(a) If C(x) is the cost of producing x units of a commodity, then the average cost per unit is c

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

Key Concepts

Marginal Cost

Marginal cost is the rate of change of the total cost function with respect to the number of units produced; mathematically, it is represented as the derivative C'(x). It indicates the additional cost incurred by producing one more unit and is crucial for decision-making in production and pricing strategies.

Optimization of Average Cost

The process of minimizing the average cost involves finding the production level at which c(x) is at its lowest. Using calculus, this is achieved by setting the derivative of the average cost function equal to zero. A key result is that at the minimum average cost, the marginal cost (C'(x)) equals the average cost, providing an important condition for optimal production analysis.

Cost Function

A cost function represents the total cost incurred in the production of x units of a commodity. It includes all costs that vary with the level of production as well as any fixed costs. This function forms the basis for analyzing spending behavior and efficiency within production decisions.

Average Cost

The average cost, defined as the total cost divided by the number of units produced (c(x) = C(x)/x), measures the cost per unit on average. It is a key metric used to assess the efficiency of production and to identify economies of scale, as it reflects how costs behave as production is increased.

Transcript

00:01 Okay here we are going to show that the marginal cost is equal to the average cost when average cost is a minimum.

00:09 And then we are going to take a example where we are given a function for the cost of producing units and then find various attributes of the cost given some units.

00:23 And then also minimize that function.

00:26 So first we have to show that average cost and marginal costs are the same when average cost is at a minimum.

00:34 So we're gonna do that using our um caution rule.

00:39 So here's the formula for average cost is given in the problem.

00:44 And so we want this to be at a minimum and to do that it has to be at a critical point.

00:50 And so the derivative must be zero.

00:52 So we're going to set let's change color here.

00:54 We're in a set derivative of average cost is equal to zero.

01:01 And so that means derivative of cost which i marked with a big c.

01:11 With like a little tip at the top to differentiate it from a normalcy.

01:18 This is also equal to zero.

01:20 So let's use our coercion rule.

01:23 So derivative of the top times the bottom minus diverted of the bottom.

01:33 That's just one.

01:35 So i'm not gonna write anything times the top all over the bottom squared and this is equal to zero.

01:46 So we're going to cancel the bottom real quick and simply add the cost to the other side and then divide by x.

01:54 So if we do all that in order we get derivative of cost or the marginal cost because the marginal cost is the cost to add another unit of production.

02:05 So it's like the growth rate of the cost and this is going to be equal to the cost over units produced, which of course is equal to our average cost.

02:21 And so there's part a done using product role.

02:25 Now we're going to tackle part b here.

02:28 So first we have to find the cost, average cost and marginal cost when production level is 1000 units a k a when x is 1000 units.

02:37 So cost is easy enough.

02:41 We just have to plug in 10,000 here.

02:45 And so if we let's do, i might need more room here, let's do cost is going to be equal to 16,000 plus 200 times 1000 is just 200,000 plus four times 1000 to the 3/2.

03:10 So i'm going to write, this is 10 to the three, three halves, which does not cancel as well as i'd like.

03:18 So let me just compute this out.

03:22 Well, the calculator is able to show, but if you just compute this out, it's going to be 247 six, .77, repeating.

03:37 So let's just round it up to 78.

03:41 So this is how much it costs to produce 1000 units.

03:47 Put a dollar.

03:47 Sign out here, let me move this down.

03:52 No way i can just move this down for now.

03:58 And so now we have to compute the average cost.

04:02 And so all we have to do to find average costs.

04:06 If you remember this is cost, total cost.

04:12 Let me use a different color again, average costs is equal to total cost, which we just computed over x.

04:27 So this is going to be 2476, we take out the comma 6 to 2.78 divided by 1000, which will be equal to you.

04:46 Just shift the decimal three points 247 and all around to 62 cents.

04:58 And so finally we have to find the marginal cost.

05:02 And if you remember in part a we showed that the average cost is equal to the marginal cost, but that's only when the average costs at a minimum.

05:15 So we can't use that.

05:17 So what we have to do to find marginal costs, let me drag this further down because i don't have enough room.

05:25 We're going to take the derivative of our cost function.

05:31 Remember the derivative of our cost function is the marginal cost.

05:36 I don't know why i changed car...

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