Exercises and how to solve them

Question 1

Determine a coutnries’ production and consumption plans in autarky when the countries always consume both goods in equal amounts

Answer 1

1. Draw the production possibility frontiers
2. Goods x1 and x2 are the same, find the amount of x1. The highest number is the country that is doing the best.

Question 2

How do you compare the opportunity costs of two countries if the possibility frontiers are given?

Answer 2

first derivative and compare the absolute values.

Question 3

Where do the curves start if you want to draw a joint PPF with the function of two countries?

Answer 3

The y-axis is the amount of good 1, if only good 1 is produced.
==> Set good 2 in the ppf to zero in the functions of both countries and add these values together.

Example: If the value is 2 then you start at point 2 of the y-axis.

The x-axis is the amount of good 2 if only good 2 is produced. ==> Set good 1 to zero in the functions of both countries and add these values together.

Example: If country 1 produces 1 and country 2 produces 4, then you start at 5 of the x-axis.

Draw the joint ppf with the two different derivatives.

There is a line dividing the quadrant in two, this shows that x1 = x2. The intersection line is the point of interest

Question 4

How do you calculate the profit with producer surplus?

Answer 4

The profit is below the producer surplus (PS) due to fixed costs.

Profit = PS (y) - FC

Question 5

Why is an equilibrium in a perfectly competitive market Pareto efficient?

Answer 5

Because it maximizes the sum of consumer and producer surplus.

Question 6

If a cost function is given, how do you calculate the marginal costs?

Answer 6

First derivative

Question 7

If a cost function is given, how do you calculate the average costs?

Answer 7

Divide by y

Question 8

If you have a cost function given, how do you know if the average costs are monotonically increasing in y or decreasing?

Answer 8

Increasing: First derivative must be positive (set it bigger than zero)

Decreasing: Negative

Question 9

How to solve:
Let the market price be p = 10. Show the producer surplus.

Answer 9

1. Find the difference between revenues (p*y) and variable costs (VC (y))
   ==> If the y axis is the price, search for the point where y is 10 (in this example it is for example 5)
   Variable costs at y are equal to average variable costs at y multiplied by y.
   ==> Half this area and it is the producer surplus.

OR
Variable costs at y are equal to the area below the function up to y. The area above the marginal cost function is the producer surplus.

Question 10

Sketch the average fixed costs AFC (y). Given are marginal costs, average costs and average variable costs of a firm operating in a perfectly competitive market.

Answer 10

The average fixed cost function is FC/y.

The first derivative is negative -> decreasing AFC.
The second derivative is positive -> convex function
This means AFC are falling less and less fast.

==> It converges to infinity for small y whereas it converges to zero for large y. Additionally we know that it is a decreasing and strictly convex function.

You can see where AFC and AVC should be equal if you solve AC(y) - AVC (y)

Question 11

How to solve:
Find the cost function C (y). What do you have to do?

Answer 11

The cost functoin C (y) consists of variable and fixed costs.

To find the function of the VC (y):
In the example it was easy to see that AVC = y and MC(y) are 2y. That means Vc (y) have to be y^2 (The derivative of the VS function must equal MC).

To find the fixed costs:
Find the fix costs for one amount (for example for 10 = y the fixed costs were 10 so FC/10 = 10 –> So FC = 100.

Put it together:
C(y) = FC + VC(y) = 100 + y^2

Question 12

How to solve:
Find the long-run supply function.

Answer 12

In case of avoidable fixed costs, the firm will only produce according to the “price-equals-marginal-costs” rule if the price is bigger or equal AC(y) at y at maximum.

In case of sunk fixed costs the firm produces according to the “price-equals-marginal-costs” rule if the price is bigger than or equal AVERAGE costs

Question 13

How to solve:
Labor (l) is the only input factor and its price is w = 2.
What is the production function Y (l) of the firm?

Answer 13

Labor = 2 * L(y)

Set cost function of this exercise (Here: 100 + y^2) equal to 2*l and solve for y (only take the positive solution).

