Noter
20. november 2014
09:59
General
p/d
Price (demand for a certain price)
TR
Total revenue
MR
Marginal revenue
TVC
Total variable cost
Cost that depends on Q
AVC
Average variable cost
VC/Q
AFC
Average fixed cost
FC/Q
ATC
Average total cost
TC/Q
TFC
Total fixed cost
Fixed cost
TC
Total cost
Fixed cost + Variable
Cost
MC
Marginal cost
sold)
TR=PQ
(revenue for next/last unit sold)
(Cost for the next/last unit
MR=R(Q)
MC=C(Q), MC=VC(Q)
- Fixed cost only exists in the short run. Fixed cost is capital K. In the long run capital can be altered.
- Fixed cost is the constant difference between TC and VC.
- Average fixed cost becomes smaller as quantity is larger. Spreading out fixed cost.
o
- Law of demand: d<0
Demand
A demand curve is a curve showing the relationship between price and quantity. Price is always per unit,
quantity number of units demanded at given price.
Demand depends on price of Q, other prices and income.
When drawing the demand curve, it is always the inverse demand curve, that means p is isolated.
The law of demand is; the demand (inverse) should always be negative. When price goes up, quantity
goes down -> price increase, less is demanded.
Changes in other faces shift the demand left/right.
Changes in price moves up/down on the curve.
When summing demand curves, it must always be the right demand curve NOT INVERSE. That means
Q=.
Supply
In the short run, supply is the MC above the AVC:
In the long run, supply is the MC above AC.
Supply only exists in a PC. In a monopoly, we always consider MC=supply.
Deriving supply curves from individual cost
Microeconomics side 1
We have a aggregate demand function D:
, and an individual cost function
. We wish to find the market equilibrium.
1. Determine if it is short run or long run. In this case, we have fixed costs thus short run.
2. In the short run supply is MC above AVC. In this case:
a. MC=10+50q
b. VC=C-FC =
c. AVC=
3. All of MC is above AVC. Thus mc=supply. We can now express individual supply as p=10+50q
In order to find total supply, we multiply the supply function with number of firms. However, this has to
be actual supply function Q=, and not invers. First isolate q.
The number of firms n, 50. Aggregate, or total, supply is therefore:
To find equilibrium set demand equal supply.
Long run equilibrium.
In the long run there are no economic profits!
Because there is a short run profit, new firms wants to enter. Therefore supply shifts to the right,
driving down the price. This lowers profit and quantity supplied by each individual firm. The
process stops when the price equals the minimum average cost, at this point profit is zero.
Therefore in the long run:
How to calculate the long run equilibrium, and number of firms? (using the example from before).
1. Calculate the average cost.
2. Find the minimum AC. We know that MC intercepts AC at the minimum of AC. MC=AC. Equal AC with
MC and calculate q. We now know the quantity supplied by each firm.
3. The price is the minimum average price.
4. Knowing the price, we can calculate total demand Q, using the demand function.
5. To calculate the number of firms:
It was claimed that long run profit is zero. This is of course economic profits. Economic profits include
opportunity costs. Opportunity costs is the cost of the second best alternative.
Price elasticity
Own price elasticity
The own price elasticity of demand denotes the percentage change (increase) in Q when P is changed
(increased) by 1 percent.
Change in quantity demanded in %, when price increases by 1%.
Microeconomics side 2
If =-5 then when price goes up with 1 % quantity decreases by 5%.
-
Elasticitys
demand is elastic.
demand is inelastic
demand is unitary elastic.
Cross price elasticity
Price elasticity of a related good.
The relationship between good Q and the other good. Cross price elasticity is positive.
When the price of the other good goes up, quantity demanded for original good goes up.
That is the case when goods are substitutes: Indifferent between A and B -> choose the cheapest.
If cross price elasticity is negative then
Goods are used together -> they are complements.
However, goods can also be unrelated.
Income elasticity
How much is quantity increased when income increases.
When income elasticity is:
Positive: Income goes up, quantity goes down.
Negative: Income goes up, quantity goes down.
Types of goods;
, goods are necessary.
Tax incidence
Constants tax = specific tax
Ad valorem = % tax
Microeconomics side 3
Ad valorem = % tax
- How does a tax affect equilibrium?
