3e) Suppose Joe has the opportunity to invest and lower his costs as follows:
C*(Qjoe) = 4 Qjoe
If Joe invests in this new technology and Sarah is stuck with her current costs
(constant marginal cost of $8), what would the new Bertrand-Nash Equilibrium be?
(3f) How much does each supplier earn under the Bertrand-Nash Equilibrium in (2e)
given that the investment cost for Joe is $500 ? Assuming that Sarah is stuck with her
current costs, what is the most that Joe would have been willing to spend for the new
technology?
(3g) Suppose Sarah has the same opportunity to invest in the lower cost technology
(at $500). If both Joe and Sarah make the investment and lower their marginal costs
to $4, what is the new Bertrand-Nash Equilibrium?
(3h) How much does each supplier earn under the Bertrand-Nash Equilibrium in (2g)
(accounting for the investment cost)? Are payoffs lower or higher in this equilibrium
than in the other two Bertrand-Nash equilibria (considered above)?
To find the answers to these questions, we need to understand the concept of Bertrand-Nash equilibrium and how it applies to the given scenario. Bertrand-Nash equilibrium refers to a situation in which two or more suppliers set their prices simultaneously, taking into account their competitors' prices, and there is no incentive for any supplier to unilaterally change their price.
Let's address each question one by one:
(3e) To find the new Bertrand-Nash equilibrium when Joe invests and lowers his costs, we need to compare the prices set by Joe and Sarah. The equation for Joe's costs is given by C*(Qjoe) = 4 Qjoe, where C*(Qjoe) represents Joe's costs, and Qjoe is the quantity he produces. On the other hand, Sarah has a constant marginal cost of $8. To determine the equilibrium, we look for the price at which both suppliers have the same cost. In this case, we equate Joe's cost equation with Sarah's constant marginal cost: 4 Qjoe = 8. Solving this equation, we find that Qjoe = 2.
Therefore, in the new Bertrand-Nash equilibrium, Joe's quantity is 2, and Sarah's quantity remains unchanged. The equilibrium price will be determined by the supplier with the lowest cost, which is Joe in this case since he invested in the new technology. Hence, the new equilibrium price will be $4.
(3f) To calculate the earnings of each supplier under the Bertrand-Nash equilibrium found in (3e), we need to multiply their quantity by the equilibrium price. Joe's earnings can be calculated by multiplying his quantity (Qjoe = 2) by the equilibrium price ($4), resulting in $8.
Since Sarah's quantity remains unchanged, her earnings will be calculated by multiplying her quantity (Qsarah) by the equilibrium price ($4) and subtracting her constant marginal cost ($8 * Qsarah). As the question does not provide information about Sarah's quantity, we cannot determine her exact earnings.
To determine the maximum amount Joe would have been willing to spend for the new technology, we need to calculate the difference in costs between his original technology and the new one. The investment cost is given as $500. Since Joe's original costs were 4 Qjoe, the maximum amount Joe would have been willing to spend for the new technology would be the cost difference divided by 4, which is $500/4 = $125.
(3g) If Sarah also invests in the lower cost technology, her cost equation becomes the same as Joe's: C*(Qsarah) = 4 Qsarah. Now both Joe and Sarah have the same costs. To find the new Bertrand-Nash equilibrium, we need to compare their prices. Since their costs are equal, they will set the same price in equilibrium.
Hence, the new equilibrium price will be $4, and both Joe and Sarah will produce the quantity at which their costs are equal. Solving the cost equation, we find that Qjoe = Qsarah = 1.
(3h) To calculate the earnings of each supplier under the Bertrand-Nash equilibrium found in (3g), we multiply their respective quantities (Qjoe = Qsarah = 1) by the equilibrium price ($4).
Joe's earnings will be $4 * 1 = $4, and Sarah's earnings will also be $4 * 1 = $4.
Comparing the earnings in the three Bertrand-Nash equilibria, we find that:
- In (3e), Joe's earnings were $8, and Sarah's earnings were not provided.
- In (3g), Joe's and Sarah's earnings are both $4.
- As we don't have earnings for Sarah in (3f), we cannot compare them directly.
However, we can see that Joe's earnings are lower in (3g) than in (3e), but the earnings for Sarah are not provided in (3e) or (3g), so we cannot make a direct comparison for her.