1. 1 What’s next? So far we have graphical estimates for our project questions We need now is some way to replace graphical estimates with more precise computations

2. 2 Differentiation Purpose- to determine instantaneous rate of change Eg: instantaneous rate of change in total cost per unit of the good We will learn Marginal Demand, Marginal Revenue, Marginal Cost, and Marginal Profit

3. 3 Marginal Cost : MC(q) What is Marginal cost? The cost per unit at a given level of production EX: Recall Dinner problem C(q) = C 0 + VC(q). = 9000+177*q 0.633 MC(q)- the marginal cost at q dinners MC(100)- gives us the marginal cost at 100 dinners This means the cost per unit at 100 dinners How to find MC(q) ? We will learn 3 plans

4. 4 Marginal Analysis First Plan Cost of one more unit

5. 5 Marginal Analysis Ex. Suppose the cost for producing a particular item is given by where q is quantity in whole units. Approximate MC(500).

6. 6 Marginal Analysis Second Plan Average cost of one more and one less unit

7. 7 Marginal Analysis Ex. Suppose the cost for producing a particular item is given by where q is quantity in whole units. Approximate MC(500).

8. 8 Marginal Analysis Final Plan Average cost of fractionally more and fractionally less units difference quotients Typically use with h = 0.001

9. 9 Marginal Analysis Ex. Suppose the cost for producing a particular item is given by where q is quantity in whole units. Approximate MC(500). In terms of money, the marginal cost at the production level of 500, $6.71 per unit

10. 10 Marginal Analysis Use “Final Plan” to determine answers All marginal functions defined similarly

11. 11 Marginal Analysis Graphs D(q) is always decreasing All the difference quotients for marginal demand are negative MD(q) is always negative

12. 12 Marginal Analysis Graphs R(q) is increasing difference quotients for marginal revenue are positive MR(q) is positive R(q) is decreasing difference quotients for marginal revenue are negative MR(q) is negative Maximum revenue Marginal revenue 0 q1q1 q1q1 This shows that the maximum revenue will occur at the value q 1 where the marginal revenue is equal to zero

13. 13 Marginal Analysis Graphs

14. 14 Differentiation, Marginal Dinner Problem- Marginal Analysis

15. 15 Marginal analysis & Profit Function Since P(q) = R(q)  C(q) profit will increase with more dinners if the increase in revenue per dinner is greater than the increase in cost per dinner. This happens where MR(q) > MC(q). Similarly, profit will decrease with more dinners if the change in revenue per dinner (positive or negative) is less than the increase in cost per dinner. This happens where MR(q) < MC(q).

16. 16 Marginal analysis & Profit Function Profit stops increasing, and starts to decrease at its maximum value. the maximum profit must occur where MR(q) = MC(q). From the plot of MR(q) and MC(q), MR(2,025) = $6.84 = MC(2,025). the maximum profit will occur at q = 2,025 dinners. Direct computation shows that D(2,025) = $22.21 and that P(2,025) = $14,052.

17. 17 Dinner Problem expect to sell 2,025 buffalo steak dinners per week priced at $22.21 per dinner (recall p=D(q)) for a total profit of $14,052

18. 18 Differentiation, Marginal This must be where MP(q) changes from positive to negative. Thus, the maximum profit occurs when the marginal profit is zero, MP(q) = 0. MP(2,025) = $0.00. (results same as in the marginal analysis of revenue and cost) Where profit, P(q), is increasing, marginal profit, MP(q), is positive. Where P(q) is decreasing, MP(q) is negative. The change from increasing to decreasing profit occurs at the maximum profit. Another method to determine the maximum profit