What is the profit equation for the monopolist

Question

Assignment:

1. A monopolist sells a product at price P and has no cost of production. The demand equation is Q = 2a - P, where Q is quantity and a is a demand shifter.

a. What is the profit equation for the monopolist?

b. What is the profit-maximizing price as a function of a? Denote the profit- maximizing price as P*.

c. What profit is earned as a function of a? Denote the optimal profit at π*.

d. Now suppose that a is unknown, and the firm must forecast it. The firm sets its price based upon this forecasted value of a,a^. The forecast error is a - a^. What is the expression for the loss, π* - π(a^)? Note that this expression will include both a and a^. What relationship does this bear to a quadratic loss function?

2. f y_t =b₀ +b₁*Time_t +e_t

And Time_n = 100 for n = 100, which is the second quarter of 2019 (quarterly data). Suppose the estimated model is

??₀ = 0.51 ??₁ = 0.02 ??² = 16

a. Construct point and normal 90% interval forecasts for the next four quarters (third quarter 2019 to second quarter 2020).

b. Suppose that the y_t in the previous part was yt = ln(Y_t). Construct point and interval forecasts for Y_t for the four quarters following 2019q2.

3. In the trend model T_t= ??₀ + ??₁*Timet, suppose that ??₁> 0.

a. Is the series expected to grow or decline in future periods?

b. Does this mean that the series grow or decline with certainty in every period?