A tax levied on consumers shift demand down. (inverse)
The producer price is the price producer get. The producer price is found from the supply curve.
Consumer price is the total price consumers pay. Found from originally demand.
Consumer price = tax+producer price.
Consumers and producers are always harmed by a tax. It does not affect equilibrium whether a tax is
applied to consumers or producers.
If we let price denot? Consumer price, then:
N=price elasticity of supply. .
The party who is most affected by a tax is the party with the relatively smallest price elasticity. When a
curve is relatively flat, elasticity is high.
When only one part is fully affected by a tax:
If supply (demand) is perfectly inelastic, (vertical) only producers are affected.
If supply (demand) is perfectly elastic only consumers (producers) pay the tax.
Consumer surplus, producer surplus and DWL.
Consumer surplus is the difference between what consumers pay, and what they are willing to pay. It is
the area under the demand curve, above the price.
Producer surplus is the difference in revenue and costs. It is the area above the supply curve, below the
price.
Total welfare is the sum of consumer and producer surplus (government)
Wellfare loss = DWL = efficiency loss -> is the decrease in total welfare. (remember to include
government)
Taxes create efficiency loss, which is a loss in welfare associated with all parties.
The efficiency loss is due to a decreasing Q demanded. Thus production decreases and thus does costs.
All consumers are able to buy less goods.
Price ceilings and floors
Regulations that puts a maximum/minimum on a price.
Ceiling = max limit
Floor = min limit
With a price ceiling, p decreases and thus Q will decrease limited by supply.
Excess demand arises
Producers and general welfare is harmed by a ceiling.
Microeconomics side 4
Whether consumers loose depends if the group of consumers who cannot buy the good is bigger
than those who can.
Welfare loss because more value is lost than costs are saved.
A price floor puts a lower limit on a price. P increases, and Q decreases limited by demand. Creates
excess supply.
Sometimes government buys excess supply, which creates further losts.
Maximizing revenue
Revenue = price * quantity
R=P*Q
Revenue is the money the firms receive.
To maximize revenue, differentiate and set equal to zero.
To argue whether it is a minimum or maximum take the second order derivative -> differentiate twice. If
the second order derivative is negative, you have maximized. If it is positive, you have minimized.
D2<0 -> max
D2>0 -> min
Revenue can also be maxed at
Revenue is maximized where
and for a linear demand curve that is at the mid-point.
Marginal revenue as product of elasticity
Da
, fr vi da:
Consumer choice
In order to show how consumers chooses which goods to buy, we make the indifference curve ->
showing combinations of bundles in which consumer will be indifferent -> they get same utility from all
the different bundles.
MRS Marginal rate of substitution
Is the slope of the indifference curve. It is how much you would give up of A, to get b. How much y
should be changed if x changes 1 unit.
Preferences
Preferences are often convex, meaning that MRS is decreasing along the IC.
Normal assumptions for preferences include:
- Completeness meaning there is an indifference curve through each point. Every consumption bundle
Microeconomics side 5
- Completeness meaning there is an indifference curve through each point. Every consumption bundle
gives utility, and consumers can rank these bundles.
- Transitivity: IC cannot cross each other. Logic. If ICs cross, you would get different utilities from same
consumption.
- More is better.
Budget constrains:
The budget constraint is:
Income is often fixed, so are prices.
The slope of the budget constraints is the marginal rate of transformation MRT.
Optimization
Optimzation occurs at the highest optainable IC inside the budget triangle.
The budget constraint should be tangent to IC. That means the slopes should be the same MRS=MRT,
However only when preferences are convex.
1. MRS=MRT is the first step
2. Insert into budget constraint.
Relationship between goods
Substitutes
- When you demand more of a good, when the price of another good goes up, they are substitutes.
For goods to be substitutes then:
,
When price goes up, demand for the other goes up.
Complements
- When you demand less of a good, when the price of another good goes up.
For goods to be complements then:
,
Unrelated
Goods are simply unrelated.
,
Income consumption curves and Engel
An income comsumption curve is a curve showing optimal consumption of bundles, when income
changes.
ICC = How consumption varies when income varies.
Microeconomics side 6
ICC = How consumption varies when income varies.
Tegning?
Engel curve
The engel curve shows for each good how consumption of that particular good varies with income.
Tegning?
