ax1, bx2} , with a> and b>0

Quasilinear utility functionsU(x1,x2) = k = v(x1) +x2Eg. U(x1,x2) = root(x1) + x2 or U(x1,x2)= lnx1 + x2

Cobb-Douglas utility fcnsU(x1,x2) = x1

c x2d or V(x1,x2) = c lnx1 + d lnx2) with c>0 and d>0

MU1 = c x1^c-1 x2^dMU2 = d x1^c x2^d-1MRS = - c x2/ d x1

MU1 = c/x1MU2= d/x2MRS= - c x2/ d x1

Marginal Utility

The marginal utility of commodity i is the rate-of-change of total utility as the quantity of commodity i consumed changes

MU i=∂U∂x i

MU i - the partial derivative of the utility fcn with respect to good i

Ex/

The marginal rate of substitution (MRS) is both:- The slope of an indifference curve at a particular point, and- The negative ratio of the MU at that particular point

The MRS can be interpreted as the rate at which a consumer is willing to substitute a small amount of x2 for x1

Optimal Choice

Economic theory assumes individuals choose their most preferred bundle, or equivalently the bundle that gives them the most utility, that is in their budget set

Consider a consumer who has the choice between 2 goods x1 and x2 at the cost p1 and p2 respectively, and an endowment m

Using Econ 201 analysis and logic, the consumer reaches an optimal bundle when he/she does not have any further incentives to shift consumption among the goods in the bundle

At the optimal choice (x*1 and x*2) the indifference curve is tangent to the budget line

The optimal bundle (x*1 and x*2) satisfies 2 conditions:1. The budget is exhausted p1x1* + p2x2* = m2. The slope of the budget constraint, -p1/p2, and the slope of the indifference

curve containing (x1* , x2*) are equal at (x1*, x2*)

While graphs are informative we often want to solve things analyticallyFor two-good analysis, for each good i, we can find analytically function xi(p1,p2,m) that map prices and endowment into an amount of that goodCan use the constrained optimization approach from Econ 211

The problem of the consumer is basically to solve:

This implies that they want to find x1* and x2* s.t. U(x1*, x2*) >= U(x1,x2) for all values of x1 and x2 that satisfy the equation p1x1 + p2x2 = m

Objective function U(x1,x2)Constraint p1x1 + p2x2 = m

There are 2 main ways to find the above optimal constrained choice:1. Write down the optimality conditions and solve the system2. Substitite the constraint into the objective function to yield an unconstrained

problem3. Use Lagrange’s method

EX/ Find the optimal choice for a consumer who has a Codd-Douglas utility fcn who has $m and p1 and p2

Lec 3

Consumer Demand Functions

In analyzing the consumer’s behaviour regarding their optimal choice of a bundle of goods, we can derive the consumer’s demand function for each of the goods in the bundle:

X1 = x1(p1,p2,m)X2 = x2(p1,p2,m)

These derived demand functions tell us all we need to know about the consumer behaviourComparative statics analysis of ordinary demand function:

- The study of how ordinary demands change as prices and income changes

There is an intimate relationship between inferior goods and giffen goods

Slutsky Equation

What happens when a commodity’s price decrases?- A change in the relative prices occurs- The total purchasing power o the consumers increase

The Slutsky equation says that the total change to demand from a price change is the sum of a pure substitution effect and an income effect

- Substitution effect: the commodity is relatively cheaper , so consumers substitute it for now relatively more expensive other commodities

- Income effect: the consumer’s budget of $m can purchase more than before, as if the consumer’s income rose, with consequent income effects on quantities demanded

When the price of good 1 change and income stays fixed, the budget line pivots around the vertical axis

We can view this adjustment as occurring in 2 stages: first pivot the budget line around the original choice, and then shift this line outward to the new demanded bundle.

X and Z denotes the initial and final choiceY denotes the optimal choice on the pivoted lineThe movement from X to Y is the substitution effectThe movement from Y to Z is the income effectThe movement from X to Z is the total change in demand

Income Effect:- Isolated change in demand due only to the change in purchasing power- Slutsky asserted that if, at the new prices, less (more) income is needed to

buy the original bundle then “real income” is increased (decreased)- Income effect can be either +ve or –ve

Pure Substitution Effect:- Isolated change in demand due only to the change in relative prices- What is the change in demand when the consumer’s income is adjusted so

that, at the new prices, they can only just buy the original bundle?- Substitution effect is always negative (opposite to the direction of price

change)