To sketch the graph, derive it once for the slope and twice for the form.

Question 14

How to solve:
Determine the individual supply (yi) of a profit-maximizing firm in the long-run equilibrium with free entry and exit.
(Cost function is given)

Answer 14

Set AC = MC

Question 15

How to solve:
Determine the market price (p*) in the long-run equilibrium with free entry and exit
(Cost function is given)

Answer 15

Determine the AC of the firm for y = individual supply (the value if AC and MC are equal)

Question 16

How to solve:
Determine the number of firms (n*) supplying a positive quantity in the long-run equilibrium with free market entry and exit.
(Cost function is given and deman)

Answer 16

Insert market price (insert individual supply (AC = MC) into AC function, this value you get is the market price) into demand function that is given.
==> This is the long-run market demand.

Divide this demand by the individual supply (In exercise by 0.5). Now you have the number of firms.

Question 17

Why do monopolists produce in the elastic part of the demand function?

Answer 17

Because it is optimal to increase prices (by reducing output) as long as they are in the inelastic part.

Question 18

Why do monopolists choose a lower quantity (higher price) than firms in perfect competition?

Answer 18

Price is a function of quantity for them.

Question 19

Assume that the seller is a monopolist who cannot discriminate prices. Determine the monopolist’s revenues at an output of y.

Answer 19

We assume a non-price discriminating monopolist ==> so revenues are equal to price times quantity. In the case of a monopolist the price is a function of quantity.
This means, the higher the quantity, that he wants to sell, the lower the price that he can charge.

R(y) * y

==> Take the Price function given and multiply this again with the supply y.

Question 20

Where can you see the Quantity effect and the price effect if the price function is given?

Answer 20

Draw the function.

The y-axis is the price and the x-axis is the quantity.

If you draw a quantity and price as a rectangle in the graph, then you can compare it with another quantity and this price.
The access on the right side is the quantity effect, on top there is the price effect.

Question 21

How do you calculate the price effect mathematically?

Answer 21

y * (P(y+differencey)-P(y))

Question 22

How do you calculate the quantity effect?

Answer 22

P(y+difference y) * difference y

Question 23

Determine the revenues at the maximum of a seller that is a monopolist and cannot discriminate prices. Draw the function as well.

Answer 23

To determine the maximum quantity:
Maximum is when marginal revenue (the first derivative of price function) is 0.

Take the second derivative to know if it’s concave or convex.

Question 24

Assume that the seller is a monopolist, who cannot discriminate prices. Determine the price elasticity of demand at the revenue-maximum

Answer 24

Price elasticity of demand:

derivative of demand function over demand function. Insert maximum price as p.

How to get demand function out of price functoin:
Inverse of price function

Question 25

Assume that the seller is a monopolist, who cannot discriminate prices. Assume that the firm’s fixed costs and marginal costs are zero. Determine the monopolist’s supply and the price elasticity of demand in optimum. How do the results change if we assume constant marginal costs?

Answer 25

If marginal costs are equal to zero, the problem of profit maximization is identical to the problem of revenue maximization since profit = revenue in this case.
(Everything stays same)

Question 26

Assume that the seller is a monopolist, who cannot discriminate prices. How do the results change if we assume constant marginal costs?

Answer 26

With marginal costs, the profit function changes.

Profit = P (y) * y - cy
(cy = marginal costs per unit)

Profit at maximum = profit’ (y) = 0 = 300-y - c
==> Thus, the profit maximizing quantity is y = 300-c and the price is p = P(y) = 300 - 1/2(300-c) = 150 + c/2

Conclusion: The higher the marginal costs, the lower the monopoly quantity at the optimum and the higher the monopoly price.

Elasticity is still derivative of demand function over demand function.
==> the conclusion here is that supply always lies in the elastic part of the demand function for c > 0

Question 27

Assume that the non-price-discriminating monopolist has the following cost function:

C(y) = FC + 100 y for y > 0

Determine the profit-maximizing supply (y*).