Inferior goods will have an negative ICC: Quantity demanded is decreased when income is increased.
All goods cannot be inferior, because then total expenses would decrease as income increases, violating
more is better.
Monopolies
Finding output, price and profit for a monopoly.
As all other profit maximizing firms, a monopoly produces the output Q such that MR=MC.
1. Monopolies produce Q such that MR=MC
a. R=PQ
b. R=(inverse demand)Q (because monopolies are price makers)
c. We now have revenue as a function of quantity. Differentiate to get MR.
d. Solve MR=MC for Q, and you have optimal production.
2. To determine the price for a monopoly, use the demand function. Plot Q into demand, and you have
price.
Price elasticity and monopolies
Monopolies always produces on the elastic part of demand.
Consumer surplus and producer surplus for monopolies and DWL.
Producer produces 5 of Q at MR=MC. The price is then 15.
Consumer surplus= ABF.
Microeconomics side 7
Producer surplus=BFED
Had supply(mc)=demand, production would be 10 Units at price 10. Consumer surpus=ACE, zero
consumer surplus.
The difference between these two is the DWL, BCD.
This is the reason why monopolies cause efficiency loss.
The effect of a subsidy (or tax) on a monopoly.
A subsidy will lower MC. Then we have
Equating
If a subsidy is set so that the produced quantity is the same as PC, there will be no welfare loss.
To locate DWL
TO find DWL locate the difference in quantities. Then calculate area under demand and supply(MC).
Taxes and monopoly
Unlike in a competitive market, the tax incidence on consumers can exceed 100% in a monopoly market.
In a monopoly the welfare losses from an ad valorem tax are less than from a specific tax.
Monopsonies - labour market
(the opposite of monopoly) - A single buyer!
The labour market
Firms hire labour until
With perfect competition PC in the labour market, the price of labour, wage, is given consequently.
.
Therefore, the inverse demand is
You hire labour until gains = costs.
It is thereby possible to find equilibrium in labour market, using the supply curve.
Assuming there is monopsony
A monopsony realizes that if it wants to hire more L wage must be increased. For a monopsony, w is not
given/constant.
The equilibrium for a monopsony can be found:
1.
2.
3.
4.
The total expenditure.
. Insert wage as function of labour,that is inverse supply.
Find marginal expenditure.
.
Equal
find amount/q of L
Find price of L, w, using TE/supply.
Welfare will increase due to a monopsony, because quantity decreases.
Compensation
If the price of one good increases, the budget line shifts.
Microeconomics side 8
If the price of one good increases, the budget line shifts.
Therefore a new equilibrium is created.
If consumer is compensated in order to get back on original IC then:
Substitution effect: Is the effect on consumption of bundles on the same IC AFTER the consumer is
compensated. The consumer will substitute in order to have constant utility.
Income effect: Is the difference between the new optimal bundle consumed, and the bundle that would
be consumed if the consumer is compensated.
If income effects lowers quantity demanded, good is normal.
Substitution effect for perfect complements will be zero.
Calculate compensations:
1. Assume there is a price increase. Does the compensation put the consumer back on IC?
2. Is the consumer able to obtain a higher IC?
3. -> Over compensated.
Market structure
Perfect competition PC:
Is when there is
A)
B)
C)
D)
E)
Large number of small firms
The firms produce identical/homogenous products (Consumers are indifferent between the products)
Everyone has full information
Negligible transition costs
Free entry and exit of firms over time
A-D implies that firms are price takers. They are not able to affect the price of a product.
There MR=MC=P
The supply curve is the portion of MC which is above AVC. SHORT RUN
In the long run it is above AC, because there is no variables.
Short run / long run
In the short run, costs are atleast as high as in the long-run.
The short run cannot produce more effective / cheaper than in the long run. At best, they can produce
exactly the same. Therefore, in the short run costs are (almost always) higher.
Maximizing in the long run
Production is maximized / costs are minimized in the long run when:
This is only for the long run!
Microeconomics side 9
The supply curve is the portion of MC which is above AVC. SHORT RUN
In the long run it is above AC, because there is no variables.
Risks
Risk adverse
Utility as function of money is concave.
Risk loving
Utility as function of money is convex.
