1. There are two firms producing the same good. Each firm has the cost structure
ID: 1250000 • Letter: 1
Question
1. There are two firms producing the same good. Each firm has the cost structure TC(q)=q2+30. These are the only two firms in the market where the MARKET demand for the good = P = -2Q + 50.
a) Find the profit of each firm if EACH firm produces 5 units of output. Show that if one firm produces 6, but the other stays at 5, the firm producing 6 earns more profit. (1 point)
b) Write the matrix for a game where the strategies for each firm are “produce 5” and “produce 6”. The numbers from (a) and (b) will help you fill in some of the payoffs but others will take some more calculation. (1 point)
c) Explain why the game is a prisoner’s dilemma. (1 point)
Explanation / Answer
a)
1. using basic algebra, solve the demand function for q=10 units (2 factories x 5 units each): P= -2(10)+50, this is your price per unit, for profit, multiply P x units produced
2. solve the supply equation for total cost, TC=5^2+30
3. subtract TC from price (P) x units to get profit
4. To show profit earned from production of 6 units, solve the TC for q=6, then solve the demand equation (P) for Q=11. TC=6^2+30; P=-2(11)+50;
show profit for 6 units multiply (Px6)-TC(6) > (P@11)x5-TC(5)
b) prisoners' delimma matrix: You already have most of the matrix, but you need one more calculation, which is the price for each when they each produce 6 and total quantity =12. This = P =-2(12)+50; profit for each = P(6)-TC(6)
To set up your matrix: numbering these from left to right, top to bottom, Block one in the matrix is the scenario when total unis = 10. So set up your price equation for 10, and TC equation for each firm at 5. For block Two (A produces 6, B produces 5) set up your profit eq for
Profit Firm A producing 5 units Firm A 6 units
Profit Firm B 5 Units Prof A&B: (P at 10 units x 5) - TC(5 units)| Prof A: (P at 11 units x 6) - TC(6 units)
| Profit b: (P at 11 units x 5) - TC(5 units)
Profit Firm B 6 units Prof B (P at 11 units x6) - TC(6 units) | Prof A&B (P at 12 units x 6) - TC(6 units)
Prof A (P at 11 units x 5)-TC(5 units) |
C)This is a prisoner's delimma because each firm, by producing an additional unit, will recieve a higher payoff or profit; no matter the situation, each firm will have a greater profit by producing an additional unit. If A doesn't produce 6 units, then B will to recieve the greater profit. The same goes for B, if they don't produce 6 units, then A certainly will, because it is in their best interest. In such cases, both firms will produce 6 units, driving price down to P(@ 12). In each case, producing the extra unit results in a higher pay-off for each firm, and thus, according to game theory, each will produce 6 units for a total supply of 12. Price per unit would be highest should each firm only produce 5, but as shown earlier, either firm is incentivized by greater profits to produce 6.
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