Hi everyone, I stuck on this question and I was wondering if ayone can help please.

Suppose there are two firms sharing a market. The firms’ products are assumed to be perfect substitutes. The demand curve for the market is P = 130 – Q, where Q is the total quantity produced. Each firm has a constant marginal cost equal to £10 and no fixed costs.

Find the Cournot output levels, price and profits

Thanks if anyone can help me out!

Suppose there are two firms sharing a market. The firms’ products are assumed to be perfect substitutes. The demand curve for the market is P = 130 – Q, where Q is the total quantity produced. Each firm has a constant marginal cost equal to £10 and no fixed costs.

Find the Cournot output levels, price and profits

Thanks if anyone can help me out!

Call these firms A and B. The way to do these questions is find the inverse residual demand curve (ie in terms of P) for the two firms, which we know are identical. Then when you have this expression use it to derive MR as we know MR=MC at the profit maximising point.

The residual demand curve faced by one firm is the market demand curve minus the supply of the rival firm.

The inverse market demand is $P = 130 - Q$ so the market demand is $Q = 130 - P$

Then the residual demand for A is $Q_A = 130 - P - Q_B$ and the residual demand for B is $Q_B = 130 - P - Q_A$.

So for both firms the inverse residual demand curves are $P = 130 - Q_B - Q_A$.

Now you have the inverse residual demand curves you can use them to find MR.

Start with firm A

$P = 130 - Q_B - Q_A \Rightarrow TR_A = PQ_A = 130Q_A - Q_A Q_B - {Q_A}^2$.

$\Rightarrow MR_A = \frac{d(TR_A)}{dQ_A}= 130 - Q_B - 2Q_A$.

You know the firm will produce where MR = MC and its MC is 10 so

$MR_A = 10 = 130 - Q_B - 2Q_A \Rightarrow Q_A = 60 - \frac{1}{2}Q_B$.

You can follow through the same working for firm B but as they are identical you are going to get the same expression reversed, $Q_B = 60 - \frac{1}{2}Q_A$. These are the best response curves, they are the equations for both firms that give them the answer 'what is our best response production level if the other firm produces a particular level', ie for firm A if firm B produces 50 then the best response for A is to produce 35.

The Cournot equilibrium is what happens if both firms are optimising their response so you solve them as simultaneous equations:

$Q_A = 60 - \frac{1}{2}Q_B$ and $Q_B = 60 - \frac{1}{2}Q_A$ so sub the second into the first and get $Q_A = 60 - \frac{1}{2}(60 - \frac{1}{2}Q_A)$ which works out as

$Q_A = 30 + \frac{1}{4}Q_A \Rightarrow \frac{3}{4}Q_A = 30 \Rightarrow Q_A = 40$.

If you sub this back into the expression for B then $Q_B = 60 - \frac{1}{2}40 = 40$.

So the Cournot output for both firms is 40.

(You might want to check that working as I did it on the fly and haven't checked it)

The residual demand curve faced by one firm is the market demand curve minus the supply of the rival firm.

The inverse market demand is $P = 130 - Q$ so the market demand is $Q = 130 - P$

Then the residual demand for A is $Q_A = 130 - P - Q_B$ and the residual demand for B is $Q_B = 130 - P - Q_A$.

So for both firms the inverse residual demand curves are $P = 130 - Q_B - Q_A$.

Now you have the inverse residual demand curves you can use them to find MR.

Start with firm A

$P = 130 - Q_B - Q_A \Rightarrow TR_A = PQ_A = 130Q_A - Q_A Q_B - {Q_A}^2$.

$\Rightarrow MR_A = \frac{d(TR_A)}{dQ_A}= 130 - Q_B - 2Q_A$.

You know the firm will produce where MR = MC and its MC is 10 so

$MR_A = 10 = 130 - Q_B - 2Q_A \Rightarrow Q_A = 60 - \frac{1}{2}Q_B$.

You can follow through the same working for firm B but as they are identical you are going to get the same expression reversed, $Q_B = 60 - \frac{1}{2}Q_A$. These are the best response curves, they are the equations for both firms that give them the answer 'what is our best response production level if the other firm produces a particular level', ie for firm A if firm B produces 50 then the best response for A is to produce 35.

The Cournot equilibrium is what happens if both firms are optimising their response so you solve them as simultaneous equations:

$Q_A = 60 - \frac{1}{2}Q_B$ and $Q_B = 60 - \frac{1}{2}Q_A$ so sub the second into the first and get $Q_A = 60 - \frac{1}{2}(60 - \frac{1}{2}Q_A)$ which works out as

$Q_A = 30 + \frac{1}{4}Q_A \Rightarrow \frac{3}{4}Q_A = 30 \Rightarrow Q_A = 40$.

If you sub this back into the expression for B then $Q_B = 60 - \frac{1}{2}40 = 40$.

So the Cournot output for both firms is 40.

(You might want to check that working as I did it on the fly and haven't checked it)

Original post by MagicNMedicine

.......

Thank you, amazing help! I then went on and got price as 50 and profit as 1600.

Sorry if you don't mind me asking, for this question it was split into different parts and there was a part that states:

"Briefly explain how the market equilibrium is determined under Cournot and Bertrand competition."

and then a second part that asks for:

"Briefly explain how equilibrium is determined under Bertrand competition"

Is that not the same question as the first one? I though the market equilbrium is the same as the Bertrand equilbrium? Am I wrong?

Thank you for your help

The market equilibrium will be different depending on whether the firms compete with each other on quantity or price.

If the firms are competing on quantity then they base their output decision (how much to produce) on how much they expect the other firm to produce. This is Cournot.