Risk neutral
Utility as function of money is linear,
Premium
Risk premium is the amount a consumer will give up in order for a consumer to be certain. For risk
adverse, this is positive. For risk, loving this is negative.
Uncertainty
To insure or not insure
Case.
A woman has valuables 200.000. There is a probability of 2% 0,02 for a burglary, in which she will lose
80.000. She can buy an insurance that covers all, for 15.000.
Her utility function as a product of money/wealth x is:
A) Find the fair price. B) Should she buy the insurance? C) Find the maximum price that she will pay for
insurance.
The fair price is the cost the insurance company faces.
The cost is therefore
The woman should buy the insurance if she is better of buying it. She is better of buying if
.
Therefore, we calculate her expected utility for both outcomes.
When she does not buy insurance, she faces a loss of 80.000, which gives her a wealth x of 120.000.
When she is insured, she has a wealth of 185.000 for certain.
Microeconomics side 10
As her utility is higher without the insurance, she should not buy it.
To determine at what price she will buy the insurance, she needs to be indifferent about buying
insurance or not. She is indifferent when she gets the same utility. Therefore we calculate her wealth,
from which she will obtain the same utility as she does without buying insurance.
The equation is solved for x by WordMat.
We now substract this from her original wealth.
The maximum price she will pay for insurance is 1799,2
Invest or not invest
Case:
Henrik has 180 DKK. He chooses to invest everything in either risk free government bond, or a risky
asset.
The government bond pays 40 % interest. For the asset, there is a 80 % change it will pay interest of 25
%, and 20 % change it will pay interest of 100 %.
A) What is his expected value of investing in the asset? B) Given a utility function over money of
, will he invest in bonds or assets? C) Suppose Henriks utility function changes. It is concave
(declining slope). Will he change behavior?
The values of the three different outcomes is:
His expected value of investing in the stock is the following:
Do note that this is the same outcome as buying the government bond.
Given his utility function over money as
invest in bonds or stocks.
, there is two ways to determine whether he will
We can determine Henriks attitude towards risk, by the slope of his utility function. We find the second
derivative:
The slope is increasing, and therefore Henrik is risk preferring. As the expected values are the same for
the two choices, he will choose the one with higher risk, facing a possible higher outcome.
Another method is to calculate his expected utility of both possibilities, and then comparing the two.
Microeconomics side 11
Another method is to calculate his expected utility of both possibilities, and then comparing the two.
As Henriks utility is higher when buying stocks, he will do this in order to maximize utility.
However, if Henriks utility function was concave
safer outcome of buying government bonds.
, he would be risk averse, and choose the
Production
Firms produce using two different inputs/factors; Labour and Kapital.
The marginal product is the increase in Q when the input is increased by 1.
The marginal product is the derivative of the production function.
Returns to scale:
What happens when you increase all inputs?
If an equal-sized proportional increase in all inputs causes an increase in output that is proportionally:
Greater, then there is increasing returns to scale
Equal, then there is constant returns to scale
Smaller, then there is decreasing returns to scale.
To determine the RTS for a Cobb-Douglas
Economies of scale
- A cost function exhibits economies of scale if the average cost of production falls as output expands.
- A cost function exhibits diseconomies of scale if the average cost of production rises as output expands.
1 firms production in two plants
1 firm produces a total of 60 units at two separate plants. How much should be produced at each plant?
. Marginal cost of plant A should equal marginal cost of plant A. This function has to
variables,
However, we know that
. Two equations two unknowns; solve.
Shut-down or not
Produce as long as revenue covers VC and some FC.
Maximizing output
The isoquant is a curve showing the combination of input, holding output constant.
The slope of the isoquant is MRTS -> marginal rate of technical substitution.
MRTS determines how K should be changed if L is changed by 1, holding Q constant.
Microeconomics side 12
MRTS determines how K should be changed if L is changed by 1, holding Q constant.
The cost is always
, called the isocost.
The slope of the cost curve is
To optimize, then
Long run average costs
- If LRAC is constant, it is because of CRTS
- If LRAC is increasing it is because of DRTS
- If LRAC is decreasing it is because of IRTS.
Maximizing output
Profits are
All firms wants to maximize profits.
Profit will always be maximized by choosing the quantity Q where MC=MR
Marginal revenue
Marginal cost
. Increase (decrease) in R if Q is increased (decreased) by 1.
