MarginsEcon.nb 1 Using Derivatives in an Economics Set- ting. LectroCopy makes photocopy machines, and sells them to make money. If they make x copy machines per week, then their total costs for that week are given by C(x)=0.000002 x 3 - 0.02 x 2 +1000 x + 120,000 in dollars. Here's what the graph of the total cost function, C(x), looks like for 0 ≤ x ≤ 8000 8·10 6 6·10 6 4·10 6 2·10 6 2000 4000 6000 8000
MarginsEcon.nb 2 Question: What is the cost of the 3001-st copier that they will produce? Ans: C(3001) - C(3000) = $933.998 With very little work we can rewrite this as a difference quotient: 933.998= C H 3001 L - C H 3000 L C H 3001 L - C H 3000 L Å Å Å Å Å Å Å Å Å Å Å Å Å Å Å Å Å Å Å Å Å Å Å Å Å Å Å Å Å Å Å Å Å Å Å = Å Å Å Å Å Å Å Å Å Å Å Å Å Å Å Å Å Å Å Å Å Å Å Å Å Å Å Å Å Å Å Å Å Å Å 3001 - 3000 1 Which is the slope of the secant line to C(x) over the interval [3000,3001] and is very very close to the slope of the tangent line to C(x) at x=3000, which in turn is given by C'(3000) = $934. (Check this by differentiating C(x), and then substituting 3000 for x.) This shouldn't be a surprise, that the slope of the
MarginsEcon.nb 3 secant line is nearly equal to the slope of a tangent line, especially for such a small value of h ( = 1 ). Conclusion: The derivative function C'(x), called the marginal cost function, provides an accurate approximation to the cost of pro- ducing the x+1-st item. So the 3000-th item costs about $934. We don't expect each of the first 3000 photocopi- ers to cost exactly $934. Question: What is the average per item cost of the first 3000 copiers? Answer: It is clearly C H 3000 L Å Å Å Å Å Å Å Å Å Å Å Å Å Å Å Å Å = $998. 3000 This motivates us to define the average cost ê ê ê function, C H x L , (say "cee bar of x") as
MarginsEcon.nb 4 ê ê ê Å Å Å Å Å Å Å Å Å Å H x L = C H x L C x ê ê ê and realize that C H x L provides us with the average per unit cost when producing x units. Now look: ê ê ê H 1000 L = 1102, C ' H 1000 L = 966 and C ê ê ê H 100 L = 2198, C ' H 100 L = 996. What's C going on here? The thousandth copier costs $966 to build but the average cost of the first 1000 built is 1102, and the hundredth copier costs $996 to build but the average cost of the first 100 built is 2198!
MarginsEcon.nb 5 How can this happen? In a word, fixed costs. The cost function incorporates the term 120,000 which are termed fixed costs, that is costs which occur even if we build no copiers. Things like rent, insurance, heat etc. must be paid even when no produc- tion is going on. These fixed costs are being figured into the average cost of building the copiers, but when finding the cost of the x+1-st copier, they are filtered out, since we are just count- ing the additional cost of the next machine.
MarginsEcon.nb 6 The marginal average cost function is the rate of change of the average cost function. It will measure the amount by which the average cost per item changes if we make just one more item. In concrete terms for our example we compute that ê ê ê ' H 3000 L = -0.02133 C which can be interpreted as the average cost of building a copier will diminish by two cents when we boost production from 3000 to 3001 per week.
MarginsEcon.nb 7 Here are our four cost related functions and their interpretations: C(x), total cost function , the total cost of building x items. C'(x), marginal cost function , the cost of building the x+1-st item after having build x items before it. ê ê ê H x L , the average cost function , C the average per unit cost when build- ing x units. ê ê ê ' H x L , the marginal average cost function , C the amount by which the average per item cost will change if we are already making x units and we make just one more unit.
MarginsEcon.nb 8 Revenues The other side of the business, besides costs, are revenues, the amount of money flowing into the business. In our simple model, revenues are easily computed to be the number (x) of items sold times the per unit price (p) in dollars. We appeal to the demand function p=f(x) developed earlier which will dictate the price once the num- ber of units demanded (x) is known. This price p will sell exactly x units with no unmet demand lingering behind. Thus our total revenue function, the amount of money brought in by sales of x items, is seen to be
MarginsEcon.nb 9 R(x) = xp = xf(x) (#units) ä (unit price) R(x) gives the total revenues generated by selling x units. As before we can find the marginal revenue function, R'(x), to be the additional revenue generated by selling the x+1-st unit. By now you should recognize that, to an economist, marginal means "derivative of". Returning to our example: LectroCopy's price/demand function is seen to be p = f(x) = 2000 - 0.04x dollars if x copiers are being sold. That would mean
MarginsEcon.nb 10 that the total revenue function is R( x ) = x p = x (2000 - 0.04 x ) = 2000 x - 0.04 x 2 and our marginal revenue function is R'(x) = 2000 - 0.08x and that the additional revenue generated by selling the 3001-st copier will be R'(3000) = $1760.
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