If they are competing on price then they base their price depending on what price they expect the other firm to set and so the output will naturally follow from that depending on what the demand curve says will be demanded at that price. This is Bertrand.

Bertrand is more appropriate when the firms can differentiate their products. If you have identical goods then the firms will just undercut each other till they produce at marginal cost so its just a price-taking competitive equilibrium.

If the firms are competing on quantity then they base their output decision (how much to produce) on how much they expect the other firm to produce. This is Cournot.

If they are competing on price then they base their price depending on what price they expect the other firm to set and so the output will naturally follow from that depending on what the demand curve says will be demanded at that price. This is Bertrand.

Bertrand is more appropriate when the firms can differentiate their products. If you have identical goods then the firms will just undercut each other till they produce at marginal cost so its just a price-taking competitive equilibrium.

Original post by MagicNMedicine

The market equilibrium will be different depending on whether the firms compete with each other on quantity or price.

If the firms are competing on quantity then they base their output decision (how much to produce) on how much they expect the other firm to produce. This is Cournot.

If they are competing on price then they base their price depending on what price they expect the other firm to set and so the output will naturally follow from that depending on what the demand curve says will be demanded at that price. This is Bertrand.

Bertrand is more appropriate when the firms can differentiate their products. If you have identical goods then the firms will just undercut each other till they produce at marginal cost so its just a price-taking competitive equilibrium.

If the firms are competing on quantity then they base their output decision (how much to produce) on how much they expect the other firm to produce. This is Cournot.

If they are competing on price then they base their price depending on what price they expect the other firm to set and so the output will naturally follow from that depending on what the demand curve says will be demanded at that price. This is Bertrand.

Bertrand is more appropriate when the firms can differentiate their products. If you have identical goods then the firms will just undercut each other till they produce at marginal cost so its just a price-taking competitive equilibrium.

Thanks again

Original post by MagicNMedicine

....

Hi, its me again. I was wondering if you could give me a hand on this question please. Its quite a long question with multiple parts. I done most of it but I don't know how to do the last parts. So here it is:

Assume a local area in which the market demand is described by the function P = 10 - Q, where Q is the quantity demanded and P is the price. In this area only one firm operates. The cost function of this firm is C = 2Q.

Compute the fixed cost (FC), the variable cost (VC), the average cost (AC), the average fixed cost (AFC) and the average variable cost (AVC). Are there economies of scale?

Cost function = aQ + b. The a is the variable cost multiplied by the quantity. Therefore here since there is no ‘b’ then there are no fixed costs. Whereas the variable cost = 2Q.

FC = 0

VC = 2Q

The average cost = Total Cost / Quantity. Total cost is FC + VC. Since there is no FC then total cost = 2Q. Therefore average cost = 2.

The average fixed cost = Since there are no fixed cost, the average fixed cost is also 0.

The average variable cost = VC / Q = 2 which just so happens to be the average cost as well since there is no fixed cost.

Economies of scale is whereby the average cost decreases as the quantity increases, if this occurs then there are economies of scale. In this case there are no signs of economies of scale as the average cost does not decrease as the quantity decrease.

An example is average cost in this case is total cost / Q. We know that TC is 2Q and say that is Q is 10 then AC is 20 / 10 = 2 and if Q is 20, AC is 40 / 20 and this is still 2 so no signs of economies of scale.

Compute the pair of profit maximizing Q and P. Solve for the profit

Total Revenue = Price X Quantity = (10 – Q)(Q) = 10Q – Q^2

Therefore Marginal revenue = 10 – 2Q

Profit maximization when MR = MC.

MC is the first derivative of TC = 2.

Therefore 10 – 2Q = 2. Therefore Q = 4.

Maximum revenue = Total Revenue – Total Cost.

= [10(4) – 4^2] – 2 = 22.

Find the level of the Herfindahl index

The Herfindahl index is a measure of market competition depending on the firm’s current market share. Since this is not given in the question, the Herfindahl index cannot be determined accurately. If by assuming that it is a monopoly since only one firm operates then the Herfindahl index will be greater than 0.6. (Is this correct)?

Assume now that another firm enters the market. Assume that this firm has the same marginal cost. These two firms become competitors and compete in prices. Find the equilibrium prices and the profit each firm gains in this equilibrium. Discuss your results.

This is the part I’m stuck on, since the two firms are competing in prices, then this is bertrand competition correct? How would I go about calculating this?

Sorry for always asking for help but my teacher really isn't that good....

(edited 11 years ago)

Original post by novadragon849

Thank you, amazing help! I then went on and got price as 50 and profit as 1600.

Sorry if you don't mind me asking, for this question it was split into different parts and there was a part that states:

"Briefly explain how the market equilibrium is determined under Cournot and Bertrand competition."

and then a second part that asks for:

"Briefly explain how equilibrium is determined under Bertrand competition"

Is that not the same question as the first one? I though the market equilbrium is the same as the Bertrand equilbrium? Am I wrong?

Thank you for your help

Sorry if you don't mind me asking, for this question it was split into different parts and there was a part that states:

"Briefly explain how the market equilibrium is determined under Cournot and Bertrand competition."

and then a second part that asks for:

"Briefly explain how equilibrium is determined under Bertrand competition"

Is that not the same question as the first one? I though the market equilbrium is the same as the Bertrand equilbrium? Am I wrong?

Thank you for your help

how did u get the profit?

so firms in cornet produce exactly the same quantity of output ??

Original post by manchesteruni33

so firms in cornet produce exactly the same quantity of output ??

Not necessarily, in this case they do as their marginal costs are equal to their average costs (£10) in both firms.

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