Chapter 5 section 5

Excel Companion Chapter 5 section 5

PointMP.xls

Open the file PointMP.xls.

Refer back to Chapter 4 and the introduction to marginal analysis. The three functions above for cost, revenue, and profit are identical to those in the file Marginal.xls. The derivative of the profit function (in the last column of PointMP.xls above) plays the same predictive role for discerning trends in the profit function P(q) as the expression P(q+1) - P(q) does in Marginal.xls. Instead of examining tables to determine where marginal cost = marginal revenue, we will use derivatives to find turning points of the profit function.

The slope of the tangent line to a profit curve at a point (q, P(q)) is usually fairly close to the slope of the secant line P(q+1) - P(q). The terminology point marginal profit is appropriate when tangent lines, instead of secant lines, are used in this way to measure rates of change. It turns out that , the derivative function, is much easier to work with algebraically than the difference function P(q+1) - P(q) which is why economists prefer to work with point marginal cost, point marginal revenue, and point marginal profit.

Looking back at a portion of the table in Marginal.xls on page 72, we see that marginal profit changes from positive at q = 98 to negative at q = 99.

quantity marginal profit

The quantity that corresponds to the theoretical maximum profit is located between 98 and 99. From the derivative of the profit function, [or profit in PointMP.xls], we observe that is positive and is negative. Solving the equation = 0 gives us a quick way of calculating the turning point. The positive solution of the equation -.006q² + 58 = 0 is

q = which is approximately 98.32. We can see that the value of q that maximizes the profit function is precisely that value of q for which crosses

the positive horizontal axis.

QUESTIONS:

1.	Where are the two break-even points ( nearest whole number)?
2.	Is the marginal profit positive or negative at the lower break-even point?
3.	Is the marginal profit positive or negative at the higher break-even point?

4 - 9 A manufacturer can fill orders for up to 80 units per day of a particular piece of office equipment. For each day that the plant is in operation, there are fixed costs of $3,600. The cost of the components to manufacture the product is $180 per unit. The manufacturer has been selling the product to a chain of office supply retail stores at a base price of $270 per unit. To boost sales, the manufacturer is considering a sales incentive discount that would lower the unit price by $50 for every 100 units that the chain buys. In other words, if q units are ordered then the unit price will be dropped by 0.50q dollars which gives us a revenue function defined by . The manufacturer's objective is to maximize daily profit.
4.	Write simplified formulas, in terms of q, for the cost and profit functions.
5.	What is the first break-even point?
6	At the lower break-even point is the marginal profit positive or negative?
7.	What is the second break-even point?
8.	At the second break-even point is the marginal profit positive or negative?
9.	Between the first and second break-even points there is a point where the derivative of the profit function is zero. What is that point?

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