. Increase (decrease) in C if Q is increased (decreased) by 1.
If MR>MC then profit goes up if quantity is increased by 1. Therefore it is not maximized.
If MR<MC then costs fall more than revenue. Profit goes up if Q is increased by 1.
Only at MR=MC is profit maximized.
NO MATTER WHAT PROFITS ARE ALWAYS MAXIMIZED WHERE QUANTITY IS MR=MC.
Marginal Rate of Substitution for CES utilty function
Marginal Rate of Transformation
Slope of the budget constraint.
Elasticity of Substitution CES
Microeconomics side 13
First find MRTS.
We not substitute in the expression for MRTS:
Rearranging we get:
We now wish to isolate K/L:
Multyplying with LN on both sides we get:
Thus the elasticity of substitution is:
Utility maximation
Utility maximized at:
Utilty maximsed at:
Utility maximation with a constraint for all functions.
Utilty:
Constraint:
Using the lagrangian multiplier.
Deriving the partial derivatives.
Microeconomics side 14
Rearranging these terms we get:
Setting the two terms equal to each other:
Utility maximation with a constraint for Cobb-Douglas function:
Constraint:
Using the lagrangian multiplier.
Deriving the partial derivatives.
Dividing these two we get:
Simplifying:
Solving for Z:
Substituting for Z in our budget constraint:
Microeconomics side 15
The equation is solved for X by WordMat.
Substituting in the budget constraint, then solving for Z:
The equation is solved for Z by WordMat.
Game theory
L
12,12
4,15
0,0
15,4
10,10
A,2
0,0
2,A
3,3
The Nash equilibrium is a strategy/choice which is optimal given what the other player chooses.
- How to find Nash equilibrium?
o Identify dominant strategy, a strategy which is always best
o Eliminate dominated strategies, a strategy which is never best.
However, Nash equilibria is not always the best opportunity.
In repeated games, players could in theory agree to the optimal strategy. As long as they do it (signal)
they continue doing it. If one cheats the other will revert.
However, this will not solve the problem in a finite game. Both players will cheat in the final, therefore
they will also cheat just before the final as so on. They will always cheat.
In a infinite game cooperation is possible, because there is no final period.
Microeconomics side 16
Alternatively if the game is finite but of unknown/uncertain duration cooperation can be possible.
Pareto efficiency - effective production - effectivity
If there is no other allocation in which 1 consumer gains utility without harming the other, it is said to be
pareto efficient.
How to locate a pareto efficiency:
Below is an Edgeworth box. This is a plot of indifference curves for two consumers.
When indifference curves are tangent, it is an pareto efficiency.
A contract curve is a curve plotting the paretos.
Effective production
- Which mix of goods should the producers produce?
The production mix is efficient if there is not another feasible production which makes at least one
consumer better of.
For production mix to be efficient
Externalities
Microeconomics side 17
Externalities
An externality is a situation where the production or consumption affects someone(s) not directly
involved in production or consumption.
Negative: Pollution, smoking
Positive: Vitamin, health care.
Emissions and pollution
Emission standards allow pollution.
- You can allow max units
- Tax emission pr. Unit
Efficient emission is found by MB=MC. Marginal benefit, the benefit of polluting (saved cost of cleaning)
should be equal the marginal cost of cleaning.
The gain for firms polluting = loss of not pollutiong.
Cost of cleaning = gain of not polluting.
Not polluting causes a net loss.
Coase theorem:
If a property right is well defined and parties can negotiate costlessly, the efficient outcome will emerge.
Information
If consumers cannot determine the value of a product, then risk neutral will pay exactly the expected
value.
Assymetric information
Assymetric information is when two traders (buyer and seller) do not hold the same information for a
given product.
The more informed party may engage in opportunistic behavior, which is taking advantage of someone
when circumstances permit.
Opportunistic behavior leads to market failures.
Adverse selection
Opportunism characterized by an informed person benefitting from trading with a less informed person
who doesnt know about an unobserved characteistic; such as the quality of a car, a persons health.
Moral hazard
Opportunism characterized by an informed person taking advantage of a less-informed person through
an unobserved action; such as engaging in risky behavior, this increasing the probability of claims against
an insurance company.
Lemon example
We have a market for used cars. We have high quality cars, and low quality cars.