Answer 27

p * y - (FC + c + y)

Question 28

Determining the monopolist’s optimal price and quantity

Answer 28

1. Determine inverse demand function. Now you have the prices of both markets.
   
   2. Determine the monopolist’s profit contingent on y1 and y2.
      ==> Price 1 * y1 + Price 2 * y2 - C(y1 + y2)
      (Add up prices of both markets minus costs of both markets)
   3. Since we want to optimize, we need to take the partial derivatives of the profit functions.
      (Be careful of the derivative rules)
   4. To have the optimal quantities, just rearrange so that y1 respectively y2 is alone on one side.
      ==> Now you have the optimal quantities.
   5. To have the optimal price, plug in the optimal quantities in the inverse funtions.

Question 29

Under which conditions will the monopolist supply both markets?

Answer 29

In order for the monopolist to supply both markets, the marginal costs have to be smaller than both prohibitive prices (the price when yi(pi) = 0).
Comparing the two demand functions, we notice that c < 100 has to be fulfilled in order for both markets to be supplied. (Because then it is smaller than price 50 and it has to be smaller than BOTH prices)

Question 30

Determine the price and the quantities on both markets for c = 50. Determine the price elasticity of demand on both markets at the optimum. Explain the relationship between the price elasticity of demand and the price on a market.

Answer 30

1. Plug in c in the inverse function of quantities (y1) and prices.
   
   2. Take the inverse function of demand and multiply with inverse function of price over inverse function of quantities.
      The more elastic the demand is at the optimum, the smaller the monopoly price.

Question 31

Determine the monopolist’s profit function assuming that price discrimination is prohibited. Graph the (aggregate) demand function.

Answer 31

1. We know the prohibitive price, it’s the price of the inverse function of the demand.
   ==> We know that if 100</= p < 150, then the total demand equals the demand in market 1.
   While for p < 100, the total demnd equals the sum over both markets.The sum of the both functions it’s just adding up both functions that are given (here 500-4p)
   2. Yet, we want to maximize the profit of the monopolist, so we need the corresponding inverse function P(y)
      (inverse function of 300-2p and 500-4p, the functions that are relevant ==> The sole function of market 2 is not relevant in this example)
   3. Get the marginal revenue with P’(y) * y + P(y)

Question 32

You have a inverse demand function P(y) and cost function with FC given. How do you calculate the monopolist’s supply?

Answer 32

With the demand function and the cost function you can get the profit function. (P (y) * y - C(y))

To get the optimal supply the marginal cost (Derivative of C(y)) and the marginal revenue (The derivative of your profit function) have to be equal. Set them equal and solve for y.

Question 33

A MC(y) function is given and the inverse demand function P(y). If a question comes “let marginal costs be x. Is the supply in optimum efficient?” What do you do?

Answer 33

Calculate the optimal supply (Marginal revenue = Marginal cost) and the number for which the demand function equals to zero. Compare these two results, if they are equal, then it’s efficient.

Explanation:
The maximum possible quantity for a monopolist is when price = marginal costs.

The efficient amount is MR = MC.

Question 34

A Cost function is given and the inverse demand function. The market supply function under perfect competition is identical to the monopolist’s marginal cost function.
Calculate the consumer surplus.

Answer 34

In an equilibrium under perfect competition, we have x = y and P (x) = MC (y)

Take the inverse demand function and set it equal to the first derivative of the cost function. Solve for y. This is your maximum supply.

Insert your maximum supply into the inverse demand function and then you have the maximum price.

The consumer surplus is calculated with 0.5 (P(0) - p) * y

(insert 0 into inverse demand function, subtract maximum price and multiply with 0.5 and maximum supply.

(Producer surplus would be 0.5maximum supplymaximum price)

Question 35

A Cost function is given and the inverse demand function. The market supply function under perfect competition is identical to the monopolist’s marginal cost function.
Calculate the producer surplus in optimum.

