1.  A person managing dry cleaning store for $30,000 per year decides to open a dry cleaning store.  The revenues of the store during the first year of operation are $100,000 and the expenses are $35,000 for salaries, $10,000 for supplies, $8,000 for rent, $2,000 for utilities, and $5,000 for interest on a bank loan.  Calculate:
 a. the explicit cost
Explicit costs = salaries + supplies + rent + utilities + interest on the bank loan = 35000 + 10000 + 8000 + 2000 + 5000 = $60000.
 b. the implicit costs
Implicit costs = entrepreneur forgone salary = $30000.
 c. the accounting profit
Accounting profit = TR - explicit costs = 100000 - 60000 = $40000
 d. the economic profit
Economic profit = TR - explicit costs - implicit costs = 100000 - 60000 - 30000 = $10000
 e. Indicate whether the person should open the dry cleaning store
 Since the person would earn an economic profit of $10000 per year, he or she should open the dry cleaning store: positive economic profits.

2. Answer the following questions:
 a. What is the slope of the line that goes through (2, 5), (8, 2)?
 m = (2 - 5) / (8 - 2) = - 3 / 6 = - 1 / 2
 slope = -0.5
 b. What is the slope of the line for which an equation is:
 6x + 3y - 12 = 0
  3y = 12 - 6x
  y = 4 - 2x
  slope = -2
 c. Is the following equation an equation for a demand or for a supply curve?
 2q + 6p - 8 = 0
  6p = 8 - 2q
  p = 4/3 -(1 / 3)q
 This is a demand curve because the slope is negative.

3. You are given the Total Cost function TC = (1/3)Q^3 - 8.5Q^2 +60Q+27
a. Derive the AC and MC functions
AC = TCQ = (1/3)Q^2 - 8.5Q + 60 + 27/Q
MC = dTR/dQ = Q^2 - 17Q +60
b. Explain the (graphical, i.e. in terms of slopes) relationship between TC and MC
MC is the slope of TC. 
 c.  Determine the level of output at which the total cost function is minimized and the level of total cost (Hint:  You will need to check the second derivative).
We need to solve: Q^2- 17Q + 60 = 0,
(Q - 12)*(Q - 5) = 0,  so the optimum are Q= 5 and 12.  Checking the second derivative to see which one of these is a minimum,
dMC / dQ = 2Q - 17
2(5) - 17 < 0, thus Q = 5 corresponds to a maximum.
2(12) - 17 > 0, thus, Q = 12 corresponds to a minimum.
At Q = 12, TC = (1 /3)*123 - 8.5*122 + 60*12 + 27 = 99.
 

4.  A market demand is such that if price is $10 per unit, the quantity demanded is 100.  For each $1 increase in price, the quantity demanded decreases by 20.  The quantity supplied is zero until the price reaches $5.  Above that price, the quantity supplied increases by 10 for each $1 increase in price. 
 a.  Determine the equation of the supply and demand curve. 
 First, one needs two data points.  An example is:  P = 10, Q = 100; P = 11, Q = 80.
Then, we can calculate the slope:   Slope = (10 - 11) / (100 - 80) = -0.05 = a.
 Using (Q = 100, P = 10), we have  P = b - 0.05Q, where b is the vertical intercept.
 Solving for b yields  b = 10 + 0.05*100 = 15.
The equation of the demand curve is therefore P = 15 - 0.05Q.
 Now we can do the same thing for the supply:
The 2 data points can be for example (Q = 0, P = 5) and (Q = 10, P = 6). 
 Slope = (6- 5) / (10 - 0) = 0.1.
We know that the vertical intercept is 5 (see above), thus, the equation of the supply curve is
 P = 5 + 0.1Q.
 b.  What are the demand and supply functions?
For the demand:
 0.05Q = 15 - P, so 
  Q = 300 - 20P. 
Similarly for the supply function, inverting the equation of the supply curve yields:
 Q = -50 + 10P.
 c.  What is the equilibrium price and quantity?
Setting demand = supply yields: 
-50 + 10P = 300 - 20P, 350 = 30P,
 P = 350 / 30 = $11.67
Plugging in the demand or supply function to get quantities yields: Q = Q = -50 + 10(11.67) = 66.67.

5.  Using the demand function
 Q = 300 - 10P  (1)
 a.  Determine the equation of the total revenue (TR), average revenue (AR) and marginal revenue (MR). 
 TR = PQ.  In order to calculate it, one needs to invert the demand function so that we can find the demand curve equation:
 10P = 300 - Q, P = 30 - 0.1Q.
Now, if we multiply both side of the demand curve equation by Q, we will have an expression for total revenue:  TR = PQ = (30 - 0.1Q)*Q = 30Q - 0.1Q2 
 Average revenue is defined as TR / Q: AR = (30Q - 0.1Q2 ) / Q = 30 - 0.1Q
 Marginal revenue is defined as the first derivative of total revenue:
  MR = dTR / dQ = 30 - 0.2Q
 b.c.  see class notes/ not included
 d.  Determine mathematically at what quantity is total revenue maximized.  Check that you have maximized not minimized total revenue.
 In order to maximize total revenue, one needs to set the first derivative of the total revenue function (the marginal revenue function) equal to 0:
 30 - 0.2Q = 0,   30 = 0.2Q,  Q = 30 / 0.2 = 150.  At quantities of 150, total revenue is optimized.  In order to know whether we have a maximum or a minimum, one needs to check the second derivative:
 d2TR / dQ2    = dMR / dQ = -0.2. 
Since -0.2 < 0, we conclude that we have maximized not minimized total revenue at Q = 150.