# cournot equilibrium and stackelberg equilibrium question

Question is as follow: there are 2 firms that want to enter the apple juice market in country A. There are no existing firms in the market or potential entrants.

They need to decide on yearly capacity Q1 and Q2, measured in liter. Starting next year, the annual demand for apple juice in country A is estimated to be D(P) = 800 – P. The cost of a liter of capacity for both firms is $500. This is the only cost of adding capacity, and once incurred, it is sunk. Assume once built, the capacity lasts forever and there is not any other future cost of production, that is, each firm can produce up to their capacity per year at zero additional cost. Both firms use a discount factor of 0.9. The market clearing price is determined by the total annual quantity in the market, that is, P = 800 – Q1 – Q2. Firms’ payoff is the Net Present Value of their profit. Assume that the firm will produce for infinitely many periods and the first year of revenue is not discounted. That is, the formula for NPV is then 1/(1-δ)*π, where π is the yearly revenue. a) Find the Cournot equilibrium (competing in setting capacities). b) Find the Stackelberg equilibrium under the assumption that Firm 1 moves first. c) Find the hypothetical monopoly capacity. I do not know how to do this question even for a.).. thanks for helping. EDITED: NPV is the net present value. In the case when all future cash flows are incoming (such as in this case the profit from selling apple juice) and the only outflow of cash is the purchase price(such as the cost incurred for entering this market in the first period), the NPV is simply the present value of future cash flows minus the purchase price. δ is the discount factor, that means the money you received next year worth only 0.9 of that value today, and the money you received 2 years later worth only 0.9^2 of that value today. Hope that would help. thanks. EDITED AGAIN: after digesting what you have given, may i ask you why i cannot get the same answer as you did? my answer is this, P = 800 - Q1 - Q2 is the Price determined by total quantity of the two firm. So i first see how firm1 would maximize its profit with expected output level of firm 2 to be Q2 π1 = (800 - Q1 - Q2) Q1 - CQ1 Take the derivative of it, resulting: π'1 = 800 - 2Q1 = Q2 - C SINCE C =$500, Q1 = 150 - Q2/2 at zero

i think this is what firm1 would choose to produce given that firm 2 will produce q2

and now firm2 expect firm1 to produce Q1 = 150 - Q2/2

it will then maximize

π2 = (800 - Q1 + Q2) Q2 - CQ2

take the derivative of it,

π'2 = 800 - Q1 + 2Q2 - 500

plug in Q1 from previous calculation

π'2 = 800 - 150 - Q2/2 -2Q2 - 500

Q2 would be

Q2 = 100 at zero

Plugging this result in Q1 we get Q1 = 100

and then we get the price P = 800 - 100 - 100 = 600

So the profit for each firm is 600 * 100 = 60000 Since we need to find the NPV, we then times 10 again, resulting in the equilibrium payoff for these 2 firms to be 60000*10 = 600000

why theres difference between the answers from yours and mine? Where am i get it wrong?

• You're talking mainly to mathematicians, not economists. Presumably NPV is Net Present Value and $\delta$ is the discount factor? – joriki Nov 29 '12 at 6:38
• yes, you re right. – Steve Nov 29 '12 at 6:49
• There's nothing to alert me to an edit in the question; I only happened to notice the edit by chance. Also it's not a good idea to write the question specifically addressed to one user. The standard way to do what you wanted to do would be to edit the question in the usual impersonal style and then leave a comment under my answer asking me to take a look at the edited question; the author of a post automatically gets notified of comments under it. – joriki Nov 29 '12 at 11:08
• To answer your question: I don't know why you expect to get the same results when you disregard the discount factor. You're comparing annual revenue with total costs, which doesn't make much sense. – joriki Nov 29 '12 at 11:11
• sorry, the character limits got me out of asking it under your answer, i will try to simplify my edited thing next time. THANKS. – Steve Nov 30 '12 at 6:57

## 1 Answer

The discounting multiplies the annual revenue by a factor of $10$. Thus the profit for firm $i$ is $10Q_i(800-Q_1-Q_2)-500Q_i=10Q_i(750-Q_1-Q_2)$. We can drop the factor ten and maximize $Q_i(750-Q_1-Q_2)$.

Given an annual capacity $Q_1$, firm $2$ wants to maximize $Q_2(750-Q_1-Q_2)$. Differentiating with respect to $Q_2$ yields $750-Q_1-2Q_2$, which is zero at $Q_2=375-Q_1/2$.

In Cournot competition, this must in turn lead to firm $1$ choosing $Q_1$, so we must have $Q_1=375-Q_2/2=375-(375-Q_1/2)/2$, and thus $Q_1=250$, and likewise $Q_2=250$.

In Stackelberg competition, firm $1$ anticipates firm $2$'s response $Q_2=375-Q_1/2$ and thus maximizes $Q_1(750-Q_1-(375-Q_1/2))=Q_1(375-Q_1/2)$. The derivative with respect to $Q_1$ is $375-Q_1$, so firm $1$ chooses $Q_1=375$, which leaves $Q_2=375-375/2=375/2$ for firm $2$.

For the monopoly capacity, (one tenth of) the profit is simply $Q(750-Q)$, the derivative with respect to $Q$ is $750-2Q$, so the optimal capacity for a monopoly is $Q=375$, the same as the capacity for the leader firm in Stackelberg equilibrium.

From the consumer perspective, the Stackelberg competition is the best, with a total annual capacity of $562.5$ being sold at a price of $237.5$, followed by the Cournot competition with total annual capacity $500$ and price $300$, and unsurprisingly the monopoly is worst with total annual capacity $375$ and price $425$.

The profits of the firms (without the factor $10$) are $375^2=140625$ for the monopolist, $250^2=62500$ per firm in Cournot competition, $375^2/2=70312.5$ for the leader firm in Stackelberg competition and $375^2/4=35156.25$ for the follower firm in Stackelberg competition. Assuming a $50/50$ chance of being the monopolist or leader firm, the expected profit per firm is $375^2/2=70312.5$ for the monopoly, $250^2=62500$ for the Cournot duopoly and $(375^2/2+375^2/4)/2=3/8\cdot375^2=52734.375$ for the Stackelberg duopoly, so the firms' preferences are in the opposite order to those of the consumers.

• may i ask is that D(P) = 800-P irrelevant? – Steve Nov 29 '12 at 8:31
• and i do not understand how you obtain those profit... what is the square of the quantity in each case implies? – Steve Nov 29 '12 at 8:32
• @Steve: No, $D(P)=800-P$ isn't irrelevant, it's where the market clearing price $P=800-Q=800-Q_1-Q_2$ comes from. I obtained the profits (without the factor $10$) directly from the formula I derived at the end of the first paragraph; the squares have no deeper significance; it just so happened that the same factor occurred twice in each expression so it was convenient to write them with squares. – joriki Nov 29 '12 at 11:23
• seems you are doing the whole thing by eliminating the factor 10, so your answer should not be the answer for this question right? the answer should be taking also the factor into account? – Steve Dec 1 '12 at 8:49
• @Steve: The question didn't explicitly ask about the profit; it asks about capacities and about equilibria, which I took to mean equilibrium capacities. If you want to include the profit at equilibrium in the answer, then yes, you'll need to include the factor of $10$ in that. – joriki Dec 1 '12 at 9:10