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Sample MBA Math Exercises - EconomicsBelow are some sample economics exercises and solutions from the MBA Math quant skills course.Click on the playback controls for audio commentary on the sample economics exercises. Similar icons below provide commentary on individual exercise solutions. Overview Commentary
Exercise
#1: Marginal Analysis with Table
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| Units of Output | Total Cost ($) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 0 | 6 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1 | 9 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 2 | 15 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 3 | 24 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 4 | 36 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 5 | 51 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 6 | 69 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 7 | 90 |
Peter Regan teaching MBA Math at Tuck
Solution Commentary
Manual Solution Each plant should produce the quantity that generates the greatest profit. This solution shows two approaches. Solution Approach 1 We can find the most profitable quantity by adding columns for Total Revenue and Total Profit to the original table. Total Revenue is the price times the number of units. Total Profit is the difference between Total Revenue and Total Cost.
The optimal quantity is 4 and the corresponding profit is $16. Solution Approach 2 Alternately, and equivalently, we can find the most profitable quantity by adding columns for Marginal Cost, Marginal Revenue, and Marginal Profit to the original table. Marginal Cost for a given quantity is the change in Total Cost from the previous unit to the current unit. Marginal Revenue in this case is the price, which is the same for each unit. Marginal Profit for a given quantity is the difference between Marginal Revenue and Marginal Cost.
We want to produce as long as the marginal profit of each successive unit is not negative. As in the first approach, the optimal quantity is 4.
Excel Solution 
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Exercise
#2: Marginal Analysis with Formula
Suppose that you can sell as much of a product
as you like at $92 per unit. Your marginal cost (MC) for producing
the qth unit is given by:
MC=10q
If fixed costs are $350, what is the optimal
output level?
Solution Commentary
Manual Solution Marginal Analysis Set marginal revenue (MR) equal to marginal cost (MC) to find the optimal output level. 92=10*q q=9.2 Total Profit is computed as Total Revenue minus Total Cost. Total Revenue is price times quantity. Total Cost is fixed cost plus the sum of the marginal costs for each unit. To determine the optimal integer quantity according to the marginal analysis, we check the total profit for the integer values above and below the non-integer q. Total profit for q of 9 is $28. Total profit for q of 10 is $20. Thus, the marginal analysis shows that the optimal q is 9 with a corresponding profit of $28. Consider Fixed Costs One last check involves comparing the result from the marginal analysis with the fixed cost of producing nothing. The result of the marginal analysis is the optimal solution. The optimal q is 9 units. Total Profit for the optimal quantity is $28.
Excel Solution The Excel solution below implements a table to serve as a visual confirmation that the formula-based solution provided above is reasonable. 
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Exercise
#3: Marginal Analysis with Calculus
Suppose a competitive firm has as its total
cost function:
TC =
24 + 2q2
Suppose the firm's output can be sold at $66
per unit.
Using
calculus and formulas (but no tables) to find a solution, how many
units should the firm produce to maximize profit?
Please specify your
answer as an integer. In the case of equal profit from rounding up
and down for a non-integer initial solution quantity, enter the
higher quantity.
Solution Commentary
Manual Solution Marginal Analysis The first derivative of the total cost function is the marginal cost function: MC = 4q Set marginal revenue (MR) equal to marginal cost (MC) to find the optimal output level. 66=4*q q=16.5 Total Profit is computed as Total Revenue minus Total Cost. Total Revenue is price times quantity. Total Cost is fixed cost plus the sum of the marginal costs for each unit. To determine the optimal integer quantity according to the marginal analysis, we check the total profit for the integer values above and below the non-integer q. Total profit for q of 16 is $520. Total profit for q of 17 is $520. Thus, the marginal analysis shows that the optimal q is 17 with a corresponding profit of $520. Consider Fixed Costs One last check involves comparing the result from the marginal analysis with the fixed cost of producing nothing. The result of the marginal analysis is the optimal solution. The optimal q is 17 units. Total Profit for the optimal quantity is $520.
Excel Solution The Excel solution below implements a table to serve as a visual confirmation that the formula-based solution provided above is reasonable. 
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Exercise
#4: Supply and Demand
Assume that the demand curve D(p) given below is the market demand for apples:
Q = D(p) = 270 - 15p, p > 0
Let the market supply of apples by given by:
Q = S(p) = 42 + 15p, p > 0
where p is the price (in dollars) and Q is the quantity. The functions D(p) and S(p) give the number of bushels (in thousands) demanded and supplied.
What is the consumer surplus at the equilibrium price and quantity?
Round the equilibrium price to the nearest cent and round the equilibrium quantity DOWN to its integer part.
Solution Commentary
Manual Solution The equilibrium point is the point where the demand and supply curves intersect. Calculate the equilibrium point algebraically by setting D(p) = S(p) and solving for p. 270 - 15p = 42 + 15p 228 = 30p Solving for p and rounding to the nearest cent yields an equilibrium price of $7.60 Plug the value for p into either the supply or demand curve, solve, and round down to the nearest integer to obtain an equilibrium quantity of 156 Consumer surplus refers to the area of the triangle above the equilibrium price and below the demand curve. The vertical side of the triangle is the distance between the equilibrium price and the p-intercept, which is the demand curve price at a quantity of zero. The horizontal side of the triangle is the equilibrium quantity. The consumer surplus is thus [(18 - 7.6)*156]/2 = $811.20.
Excel Solution No Excel solution provided. The solution is based on simple arithmetic. |