Answer 35

Producer surplus would be 0.5maximum supplymaximum price BUT when “in optimum” is written in the task you have to use MR(y) = MC(y) to get to the optimal supply and optimal price.

With the new values use optimal price * optimal supply - Costs when you insert optimal supply in cost function.

==> If you have fixed costs, it’s optimal profit + fixed costs

Question 36

Calculate the deadweight loss (DWL).

Answer 36

Deadweight loss =

0.5 * (p* - MC (y)) * (y - y)

With y being the Pareto-efficient quantity y, which is determined by the intersectio of the marginal costs with the (inverse) demand function:

MC (y) = P (y)

Question 37

The inverse demand function in a Cournot Duopoly is given by P(y) = 100 - y.

The cost function of the firms in the markets is given by C1 (y1) = cy1 and C2 (y2) = cy2 with c = 10.

What is the profit function of firm 1? What is important to notice?

Answer 37

P(y1, y2) * y1 - C1(y1) =

(100 - (y1 + y2)) * y1 - 10 * y1

It is key that you have to insert y1 AND y2. The price company charges depends on what the other company does.

Question 38

The inverse demand function in a Cournot Duopoly is given by P(y) = 100 - y.

The cost function of the firms in the markets is given by C1 (y1) = cy1 and C2 (y2) = cy2 with c = 10.

Optimize the profit of y1 depending on the other guy’s quantity

Answer 38

You have to find the reaction function.

This is the derivative of the profit function and then solve for y1.

Here:
y1 = 45 - 1/2*y2

With this you can insert any quantity for the demand of y2.

Question 39

Determine the price and market supply at the Cournot-Nash equilibrium

Answer 39

First determine the market supply.
This would be
y1* = Y1(Y2(y1*))

In other words, insert in the reaction function of Y2 y1* and insert this into the reaction function of Y1. Solve for y1. (Reason: the best response y2 is a best response for y1)

For Price:
Insert the value for the supply into the Price function (P(y1, y2) and dont forget to take both y1 AND y2).

Question 40

Determine the producer surplus and the consumer surplus at the Cournot-Nash equilibrium

Answer 40

The producer surplus equals the sum of both firm’s profits at (y1, y2)
Start by determining the equilibrium profit of one firm:

Formula:
y1* (pCN - c)

y1* = the supply at the Cournot Nash equilibrium

pCN = Price at the cournot nash equilibrium.

Take this value times 2 and you have the producer surplus.

Consumer surplus:
1/2(y1 + y2*) (P(pCN))

Question 41

Is the market equilibrium Pareto-efficient?

Answer 41

Is pCN > c? If yes, then it’s not pareto-efficient.

Question 42

Suppose that starting from the Nash equilibrium in the Cournot duopoly firms can only move along lines I and II. In which direction should the two firms move in order to increase their profits? What has to be taken into consideration when doing so?

Answer 42

If both firms move along line I, then
(i) The total supply in the market remains constant, i.e., y1 + y2 = yCN on
line I.
(ii) The quantity supplied by one firm increases, while the quantity supplied
by the other firm decreases by the exact same amount.
Therefore, moving along line I has no impact on the price as pCN =
P(yCN).
However, profits are redistributed from one firm to the other, while the
sum of profits remains constant.
(b) If both firms move along line II in a northeastern direction, then
(i) The total supply in the market increases.
(ii) The profit shares of both firms do not change.
Since the quantity supplied in the Nash equilibrium of the Cournot duopoly
(yCN = 60) is already larger than the quantity supplied in monopoly (yM =
45, see Question 2(a)), a further increase in supply will reduce total profits.
(c) Moving along line II in a southwestern direction does not change the
profit shares of both firms, but reduces the total quantity supplied in the
market. Thus, it increases the profits of both firms. It has to be taken into
consideration that the movement along line II has no monotonic impact on
profits. To illustrate, note that the profit of both firms in point y1 + y2 = 0 is
smaller than the profit in the Cournot-Nash equilibrium.

