Question

Assume that total benefit from a continuous decision, involving a control variable Q, is: TB =...

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.

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