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)

(1) To compute the optimal prices, we need to set the marginal costs equal to the marginal revenues for each good.

For good 1:
Marginal cost (MC1) = p1
Marginal revenue (MR1) = ∂(pq1)/∂q1 = b - 2bp1 + ep2

Setting MC1 = MR1:
p1 = b - 2bp1 + ep2

Rearranging the equation:
2bp1 + p1 = b + ep2
p1(2b + 1) = b + ep2
p1 = (b + ep2) / (2b + 1)

Inverse elasticity of demand for good 1:
e1 = ∂q1/∂p1 * p1 / q1
e1 = -(b - 2bp1 + ep2) * p1 / (a - bp1 + ep2) * p1
e1 = -(b + ep2) * p1 / (a - bp1 + ep2) * p1

Lerner index for good 1:
L1 = (p1 - MC1) / p1
L1 = (p1 - p1) / p1
L1 = 0

Similarly, for good 2:
p2 = (b + ep1) / (2b + 1)
e2 = -(b + ep1) * p2 / (a - bp2 + ep1) * p2
L2 = 0

(2) In a Nash equilibrium, each firm chooses its price simultaneously, considering the actions of the other firm. Given the demand functions and the assumption of simultaneous decision-making, we can determine the Nash equilibrium.

To find the Nash equilibrium, we set up the system of equations using the optimal prices derived in part (1):

p1 = (b + ep2) / (2b + 1)
p2 = (b + ep1) / (2b + 1)

Solving the system of equations simultaneously will give us the Nash equilibrium prices for both goods.

We can compare these Nash equilibrium prices with the optimal prices derived in part (1) to see if they are the same or not.

Please note that without specific numerical values for a, b, and e, we cannot provide the exact numerical solution, but you can use the above equations to find the Nash equilibrium for your specific values.

To compute the optimal prices, we need to find the prices that maximize the total revenue, given the demand functions. We can do this by taking the partial derivatives of the total revenue function with respect to each price and setting them equal to zero.

(1) Optimal Prices:

First, let's solve for the optimal price for good 1:

Total Revenue for Good 1, TR1 = p1 * q1
TR1 = p1 * (a - b * p1 + e * p2)

Taking the derivative with respect to p1 and setting it equal to zero:
d(TR1) / d(p1) = a - 2 * b * p1 + e * p2 = 0

Solving for p1, we get:
p1 = (a + e * p2) / (2 * b)

Next, let's solve for the optimal price for good 2:

Total Revenue for Good 2, TR2 = p2 * q2
TR2 = p2 * (a - b * p2 + e * p1)

Taking the derivative with respect to p2 and setting it equal to zero:
d(TR2) / d(p2) = a - 2 * b * p2 + e * p1 = 0

Substituting the value of p1 from above and solving for p2, we get:
p2 = (a + e * p1) / (2 * b)

Lerner Index:

The Lerner index measures the extent of market power for a firm and is calculated as the inverse of the elasticity of demand. The formula for the Lerner index is:

Lerner index = (p - MC) / p,
where MC is the marginal cost.

The inverse elasticity of demand is the reciprocal of the price elasticity of demand and is given by:

Inverse elasticity of demand = 1 / (d(q) / d(p)).

For good 1:

Inverse elasticity of demand for good 1 = 1 / (d(q1) / d(p1))
= 1 / (-b + e * p2)

For good 2:

Inverse elasticity of demand for good 2 = 1 / (d(q2) / d(p2))
= 1 / (-b + e * p1)

It is important to note that the Lerner index and inverse elasticity of demand can only be calculated once the optimal prices are known.

(2) Nash Equilibrium:

In a simultaneous pricing game, the Nash equilibrium occurs when neither firm can unilaterally change its price to obtain more profit.

From our previous calculations, we have the optimal prices for both goods:

p1 = (a + e * p2) / (2 * b)
p2 = (a + e * p1) / (2 * b)

To find the Nash equilibrium, we need to solve these two equations simultaneously. However, there is no general solution for these equations as they are nonlinear.

