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Econ 301 MT 1 motes
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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