Deeper understanding Flashcards
Consider a Bertrand model with homogenous products and 2 firms with the following demand functions:
Q1 = 12 - 2P1 + P2
Q2 = 12 - 2P2 + P1
Calculate deadweight loss of this equilibrium
this is entire task relies on considering market demand in oligopolies as perfectly INELASTIC. This means, the demand is X regardless.
becasue of demand inelasticity, we can calculate the market demand function:
Q = Q1+Q2 = 24 - P1 - P2
From the Bertrand equilibrium, we get the following prices and quantas:
p1 = p2 = 4
Q1=Q2 = 8
Q = 16
Since prices are equal, the market demand function becomes:
Q = 24 - 2P
P = 12 - 1/2 Q
This demand curve will be used to calculate consumer surplus.
for simplicity, we consider MC = 0. Then, producer surplus is simply pricequanta = 416=64.
For perfect competition, producer surplus is 0, since price = MC, which equals 0. 0*24 = 0. 24 is the perfect competition quantity. We find this by setting supply equal to demand:
12 - 1/2 Q = 0
12 = 1/2 Q
Q = 24
When calculating the surpluses, we get deadweight loss of 16. Indication that oligopolies are not efficient markets.
key takeaway here is the derivation of the market demand curve. assuming we have enough information to make the supply curve, we only need to do work to get the demand curve for the entire market.
Consider advertisement. What to we do with Price?
Price is assumed to be fixed. This is a simplification.
Draw up the profit function when advertising is included. Find first order conditions and interpret them.
pi = Revenue - Costs - AdCosts
pi = P*Q(P,A) - C(Q) - A
IMPORTANT: Advertisement costs are always fixed. At least, usually fixed.
∂pi/∂A = 0
P*∂Q/∂A - ∂C/∂Q * ∂Q/∂A -∂A/∂A = 0
P∂Q/∂A - MC*∂Q/∂A - 1 = 0
P∂Q/∂A = 1 + MC∂Q/∂A
This essentially means: MR_ads = 1 + MC*marginal product of advertisement. So, it tells us that we maximize profits (when ads are included and price is fixed) whenever we set the marginal revenue we get from the ad-boost to be equal to the extra costs we get by using ads/additional expenditure. To be more precise, we spend some amount on ads all the way until we reach this point.
Elaborate on the variable cost component of the advertisement
There is none. Ad-cost is fixed only.
Use first order conditions of the profit maximizing firm that use ads to find the rule of thumb for ad-cost
From the first order conditions, we get:
P∂Q/∂A = 1 + MC∂Q/∂A
We move the MC part over and factor it out:
∂Q/∂A * (P - MC) = 1
Now we multiply by the factor of A/Q. This we do to get advertising elasticity of demand. We get:
(P - MC)*∂Q/∂A * A/Q = A/Q
We multiply by the factor of 1/P as well
(P-MC)/P * E_ads = A/(PQ)
We know from the rule of thumb for pricing when we have monopoly power that P-MC)/P = -1/E_d. we use this.
-E_ads/E_d = A/(PQ)
This result tells us that the ratio of elasticity of ads and demand should be equal to the ratio of advertisement cost in relation to total revenue.
From this result, we know that if elasticity of ads is large, it means that small changes in ad-cost leads to LARGE change in output, which means that we should absolutely spend more money on ads. In essence, if the market responds very well to ads, we utilize this. If the opposite is the case, and the ratio is extremely small, say 1%, we only use 1 percent of the total revenue on ads.
It is worth noting that this assumes a fixed price.
Regarding advertising, what is the common mistake?
The common mistake is to only consider the fixed costs of advertising vs the gain. This is WRONG because it fails to capture how increased output as a result of ads actually leads to higher costs as well. We need to take the entire marginal expenditure into account.
Elaborate on double marginalization
Double marginalization refers to cases where markup of price happens several times. This is as a result of vertical chains. Recall that vertical chains are firms that supply parts of products to other firms. Vertical refers to the fact that different divisions or organizations make different parts. Horizontal means that firms are producing the same shit.
