Question 1. Mathematical Model of Production.

A company produces x amount TV sets. The total cost of a company per day is given by C(x) = Sx + SO and the price-demand function is given by p(x) = 11 - 0.01x. Find:

a) How many TV sets should the company manufacture each day to maximize the revenue?
b) What is the maximal revenue?
c) How many TV sets should the company manufacture each day to maximize the profit?
d) What is the maximal profit?
e) What is the selling price of one TV set when the profit is maximal?

Question 2. The Total Derivative. Cobb-Douglas Production Function

Consider the Cobb-Douglas production function is Q (K , L) = 6K^1/3L^2/3

a) Find total output when K=1000 and L=125.
b) Derive marginal productivity of capital.
c) Derive marginal productivity of labor.
d) Calculate these partial derivatives when K=1000 and L=125.
e) Using marginal analysis estimate Q(1003, 122)

Question 3. Price Elasticity of Demand. One Variable Case

A demand function is given by Q(P) = 600 - 12P

a) Find the elasticity.
b) Find the elasticity at a price of $20, stating whether the demand is elastic or inelastic.
c) Find the elasticity at a price of $25, stating whether the demand is elastic or inelastic.
d) Find the elasticity at a price of $30, stating whether the demand is elastic or inelastic.
e) At a price of $30, will a small increase in price cause the total revenue to increase or decrease? Why?

Question 4. Unconstraint Optimization

Profit function of the firm depending on x and y product is as following:

P(x,y)= 12x - 4y - 2x² + 2xy - y² - 7

Find the amount of each type of products (x and y) that must be produced in order to maximize profit.

Check the second order condition (SOC) to make sure that the solution is maximum point.