We can try to solve them using an iterative method or by plugging in specific values for a, b, and e.

Comparing Part (1) and (2):

It is not possible to directly compare the Nash equilibrium with the optimal prices derived in part (1) without specific numerical values for a, b, and e. However, in general, the Nash equilibrium may or may not coincide with the optimal prices. The Nash equilibrium represents a stable solution where neither firm has an incentive to change its price unilaterally. The optimal prices, on the other hand, increase total revenue and maximize the profits of each firm individually.

To compute the optimal prices, Lerner index, and inverse elasticity of demand for each good, we need to maximize the utility of consumers given the demand functions and the constraints.

(1) Compute the optimal prices:
To find the optimal prices, we need to solve the utility maximization problem. The utility-maximizing prices will be the ones that maximize consumer satisfaction.

The utility maximization problem can be formulated as follows:
Maximize U = q1*q2, subject to q1 = a - bp1 + ep2 and q2 = a - bp2 + ep1.

To solve this problem, we need to find the first-order conditions by taking the derivative of the utility function with respect to each price and setting them equal to zero.

For good 1:
∂U/∂p1 = q2* (∂q1/∂p1) + q1* (∂q2/∂p1) = 0

And for good 2:
∂U/∂p2 = q1* (∂q2/∂p2) + q2* (∂q1/∂p2) = 0

Substituting the expressions for q1 and q2 from the demand functions:
∂U/∂p1 = (a - bp1 + ep2)*e - (a - bp2 + ep1)*b = 0
∂U/∂p2 = (a - bp2 + ep1)*e - (a - bp1 + ep2)*b = 0

We can solve these two equations simultaneously to find the optimal prices p1* and p2*.

The Lerner index is a measure of market power and is defined as the ratio of the difference between price and marginal cost to price. In this case, the marginal cost is assumed to be zero. The Lerner index for good 1 is given by:
L1 = (p1* - 0) / p1* = 1

Similarly, the Lerner index for good 2 is:
L2 = (p2* - 0) / p2* = 1

The inverse elasticity of demand for good 1 is defined as the ratio of the percentage change in price to the percentage change in quantity demanded. It measures the responsiveness of demand to a change in price. In this case, it can be computed as:
E1 = - (∂q1/∂p1) * (p1 / q1)

Similarly, the inverse elasticity of demand for good 2 is given by:
E2 = - (∂q2/∂p2) * (p2 / q2)

Now, we have the optimal prices, Lerner index, and inverse elasticity of demand for each good.

(2) Compute the Nash equilibrium:
In a simultaneous price-setting game between two firms, we need to find the Nash equilibrium, which is the stable solution in which neither firm has an incentive to change its price given the other firm's price.

To find the Nash equilibrium, we need to solve the reaction functions, which represent the best response of each firm to the other firm's price.

The reaction function for firm 1 is obtained by assuming that firm 2 sets the price p2, and maximizing firm 1's profit given this price. Similarly, the reaction function for firm 2 is obtained by assuming that firm 1 sets the price p1, and maximizing firm 2's profit given this price.

The reaction function for firm 1 is:
p1 = (a - ep2 + b*p2) / 2b

And the reaction function for firm 2 is:
p2 = (a - ep1 + b*p1) / 2b

To find the Nash equilibrium, we need to find the intersection point of these two reaction functions, which gives us the prices at which neither firm has an incentive to change its price.

By substituting the reaction functions into each other, we get:
p1 = (a - e((a - ep1 + b*p1) / 2b) + b*((a - ep1 + b*p1) / 2b)) / 2b

Simplifying and solving for p1, we find the value of p1*.
Similarly, we can solve for p2* using the reaction function for firm 2.

Then, we can compare the Nash equilibrium prices with the optimal prices found in part (1) to analyze any differences or similarities.

This is how you can compute the optimal prices, Lerner index, inverse elasticity of demand, and Nash equilibrium in this scenario.