So, when someone with monopoly on engines sells their engines to car maker, who has a monopoly on cars, the consumer ends up paying a price that is essentially marked up twice. This is because the engine-firm marks up price according to something like P=MC/(1+1/E_d)
This price, which is obviously marked up, becomes the marginal cost of the firm that buys the engine. Then, they will mark up: P_consumer = (MC_eng/(1+1/E_eng)) / (1+1/E_car)
The result is that the price is much higher than it would be if the firms were vertically integrated into the same firm.
The interesting part is that both firms actually benefit in terms of profit by vertically integrating into a single unit.
Elaborate on Bertrand model.
EXTREMELY IMPORTANT: We cannot use some regular cost function do usual way, as we have a function that is dependent on PRICE, not quantity. Therefore, we need to use MC WITH the price, like this:
pi = (P1 - MC1)*Q_1(P_1)
This is crucial. If we dont do this, we end up with the opposite result that says that marginal cost decrease for either firm leads to higher prices, which is entirely wrong!
When we use the correct Bertrand method, we find that if either firm were to invest and reduce marginal costs, the price of the goods decrease. So, if one firm takes the aggressive action of reducing costs, the prices in the market will fall. This mimic a tit-for-tat strategy for the firms.
We can also see how this differs from the Cournot case. in the Cournot case, if one firm reduce MC, it will produce a higher output. Since the COurnot is followed, the competitor will maximize his profits, GIVEN what the other one just did, by reducing his market share, and thus increasing the market price, gaining him a little more. So, in Cournot, reduced MC leads to increased quantity for the firm, while the competitor responds by reducing his quantity. In Bertrand, reduced mc for the firm leads to lower price for both.
elaborate on the different types of goods, in regards to income
We categorize goods by considering the income elasticity of demand. We have the broad classes of inferior and normal goods. Inferior goods drop demand as income increase. thus, the income elasticity of demand shows a negative correspondence.
Normal goods have income elasticity of demand greater than 0. However, it is useful to divide normal goods into 2 groups:
1) Necessity goods
2) Luxury goods
Necessity goods have income elastcitiy of demand of between 0 and 1.
If the good have a income elasticity of demand greater than 1, we’re talking luxury goods.
Consider a Cobb-Douglas utility function. What can we say about elasticity of demand? Why?
It has constant elasticity of demand for both goods. We find this result by maximizing utility with Lagrange, and then calculating the demand function of one of the goods.
dy/dp_y = -Income/(p_y^2 * (alpha/beta + 1))
Then we get, by using the fact that p_yy = Incomebeta:
E_p = -1/(alpha+beta)
This means, constant elasticity of demand. If alpha + beta is 1, we get UNIT ELASTIC demand.
Explain detailed on what risk premium is, and how we find it.
Risk premium is what a risk averse consumer would be willing to “pay” maximum to avoid a risk. This means, at what point is the consumer indifferent between the risky income and the riskfree income? We are looking at the point where we get the same amount of utility for a risk FREE income vs the expected utility we get from a risky option.
If the utility function is given as u=ln(Income), and we’re offered risk free income of 100k, we would do the following:
u = ln(100k) = 11.5
Say the risky option is 200k with 50% probability or else 0. Then, we’d calcualte expected iutiliry:
EU = 0.5*ln(200000)=6.1
So, we are interested in the point where we’d be indiffernet between choosing this risky option, and some risk free option. Thus,
6.1 = ln(x)
x = 447.
So, we would be indiffernet between choosing 447 riskfree and 200k with 50% probability. This is extreme, but a risk averse investor will not be happy with risks.
Explain the process of finding risk premium
We calcualte the Expected Utility from the risky option. Then we are looking at what income we would REQUIRE to be indiffernet between the risky option and risk free option. The difference between the risky, expected income and the risk-free incoem with same utility is the risk premium.
So, we dont use the utility of the current risk free option at all. We only use the expected utility, and then use this value to find a corresponding income.
