207 ECONOMICS SPRING 2008 LABORATORY EXERCISE 6
Problem 1. Find the derivatives of each of the following functions with respect to x. a. y = 24x 1/3 + 3x 2 e2x
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b. y = log[(2x 3 - 4x 2 )4 ]
Date: 17 February 2008.
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ECONOMICS 207 SPRING 2008 LABORATORY EXERCISE 6
c. f ( x ) =
2x 3 ( x 2 +2x )3 4x 2 -2x -7
d. Find the derivative with respect to x1 .
1/5 2/5 y = 1620x1 x2 - 16x1 - 243x2
ECONOMICS 207
SPRING 2008
LABORATORY EXERCISE 6
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Problem 2. Find the second derivative of each of the following functions with respect to x a. y = 5x 3 + 4x 2 - 10x
b. y = ( x 2 + 4x )3
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ECONOMICS 207 SPRING 2008 LABORATORY EXERCISE 6
3 +4x
c. y = 5xe2x
d. y = 20(100x + 20x 2 - x 3 ) - 500x
ECONOMICS 207
SPRING 2008
LABORATORY EXERCISE 6
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Problem 3. In the following problems you are given a production function for a firm where y is the level of output and x is the level of the variable input. You are given the price (p) of the output and the price (w) of the single variable input. For each problem 1) Write down an equation that represents profit for the firm. 2) Maximize this function by taking its derivative with respect to the variable input x and setting it equal to zero. 3) Solve for the profit maximizing level of x. 4) Find the optimal level of output. 5) What is revenue at the optimal level of output? 6) What is cost at the optimal level of the input x? a. output price = p = 20 input price = w = 500 y = output = f ( x ) = 100x + 20x 2 - x 3 For this example, profit is given by Pro f it = 20(100x + 20x 2 - x 3 ) - 500x
= 2000x + 400x2 - 20x3 - 500x = 1500x + 400x2 - 20x3
For this example, the derivative of profit is given by Pro f it = 1500x + 400x 2 - 20x 3 d Pro f it = 1500 + 800x - 60x2 dx d2 Pro f it = 800 - 120x dx 2 Now set the derivative of profit equal to zero and solve for x as follows. d Pro f it = 1500 + 800x - 60x2 = 0 dx 3x2 - 40x - 75 = 0
(3x + 5)( x - 15) = 0 5 x = - or x = 15 3
d2 Pro f it dx 2
At x = - 5 , 3
d2 Pro f it dx 2
= 800 - 120(15) = 800 - 1800 = -1000 < 0. The optimal x =15.
= 800 - 120(-5/3) = 800 + 200 = 1000 > 0. At x = 15,
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ECONOMICS 207 SPRING 2008 LABORATORY EXERCISE 6
The optimal output is given by y = 100x + 20x 2 - x 3
= (100)(15) + 20(152) - 153
Revenue is equal to the price of output multiplied by the output price. revenue = py = (20)(2625) = 52500 Cost of production is equal to the level of the input used multiplied by the input price. cost = wx (500)(15) = = 7500 Profit is equal to revenue minus cost. pro f it = revenue - cost = 52500 - 7500 = 45000. A graph of revenue, cost and profit is given in figure 1 F IGURE 1. Revenue, Cost and Profit
$
= 1500 + (20)(225) - 3375 = 1500 + 4500 - 3375 = 6000 - 3375 = 2625
50 000
40 000
30 000
Profit x
20 000
10 000
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x
ECONOMICS 207
SPRING 2008
LABORATORY EXERCISE 6
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b.
output price = p = 2 input price = w = 1998 y = output = f ( x ) = 200x + 100x 2 - 3x 3
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ECONOMICS 207 SPRING 2008 LABORATORY EXERCISE 6
Problem 4. In this problem (Problem 4), you will be given the price (p) for a competitive firm and the total cost function (TC[y]) for the firm. TC[y] specifies cost as a function of output which is denoted by y. Total revenue is given by TR[y] = py. Marginal cost (MC) is the derivative of the cost function with respect to output. At the optimal level of output, marginal cost will be equal to price. A graph of total revenue (TR), total cost (TC) and profit for such a firm is given in figure 2. Maximum profit is at the point where total revenue is at the highest level above total cost. F IGURE 2. Revenue, Cost and Profit
$ 14 000 12 000 10 000 8000 6000 4000 2000 TC TR
Profit y y
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For each problem write down the equation for profit. Then find the profit maximizing level of output. a. price = p = 550 Total Cost = TC (y) = 500 + 400y - 40y2 + 2y3
ECONOMICS 207
SPRING 2008
LABORATORY EXERCISE 6
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