(a) If C(x) is the cost of producing x units of a commodity,...

(a) If $C(x)$ is the cost of producing $x$ units of a commodity, then the average cost per unit is $c(x)=C(x) / x .$ Show that if the average cost is a minimum, then the marginal cost equals the average cost. (b) If $C(x)=16,000+200 x+4 x^{3 / 2},$ in dollars, find (i) the cost, average cost, and marginal cost at a production level of 1000 units; (ii) the production level that will minimize the average cost; and (iii) the minimum average cost.

(a) If $C(x)$ is the cost of producing $x$ units of a commodity, then the average cost per unit is $c(x)=C(x) / x .$ Show that if the average cost is a minimum, then the marginal cost equals the average cost.
(b) If $C(x)=16,000+200 x+4 x^{3 / 2},$ in dollars, find (i) the cost, average cost, and marginal cost at a production level of 1000 units; (ii) the production level that will minimize the average cost; and (iii) the minimum average cost.

Key Concepts

Differentiation and Derivative Analysis

Differentiation is a mathematical technique used to compute the rate of change of the cost function with respect to production. By finding derivatives, one can determine marginal costs and set up necessary conditions to identify optimal production levels, such as when the marginal cost equals the average cost at a minimum.

Optimization

Optimization in this context entails finding the production level that minimizes the average cost. It involves calculus, particularly setting the derivative of the average cost function to zero to identify the critical point, at which conditions such as marginal cost equaling average cost can be analyzed.

Cost Function

The cost function represents the total cost incurred in the production of a given number of units, encompassing both fixed and variable costs. It lays the foundation for understanding various economic behaviors and decision-making processes related to production, pricing, and control over expenses.

Average Cost

Average cost is determined by dividing the total cost by the number of units produced. It indicates the cost per unit and is instrumental when evaluating production efficiency, economies of scale, and pricing strategies in an economic analysis.

Marginal Cost

Marginal cost is defined as the additional cost incurred from producing one more unit of a commodity. It is the derivative of the cost function with respect to the number of units and plays a critical role in optimizing production levels and in setting production policies.

Transcript

00:01 They tell us the average cost per unit, little c, is the cost of producing x units of something divided by the number of units produced.

00:13 So it's just total cost divided by the units, and so this is the average.

00:19 So we're going to actually, you know what i'm going to do.

00:21 I'm going to put bars over this little c to make it a little more clear about which is little c and which is big c.

00:31 All right, i think, well, i'll get more of them as we go through the problem.

00:37 So they want us to show that the average cost is minimum.

00:43 If the average cost is minimum, then the marginal cost equals the average cost.

00:48 So the marginal cost is the derivative of this.

00:55 So we can take the derivative, and again, we apply that, use the chain rule here, and use the quotient rule.

01:01 So we get the bottom times the derivative of the top, minus the top times the derivative of the bottom, which is just one, divided by the bottom square.

01:14 Now we want to set this equal to zero.

01:17 Well, doing that means we need to set this equal to zero.

01:21 And that means c prime at x not, our optimal value, is c at x not divided by x not, which is simply the average cost.

01:34 At x not.

01:35 So we can see that our relationship here is, as they showed.

01:41 So now the next question, part of the problem is given an actual cost function here.

01:48 And they ask, we want to find, for 1 ,000 units, the cost, the average cost, and the marginal cost.

01:56 So 1 ,000 units, plug in x equals 1 ,000, and we get, for the cost is $3402 ,500.

02:06 The average cost, the average cost, is just this divided by $1 ,000, $342 .5.

02:14 And the marginal cost, we can take the derivative of this.

02:23 And so from here that is we get nothing, and here we get 200, and here we get 6x times a square of x.

02:33 So we plug in 1 ,000 into here, and we get 389.

02:40 $189 .7.

02:43 It's the marginal cost.

02:45 Now they say the production level that will minimize the average cost.

02:51 Well, as we know from up here, the production level that minimizes the average cost is when the marginal cost equals the average cost...

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