Question 43

Determine the production possibility frontier.

Answer 43

Use y - mx + c

With y value when x = 0 and with m being the slope
Leave x as it is

Question 44

For which values does Anna have a comparative advantage in the production?

Answer 44

Compare slopes

Question 45

Determine the optimal consumption of both goods for Beat in the autarky case

Answer 45

Set both goods as the same variable (They are equal) and solve for the good.

Question 46

Determine the trade surplus in trade agreement compared to the autarky case

Answer 46

Search for the graph of trade agreement. This is just y-mx+c while y is Annas y when x is 0 plus Beats y when x is 0.
Set this equal to the variable of the good and compare the optimal consumption of both goods in auatrky case (setting graph for Beat equal to variable of good and the same for Anna)

Question 47

Determine the subgame-perfect Nash equilibrium of the game in extensive form.

Answer 47

What is the best response of firm 1? Firm 2 then has only the option to react. Determine the profit of firm 1, when it chooses the best option and subtract the best option from the result then you get the reaction of firm 2.

Question 48

Determine the aggregate demand function for the joint market (x(p))

Answer 48

Combine both demands (Add them up. This is the first solution.

The second solution is if market 1 is 0.

Question 49

Determine the equilibrium quantity in country B.

Answer 49

Set the market demand and market supply function equal and solve for p. Then insert p into the market demand function and solve for x.

Question 50

Determine the aggregated supply function and the aggregated demand function of the integrated market.

Answer 50

Bring the quantities to the left side of the market supply and demand function. (Add up functions if there are several market supply and several demand functions.)

Question 51

Determine the welfare loss due to the non-internalized interdependence.

Answer 51

Find the equilibrium without internalization. For this set demand and supply function equal. Then you get p. Insert this in the demand function. Now you have x

Find the equilibrium with internalization. This is just with adding -q (q is given) into the supply function. (E.g. -10’000 + 4(p-q). Find the new supply function by inserting q so that p is the only unknown variable. With this set this equal to the demand function again. Now you have p with internalization. Insert this p in the demand function now you have x with internalization.

To find welfare loss use
q * (x* - xS)

While xS being the value you got with internalization and x* without internalization.

Question 52

What is the supply equal to in the Cournot Nash-Equilibrium?

Answer 52

Use formula for x1’s profit

P(y)x1-MCx1

P(y) = demand for total quantity

Derive this to find the reaction functions.

Set the first derivative to 0 and solve for x1. This is your reaction function for x1. Do the same but solve for x2. This is the reaction function of firm x2.

Insert the reaction function of firm 2 into the reaction function of firm 1 (replace x2, so that x1 is the only unknown variable)

Solve for x1, then you have the quantity that is supplies by firm 1. Because this is an equilibrium, they probably produce the same amount (unless other MCs or stated otherwise) double this amount so that you get the whole supply in Cournot Equilibrium.

Question 53

How do you compute the Consumer surplus in the Bertrand Equilibrium?

Answer 53

The consumer surplus is always the area of the triangle between the demand curve and the price level.

To compute it you need the height and the base of the triangle.

Height = difference between the maximum price consumers are willing to pay (from the demand curve at y =0) and the actual price they pay (This is the Marginal costs in a Bertrand Equilibrium).

To calculate the base insert the Marginal costs (e.g. 4) as the result of the demand curve (e.g. 4 = 100-2y) and solve for y.

To calculate the height insert zero into the demand curve and subtract the actual price from the result (e.g. here 100-2*0 - 4 = 96). This is the height.

Multiply the height with the base and multiply the result again with 0.5.

Question 54

How do you calculate the Producer Surplus in the Nashequilibrium of the Cournot Duopoly?

Answer 54

Producer surplus is total revenue minus total variable costs.

Total revenue = amount supplied at equilibrium * price.

Total costs = MC * total amount supplied

Subtract the costs from the total revenue