Slutsky’s Effect for Normal Goods- The substitution and income effects reinforce each other when a normal

good’s own price changes- The Law of Downward-Sloping Demand therefore always applies to normal

goods

Slutsky’s Effect for Income-Inferior Goods:- The substitution and income effects oppose each other when an income-

inferior good’s own price changes- The income effect may be larger in size than the substitution effect, causing

quantity demanded to increase as own-price rises => Giffen goods

Example- Demand change, Sub and Income effects

Lec 4

Profit Maximization

Short-Run PM

The firm’s problem is to locate the production plan that attains the highest possible isoprofit line, given the firm’s constraint on choices of production plans

A $ Π isoprofit line’s equation is Π=py−w1 x1−w2 x́2

Output level is y and input levels are x1 and x2- barProduct price is p and input prices are w1 and w2

The constraint is the production function

Given p, w1 and x2 = x2-bar the short-run profit-maximizing plan is

The profit-maximizing problem facing the firm can be written as:

FOC:

The condition for the optimal choice of factor 1 is

Profit Maximization in the Long Run

Now the firm can vary both input levels and the profit-maximization problem can be posed as

FOCs:

The same condition must hold for each factor choice

The choices of inputs that yield the maximum profit for the firm, x1*(w1,w2,p) and x2*(w1,w2,p), are the profit maximizing factor demand functions

Cost Minimization

Standard assumption: firms make decisions to maximize profits, or maximize total revenue minus total costsDecision process can be broken up into 2 parts:1. For any given level of output, what combination of inputs should the firm use? -Minimizing costs of production a given level of output2. Given the optimal combination of inputs, how much should the firm produce/supply?

For given w1, w2, and y, the firm’s cost-minimization problem is to solve

The levels x1*(w1,w2,y) and x2*(w1,w2,y) in the least-costly input bundle are the firm’s conditional demands for inputs 1 and 2The (smallest possible) total cost for production y output units is therefore

Given w1, w2 and y, how is the least costly input bundle located?And how is the total cost function computed?

A curve that contains all of the input bundles that cost the same amount is an isocost curveThe equation of the $C isocost line is

C = w1x1 + w2x2It can also be rearranged to give

Slope is –w1/w2Of all input bundles yielding y units of output. Which one is the cheapest?

At an interior cost-min input bundle:

Tangency condition:

The technical rate of substitution must equal the factor price ratio

The choices of inputs that yield minimal costs for the firm, x1*(w1,w2,y) and x2*(w1,w2,y), are the conditional factor demand functions or derived factor demands

Recall that we can use several analytical techniques to solve this kind of problem1. Write down the optimality conditions and solve the system2. Substitute the constraint into the objective function to yield an

unconstrained problem3. Use Lagrange’s method

Short-Run & Long-Run Total Costs

In the long-run a firm can vary all of its input levelsThe long-run cost-minimization problem is

Consider a firm that can’t change its input 2 level from x2-bar unitsThe short-run cost-minimization problem is

How does the short-run total cost of producing y output units compare to the long-run total cost of producing y units of output?

- In general the short-run total cost exceeds the long-run total cost of producing y output units

The short-run cost-min problem is the long-run problem subject to the extra constraint that x2=x2-bar

- If long-run choice for x2 was x2-bar then the long-run and short-run total costs of production y output units are the same

- If long-run choice for x2 != x2-bar then the extra constraint x2=x2-bar prevents the firm in the short-run from achieving its long-run production cost

Lec 5-6-7

General Equilibrium of Exchange Economy

Exchange

Consider an island with 2 consumers A and B, 2 goods 1 and 2Their endowments of goods 1 and 2 are wA= (w1A, w2A) and wB=(w1B, w2B)

Then that means on the whole island, there are6+2=8 units of good 14+2=6 units of good 2Consider first each person’s well-being in the absence of any marketEach person must simply consume his endowmentWhat is “wrong” with this allocation of island resources?

Edgeworth Box

Are there feasible allocations that make both individuals better off than simply consuming what they are endowed with?How do we picture all of the feasible allocations?Edgeworth and Bowley devised a diagram, called an Edgeworth box, to show all possible allocations of the available quantities of goods 1 and 2 between two consumersWhat allocations of the 8 units of good 1 and the 6 of good 2 are feasible?

Starting an Edgeworth Box

How can all of the feasible allocations be depicted by the Edgeworth box diagram?

Endowment allocation

One feasible allocation is the before-trade allocation, the endowment allocation

Other feasible allocation

All points in the box, including the box boundary, represent feasible allocation of the combined endowmentsWhich allocations will be blocked by one or both consumers?Which allocations make both consumers better off?

Adding preferences to the box

Trade

What is the region of the box where A and B are both made better off?Presumable, A and B will trade to some point in this region

Pareto Efficient Allocations

An allocation of the endowment that improves the welfare of a consumer without reduction the welfare of another is a Pareto-improving allocation

Trade improves both A’s and B’s welfare; this is a Pareto-improvement over the endowment allocation

Where are the Pareto-improving allocations?- Since each consumer can refuse to trade, the only possible outcomes from

exchange are Pareto-improving allocations

But which particular Pareto-improving allocation will be the outcome of trade?