Microeconomics side 18
The sellers think/act as the following:
Furthermore we assume that:
In this example, buyers and sellers have the same, full info.
Seller 1 values his car at 5000. Will not sell for less.
Seller 2 values his car at 3000. Will not sell for less.
All cars are sold when:
All cars are sold, because the price (average) is higher than the value.
However, if buyers cannot differentiate quality cars from lemons, they will not purchase cars higher than
market price. In our case:
Thus, when all cars are sold the average price is 6000. With asymmetric information, this harms the
seller, because all cars will be sold at 6000.
Solving asymmetric information
Asymmetric information can be solved by the seller providing some sort of costly signal that the good is
flawless. A signal has to be costly, because talk is cheap. Examples of a costly signal is a money back
guaranty or a warranty.
Education can also be considered as a signal. A uni diploma is a costly signal of abilities, or the will to
work hard.
Signals have to be credible.
Langragian
Utility maximation with a constraint for
Cobb-Douglas function:
Microeconomics side 19
Cobb-Douglas function:
Constraint:
Using the langragian multiplier.
Deriving the partial derivatives.
Dividing these two we get:
Simplifying:
Solving for Z:
Substituting for Z in our budget constraint:
The equation is solved for X by WordMat.
Substituting in the budget constraint, then solving for Z:
The equation is solved for Z by WordMat.
Minimizing cost for a Cobb-Douglas
Case:
A firm has a Cobb-Douglas production function
, in which K is capital, and L is labour.
Microeconomics side 20
A firm has a Cobb-Douglas production function
, in which K is capital, and L is labour.
The firm wishes to produce 10 units Q. The price of rent r is 1, while the price of wage w is 4.
The firms cost function is
Therefore, we wish to minimize our cost
, subject to our constraint
We do this using the langragian multiplier.
We find the partial derivatives for K and L.
Dividing these two, we get:
Thus
Solving for K we get.
We now substitute the expression for K in out constraint
, thus the minimizing cost use of labour is 2,5. We now substitute to isolate for K.
Opsumising, at L=2,5 and K=10, when producing 10 units the cost is minimized. At this output the cost is:
Utility maximation with a budget
constraint for Cobb-Douglas
Utility function:
Microeconomics side 21
Price of
, price of
, income=y
The budget constraint is therefore
If
what is the optimal bundle / optimal allocation of goods?
We wish to maximize our utility function subject to our budget constraint. Using the langragian
multiplier we get:
We take the partial derivatives:
Solving both equations for lambda, we get:
Since we now have two equations for lambda, we can equal the two terms:
Rearranging we get:
When MRT=MRS we have optimal conditions.
We now simplify the right hand side of the equation:
Rembereing the left hand side of the equation we get:
Isolating for one of the unknowns, q2, we get:
Microeconomics side 22
We now have an expression for q2. We substitute this in our budget constraint, to solve for q1:
When 10=q1, we can now solve for q2:
Expenditure minimization for CobbDouglas
We wish to minimize expenditure given a certain level of utility .
Our expenditure function is
When
and out utility function
the langragian multiplier is:
We take the partial derivatives:
Dividing both equations we get:
Microeconomics side 23
Isolating for q2 we get:
We now substitute into our budget constraint (the utility function)
Maximize output for Cobb-Douglas
We wish to maximize output, given a production function and budget constraint = cost function.
Production function
Given the values above, using the langragian multiplier we get:
The partial derivatives with respect to L and K are:
Rearranging we get:
Equalling these two we get:
Rearranging we get:
Simplifying for the right hand side we get:
Thus:
Isolating for K we get
Substituting into our budget constraint, the cost function, we get:
Microeconomics side 24
Solving for
Now we can maximize quantity.
Minimizing cost for a Cobb-Douglas
We wish to minimize cost given constant production/quantity output.
Production function
, OBS: Q is fixed
Given the values above, using the langragian multiplier we get:
The partial derivatives with respect to L and K are:
Dividing we get:
Solve for one, then substitute into the constraint (production function).
Utility maximation with a constraint for
all functions.
Utilty:
Constraint:
Using the lagrangian multiplier.
Deriving the partial derivatives.
Microeconomics side 25
Rearranging these terms we get:
Setting the two terms equal to each other:
Microeconomics side 26
