Algebraic methods to find the equilibrium point

Question

Problem 1: Given the demand and supply functions below
   
Demand: P =  -0.0056Q + 12
Supply: P = 0.0033Q + 8
   
(Assume that P represents the price per unit, and Q represents the quantity of units demanded at that price.)
   
(a) Use algebraic methods to find the Equilibrium Point. 

(b) Then use Excel to sketch both functions on the same graph with price on the vertical axis, and quantity on the horizontal axis.
   
Problem 2: A manufacturer finds that the total cost, TC, of producing a product is given by the cost function:

TC = q³ - 12q² + 60q

a. Determine the average cost per unit if 3 units are produced, AC(3).

b. Determine the marginal cost if 3 units are produced.  This is MC(3). This will give the approximate cost of the next unit (namely, the 4th).

c. Use Calculus: At what level of  q will average cost per unit, AC, be a minimum? 

Problem 3: Suppose demand for Wine A (QA) is a function of the price of Wine A (PA) and the price of Wine B (PB)

QA = f(P_A, P_B) = 1000 – 50P_A + 2P_B  bottles

(a) Calculate demand for Wine A if the price of Wine A is $3 and the price of Wine B is  $10.
(b) Suppose that the price of Wine B is fixed at $10. Explain the significance of the -50
(c) Suppose that the price of Wine A is fixed at  $3. Explain the significance of the  2
   
Time Value of Money:
   
Problem 4: Suppose that a project requires an initial investment of $40,000.  The project guarantees the following cash flows at the end of the indicated periods:

Year	Cash Flow
1	$ 4,000
3	$ 6,500
4	$ 14,000
6	$ 16,000
7	$19,000

We suggest that you use mathematical formulas (rather than Excel’s built-in functions or your finance calculator) in this exercise.   You will need to be familiar with these formulas.
   
(a) Find the Net Present Value of the project. Is the project profitable? Assume an interest rate of  8 % per year compounded annually.

(b) Does profitability change [from (a)] if all payments are delayed by a year? Explain your answer.

(c) Does profitability change [from (a)] if the interest rate increases to 9% per year? Explain your answer.