Econ 401 Final

Markup pricing refers to the idea that output price is the marginal cost of production plus a markup , where the markup is equal to -k/(1+e) where e is the elasticity and k is the constant cost function

Given a quantity tax is imposed, the optimal monopoly price will be ((k+t)e)/(1+e) where k is the marginal cost, t is the tax and e is elasticity.

(te)/(1+e) is the amount of the tax passed on to the buyers. This amount can be greater than the tax itself

A monopoly will produce where MR = MC.

1.) Find MC function by taking the derivative of the total cost function. This will likely be a constant value.

2.) To find MR function, get total revenue function by taking the inverse of the demand function and multiplying it by Q (P X Q) and then take the derivative of this function

3.) Set MR=MC and find Q

4.) To find price, plug our found Q into out demand function

A monopoly will produce where MR = MC

1.) Find the total revenue function by multiplying the inverse demand function by Q and then find the MR function by taking the derivative or this.

2.) Find the marginal cost function by taking the derivative of the total cost function. Given a $2 quantity tax, we add 2 to our MC function.

3.) Set the MR equation equal to the MC + 2 equation and solve for Q. Plug Q back into the demand function to find price

4.) If the government imposes a $3 quantity tax, the equilibrium will not change from MR = MC as the profit is what it is regardless of how much the govt takes

Each output unit unit is sold at a different price as to exactly satisfy the marginal consumers willingness to pay. Prices will differ across buyers and quantities. (Perfect price discrimination)

Requires a lot of information about buyers

Price paid by buyers in a given group is the same for all units but prices may differ across buyer groups (eg. Senior discounts)

1.) To find the firms profits when the markets are separate, we can simply treat each market like a monopoly situation. We find set MR=MC and solve for P and Q and solve for profit in each market.

2.) to find the firm's profit when barriers between the two markets are removed, we invert the two price functions to get demand functions, we add these two functions together, we re-inverse this 1 equation and we are left with the market price equation. We then use this price equation to get the market MR equation (multiply by q then derive) and we set this equal to the marginal price to get a Q value which can then be used to get get price. We then simply multiply P and Q to get profit.

The profit maximizing outcome occurs where MR = MC. Where MC is the derivative of our cost function and MR is the derivative of our p(y)y=TC funciton.

1.) Lump sum tax does not affect our optimal outcome

2.) a specific tax is a per unit tax so we simply factor this into our cost curve in finding mc (if c(y)= x^2 and specific tax is $20/unit then new c(y) = x^2 +20x)

P1 = Lump sum fee = total consumer surplus

P2 = Marginal cost (as this maximizes surplus and the consumer will get all this surplus anyways

A firm is monopolistically competitive when they have slight market power and can enter the market freely. This firm will charge MR = MC

Dominant strategy equilibrium is always nash equilibrium but nash equilibrium is not always a dominant strategy equilibrium

A tool for finding optimal output level in a duopoly setting. Firms are in cournot equilibrium when y1 = y2 where these equations are derived from R1(y1,y2) and R2(y1,y2)

Firms are choosing simultaneously and independently

Firm 1: Final result should have y1 as dependent variable, y2 as independent variable.

Start with the formula: R(y1,y2) = p(y1, y2)*y1 + TC(y1)

Important note: any Y in market price becomes y1+y2

After finding and simplifying this equation, we derive this function with respect to y1.

Firm 2: Same process yet y1, y2 are flipped

A reaction firm for firm 1 should have an independent variable of q2 and vice versa

Cournot nash firms don't achieve the highest profits possible due to competition

Can maximize profits via collusion

Collusion will certainly lead to higher profits among both firms as they can always choose their C-N output level (but probably opt for something more collusive)

Both firms are incentivized to cheat on their agreement

We have player 1 (leader) with R(q2) and player 2 with R(q1) (follower).

We start by finding player 1's reaction function:

R(q2) = p(q1, q2)*q1 - c(q1) -> make q1 alone on one side

Plug our information into:

P(q2, R(q2))q1 - c1(q1)

Take this derivative and set it equal to 0 to get q2

Plug q2 back into R(q1) to get q1, or the response of player 1 to player 2's output of q2

Sum q1+q2 and plug these into the price equation

- Item must be sold to an individual who values good more than seller

- Item is sold to individual with the highest valuation

Yes

If a low-price bidder bids, the high price bidder will outbid

No

Highest bidder may not be able to get product if reserve price is too high

No

Highest bidder doesn't know how high to bid as they cant see other bids so they may not win the auction

Yes

In this auction it is the highest bidders dominant strategy to bid their valuation

No

Highest bidder doesn't know others valuation and so might let clock go too long

The high quality version of the good will be sold as long as the expected value of the good is greater than the cost of the high quality goods acquisition.

To find this, set the expected value of the good equal to the price to acquire the high quality good and solve for q

The market will unravel until price v, where v can be found in:

(lowest car price) + (v+buyer premium=highest car price+buyer premium) = v

If a workers skill level is not known, they will be paid the pooling equilibrium which is their expected productivity:

W_p = (1-h)aL+haH

h = percent chance worker is high productivity

aL = low ability marginal revenue product

aH = High ability marginal revenue product

High abil workers will get an education if

1) cost of education is lower than the benefits of an education for high ability workers

2) cost of education is higher than benefits for low ability workers

To find elasticity of demand, we use the following formula: P/Q * (dQ/dP)

We find P and Q using the simple process to determine price and quantity equilibrium in a monopoly market:

- Finding TR by multiplying p(y) by y

- Finding MR by taking the derivative of TR

- Set MR equal to MC

- Solve for P and plug this into demand to find Q

Then we simply derive the demand function wrt p to get the second part of out equation.

Plug in all found parts

Best to use a graph for this function:

A monopoly using a 2 part tariff will charge where P=MC as they want to maximize the surplus they capture with the tariff

Using P=MC, we can find the quantity of a good transacted per person and the price would be MC. Then, using a graph, we can find the amount of consumer surplus the firm would capture with the tariff per person. These added together will give us the total revenue collected by the firm. We then subtract total cost per person (MCXQ) and subtract to get profit per person. Them we multiply all this by the # of people in the market

We can use the equation (te)/(e+1) where e is elasticity and t is the given tax

We can find e with the equation P/Q * dQ/dP

The profit function in a competitive fringe situation is:

P(D(p)-Yf(p)) - C(D(P)-Yf(p)) where D(p) is the entire market's demand function and Yf(p) is the followers collective supply function. At this level, leader chooses price p and followers collectively supply Yf(p) units while the leader supplies the rest, or D(p)-Yf(p).