Assume that total benefit from a continuous decision, involving a control variab
ID: 1144978 • Letter: A
Question
Assume that total benefit from a continuous decision, involving a control variable Q, is: TB = 25Q – 2.5Q2 and the associated total cost is: TC(Q) = 5 + 4Q2.
With the total benefit and total cost functions (equations) given above:
Estimate the value of total benefit when the control variable Q is 3. Clearly show your steps and calculations.
Estimate the marginal benefit when the control variable Q is 3. Clearly show your steps and calculations.
Determine the level of control variable Q that maximizes total benefit. Clearly show your steps and calculations.
Provide a justification that total benefit is maximized at your determined level of control variable Q in (c) above.
Estimate the level of control variable Q that maximizes net benefits. Clearly show your steps and calculations.
Explanation / Answer
The value of total benefit when the control variable Q is 3
TB = 25Q - 2.5Q2
TB = 25 * 3 - 2.5 32
TB = 52.50
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TB = 25Q - 2.5Q2
Marginal benefit is found by taking the derivative of total benefit
Marginal benefit = 25 - 5Q
Marginal benefit = 25 - 5 * 3
MB = 10
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The level of control variable Q that maximizes total benefit is found by equating marginal benefit and marginal cost equation
TC = 5 + 4Q2
Marginal cost =8Q
8Q = 25 - 5Q
13Q = 25
Q = 2 units
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Total benefit at the determined level of control variable Q = 25Q - 2.5Q2
TB = 25 * 2 - 2.5*22
TB = 40