Pareto Optimality

New mutual gains-to-trade region is the set of all further Pareto-improving reallocationsAn allocation is Pareto-optimal(or Pareto efficient) when the only was one consumer’s welfare can be increased is to decrease the welfare of the other consumerThe allocation is Pareto-optimal since the only way one consumer’s welfare can be increased is to decrease the welfare of the other consumerAt any Pareto efficient allocation, the indifference curves of the 2 agents must be tangent in the interior of the box

Why is that the case?Further trade from point M can’t improve both A and B’s welfares

An allocation where convex indifference curves are “ only just back-to-back” is Pareto-optimal

Where are all of the Pareto-optimal allocations of the endowment?

All the allocations marked by a dot are Pareto-optimal

The contract curve is the set of all Pareto-optimal allocations

In a typical case, the contract curve will stretch from A’s origin (OA) to B’s origin (Ob

But to which of the many allocations on the contract curve will consumers trade?That depends upon how trade is conductedIn perfectly competitive markets? By one-on-one bargaining?The core is the set of Pareto-optimal allocations that are welfare-improving for both consumers relative to their own endowmentsRational trade should achieve a core allocationBut which core allocation?

- Again , that depends upon the manner in which trade is conducted

Core allocations

Trade in Competitive Markets

Each consumer is a price-taker trying to maximize her own utility given p1, p2 and her own endowment

For consumer A, that is

And similarly for consumer B

The gross demand of agent A for goods 1 is the total amount of the good that he wants at the going pricesThe net demand of agent A for good 1 is the difference between his total demand and the initial endowment of good 1 that agent A holdsSo given p1 and ps

A general equilibrium occurs when prices p1 and p2 cause both the markets for commodities 1 and 2 to clear

For any prices p1 and p2 there is no guarantee that an equilibrium will always occur

At the given prices p1 and p2 there is an excess supply of commodity 1 and an excess demand for commodity 2Neither market clears so the prices p1 and p2 do not cause a general eqmSince there is an excess demand for commodity 2, p2 will riseSince there is an excess supply of commodity 1, p1 will fallThe slope of the budget constraints is –p1/p2 so the budget constraints will pivot about he endowment point and become less steepWhich Pareto Optimal allocations can be achieved by competitive trading?

Walrasian equilibrium

At the new prices p1 and p2 both markets clear, there is a general equilibrium

Trading in competitive markets achieves a particular Pareto-optimal allocation of the endowmentsWe can describe the equilibrium as a set of prices (p1,p2) such that

This is equivalent to

Lets denote the net demand functions or excess demand for good 1 by consumer A and B as follows

Summing together the above excess demand functions gives the aggregate excess demand for good 1

We can derive the aggregate excess demand for good 2 in a similar way and describe an equilibrium (p1,p2), by saying that aggregate demand for each good is zero

Walras’ Law Walras’ Law states that

Walras’ Law is an identityEvery consumer’s preferences are well-behaved so, for any positive prices (p1,p2), each consumer spends all of his budget

Summing the above equations and some manipulation gives:

This is Walras’ Law- it says that the summed market value of excess demands is zero for any positive prices p1 and p2

Implications of Walras’ Law

One implication of Walras’ Law for a 2 commodity exchange economy is that is one market is in equilibrium then the other market must also be in equilibriumProof:

A second implication of Walras’ Law for a 2 commodity ex-change economy is than an excess supply in one market implies an excess demand in the other market

Equilibrium and Efficiency

All competitive market equilibria are Pareto efficient: a result known as the first fundamental thm of Welfare Economics

First Fundamental Theorem of Welfare EconomicsGiven that consumers’ preferences are well-behaved, trading in perfectly competitive markets implements a Pareto-optimal allocation of the economy’s endowment

An easy proof can be based on the previous graph of the equilibrium in the Edgeworth box

Second Fundamental Theorem of Welfare EconomicsIf all agents have convex preferences, any Pareto-optimal allocation can be achieved by trading in competitive markets provided that endowments are first appropriately rearranged amongst the consumers

This theorm is illustrated in the following graph:

Implication of the two Welfare Theorems

The First Welfare thm shows that the particular structure of competitive market has the desiriable property of achieveing a Pareto efficient allocation

- competitive markets economize on the info needed for efficient resource allocation

The Second Welfare thm implie that the problems of efficiency and distribution can be separated- Prices should be used to reflect scarcity and income transfers should be used to adjust for distributional goals