Round your answers to two digits after the decimal(the nearest 1/100th). The following question consists of a function in one variable. Moreover, the objective function is quadratic. So the...

4

Extracted text: Round your answers to two digits after the decimal(the nearest 1/100th). The following question consists of a function in one variable. Moreover, the objective function is quadratic. So the first-order condition is as simple as can be, namely linear. As we saw in 1530, this is the easiest possible kind of optimization problem. As such, it is a great setting to practice working with the Envelope theorem. A firm produces a single commodity (read: the firm operates in a competitive market) and receives p = 35 for each unit sold. The cost of producing K units is 8K + 5K? and the tax per unit is 1. The firm wants to maximize its profit. 1. Find the firm's profit n as a function of K and express as: T= a · K+b· K². • The coefficient a is equal to • The coefficient b is equal to 2. Find the optimal production K* that maximizes profit: K* 3. Find the optimal profit a*: 7* 4. Next, the price of the commodity p increases. As economists, we are interested in how the profit the firm realizes changes as a result in the change in price of the commodity. To assess this, in principal we need to consider the direct effect a change in p has on the profit and the • x effect a change in p has through it's effect on optimal quantity K* the firm produes, i.e, d | dpK dK* + aK\K dp This calculation, however, can be simplified: The Envelope theorem states that the second term on the right hand side is equal to because it is the derivative with regard to al an * x variable. We therefore only need to consider the * x effect of a change in p on the profit. Specifically in the example we are considering here, the change in the profit as p changes can be expressed as: dn* a +b- K +c·K². dp where o the coefficient a is equal to o the coefficient b is equal to o the coefficient c is equal to