Summary C(x) is the cost function C(x)/x is the average cost C’(x) is the marginal cost p(x) is the demand function which is the price per unit if we sell.

Presentation on theme: "Summary C(x) is the cost function C(x)/x is the average cost C’(x) is the marginal cost p(x) is the demand function which is the price per unit if we sell."— Presentation transcript:

1 Summary C(x) is the cost function C(x)/x is the average cost C’(x) is the marginal cost p(x) is the demand function which is the price per unit if we sell x units. R(x) = xp(x) is the revenue function. R’(x) is the marginal revenue P(x) = R(x) – C(x) is the profit function P’(x) is the marginal profit.

2 ECONOMICS Suppose C(x) is the total cost that a company incurs in producing x units of a certain commodity. The function C is called a cost function.

3 If the number of items produced is increased from x 1 to x 2, then the additional cost is ∆C = C(x 2 ) - C(x 1 ) and the average rate of change of the cost is: AVERAGE RATE

4 The limit of this quantity as ∆x → 0, that is, the instantaneous rate of change of cost with respect to the number of items produced, is called the marginal cost by economists: MARGINAL COST

5 As x often takes on only integer values, it may not make literal sense to let ∆x approach 0.  However, we can always replace C(x) by a smooth approximating function—as in Example 6. ECONOMICS

6 Taking ∆x = 1 and n large (so that ∆x is small compared to n), we have: C’(n) ≈ C(n + 1) – C(n)  Thus, the marginal cost of producing n units is approximately equal to the cost of producing one more unit [the (n + 1)st unit]. ECONOMICS

7 It is often appropriate to represent a total cost function by a polynomial C(x) = a + bx + cx 2 + dx 3 where a represents the overhead cost (rent, heat, and maintenance) and the other terms represent the cost of raw materials, labor, and so on. ECONOMICS

8 The cost of raw materials may be proportional to x. However, labor costs might depend partly on higher powers of x because of overtime costs and inefficiencies involved in large-scale operations. ECONOMICS

9 For instance, suppose a company has estimated that the cost (in dollars) of producing x items is: C(x) = 10,000 + 5x + 0.01x 2  Then, the marginal cost function is: C’(x) = 5 + 0.02x ECONOMICS

10 The marginal cost at the production level of 500 items is: C’(500) = 5 + 0.02(500) = $15/item  This gives the rate at which costs are increasing with respect to the production level when x = 500 and predicts the cost of the 501st item. ECONOMICS

11 The actual cost of producing the 501st item is: C(501) – C(500) = [10,000 + 5(501) + 0.01(501) 2 ] – [10,000 + 5(500) + 0.01(500) 2 ] =$15.01  Notice that C’(500) ≈ C(501) – C(500) ECONOMICS

12 Economists also study marginal demand, marginal revenue, and marginal profit—which are the derivatives of the demand, revenue, and profit functions. ECONOMICS

13 MARGINAL COST FUNCTION  Recall that if C(x), the cost function, is the cost of producing x units of a certain product, then the marginal cost is the rate of change of C with respect to x.  In other words, the marginal cost function is the derivative, C’(x), of the cost function.

14 DEMAND FUNCTION Now, let’s consider marketing.  Let p(x) be the price per unit that the company can charge if it sells x units.  Then, p is called the demand function (or price function), and we would expect it to be a decreasing function of x.

15 If x units are sold and the price per unit is p(x), then the total revenue is: R(x) = xp(x)  This is called the revenue function. REVENUE FUNCTION

16 The derivative R’ of the revenue function is called the marginal revenue function.  It is the rate of change of revenue with respect to the number of units sold. MARGINAL REVENUE FUNCTION

17 If x units are sold, then the total profit is P(x) = R(x) – C(x) and is called the profit function. The marginal profit function is P’, the derivative of the profit function. MARGINAL PROFIT FUNCTION

18 Taking ∆x = 1 and n large C’(n) ≈ C(n + 1) – C(n) Cost of producing the (n+1)st unit R’(n) ≈ R(n + 1) – R(n) Revenue from the (n+1)st unit. P’(n) ≈ P(n + 1) – P(n) Profit from the (n+1)st unit Interpretation of Mariginals

19 Summary C(x) is the cost function C(x)/x is the average cost C’(x) is the marginal cost p(x) is the demand function which is the price per unit if we sell x units. R(x) = xp(x) is the revenue function. R’(x) is the marginal revenue P(x) = R(x) – C(x) is the profit function P’(x) is the marginal profit.