Suppose there are two goods. The demand for good 1 is q1=a-bp1+ep2 and the demand for good two is q2=a-bp2+ep1.
a and b are strictly positive, |e|<b
(1)Compute the optimal prices, and the Lerner index and inverse elasticity of demand for each good.
(2)Now suppose the goods are produced by two firms that choose prices simultaneously. Compute the Nash equilibrium and compare it to the answer in part (1)
I.D.K. that's hard.
To find the optimal prices, we need to maximize the total revenue for each good. The total revenue is given by the product of price (p) and quantity demanded (q). So, we need to find a price that maximizes the total revenue for each good.
(1) Computing optimal prices, Lerner index, and inverse elasticity of demand for each good:
Step 1: Start with the demand function for good 1:
q1 = a - bp1 + ep2
Step 2: Write the total revenue function for good 1:
TR1 = p1 * q1 = p1 * (a - bp1 + ep2)
Step 3: Differentiate TR1 with respect to p1 and set it equal to zero to find the price that maximizes revenue:
d(TR1)/dp1 = a - 2bp1 + ep2 = 0
Solving for p1, we get:
p1 = (a + ep2) / (2b)
This is the optimal price for good 1.
Step 4: Calculate the Lerner Index for good 1. The Lerner Index measures the extent of the firm's market power and is given by:
L1 = (p1 - MC1) / p1
The Marginal Cost (MC) is not given in the question, so we can't calculate the exact Lerner Index. However, you can substitute the given values to compute the Lerner Index if you have the Marginal Cost value.
Step 5: Calculate the inverse elasticity of demand for good 1. The elasticity of demand is given by:
E1 = (dq1/dp1) * (p1/q1)
To calculate dq1/dp1, differentiate the demand function of good 1 with respect to p1:
dq1/dp1 = -b
Substituting this into the elasticity formula, we get:
E1 = (-b) * (p1/q1)
This is the inverse elasticity of demand for good 1.
Repeat the above steps for good 2:
Step 1: Demand function for good 2:
q2 = a - bp2 + ep1
Step 2: Total revenue function for good 2:
TR2 = p2 * q2 = p2 * (a - bp2 + ep1)
Step 3: Differentiating TR2 with respect to p2 and setting it equal to zero gives:
d(TR2)/dp2 = a - 2bp2 + ep1 = 0
Solving for p2, we get:
p2 = (a + ep1) / (2b)
This is the optimal price for good 2.
Step 4: Since the Marginal Cost (MC2) is not given, we cannot calculate the exact Lerner Index for good 2.
Step 5: The inverse elasticity of demand for good 2 can be calculated similarly to good 1. The elasticity of demand is given by:
E2 = (dq2/dp2) * (p2/q2)
Differentiating the demand function of good 2 with respect to p2 gives:
dq2/dp2 = -b
Substituting this into the elasticity formula, we get:
E2 = (-b) * (p2/q2)
This is the inverse elasticity of demand for good 2.
(2) Computing the Nash equilibrium:
In a Nash equilibrium, both firms maximize their profits simultaneously. To find the Nash equilibrium, we need to solve for the prices chosen by both firms.
Let's assume that firm 1 chooses price p1 and firm 2 chooses price p2.
The profit for firm 1 is given by:
Profit1 = (p1 - MC1) * q1
(where MC1 represents the marginal cost for firm 1)
Similarly, the profit for firm 2 is given by:
Profit2 = (p2 - MC2) * q2
(where MC2 represents the marginal cost for firm 2)
To find the Nash equilibrium, we need to maximize both firm's profits simultaneously. We can do this by setting the first-order derivatives of both profits equal to zero.
Solve for dp1 and dp2 to find the Nash equilibrium. The resulting prices will be the Nash equilibrium prices for both firms. Compare these prices with the optimal prices obtained in part (1) to analyze the differences.