Why can't marginal costs be zero?

Total cost, marginal cost, average cost

This course text deals with total costs, marginal costs and average costs. We deal with each type of cost in turn.

total cost

The total costs represent the sum of fixed and variable costs. This cost function $ K (x) $ is thus composed of the fixed costs $ K_f $ and the variable costs $ K_v $.

 

Let us assume that a company produces a product $ x_1 $. To manufacture this product, 2 input factors $ r_1 $ and $ r_2 $ are required. The costs for the procurement of these input factors are denoted by $ q_1 $ and $ q_2 $. The cost function then looks like this:

$ K (x) = q_1r_1 (x_1) + q_2r_2 (x_1) + K_f $

with $ K_v = q_1r_1 (x_1) + q_2r_2 (x_1) $

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Total costs: $ K (x) = K_v + K_f $

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The company from the previous example needs two input factors to produce the soccer balls. That is one unit of rubber $ r_1 = 1 $ and two units of leather $ r_2 = 2 $. The cost of a unit of rubber is $ q_1 = € 0.5 $ and a unit of leather is $ q_2 = € 0.75. The fixed costs are € 50,000. What does the cost function look like?

$ K_v = q_1r_1 (x_1) + q_2r_2 (x_1) = 2 $ € / piece

$ K_f = € 50,000

$ K (x) = 2x + $ 50,000

Marginal cost

Marginal costs (or marginal costs) are those costs that arise from the production of an additional unit.

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The above example is given.

With a production of 10,000 footballs, the company has a cost of $ K (x) = € 70,000, if the company produces 10,001 footballs the cost increases to $ K (x) = € 70,002, for 10,002 pieces the cost is $ K (x) = € 70,004, etc.

The marginal cost is the cost of producing one more unit. In this example 2 €.

Mathematically, the marginal costs are calculated by deriving the cost function according to $ x $:

$ K (x) = 2x + $ 50,000

$ K´ (x) = 2 $

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Marginal costs: $ K´ (x) $

Falling marginal costs

Companies with high fixed costs tend to produce large quantities. The reason is the distribution of fixed costs over a large amount, but also, for example, the use of discounts for input factors. The latter leads to marginal costs falling.

In the example above, the marginal cost is constant at $ 2 each. So with every soccer ball produced, the costs increase by $ 2. Let us now assume that if the company produces 20,000 footballs or more, they receive a discount from the leather supplier. The leather can now be bought for $ q_2´ = 0.5 € $ instead of $ q_2 = 0.75 € $. Now that means that its cost function and its marginal cost look like this:

$ K (x) = 1.5x + $ 50,000

$ K´ (x) = 1.5 $

From a production of 20,000 soccer balls, the marginal costs are only $ 1.5 instead of $ 2.

Rising marginal costs

It can also be the case that the marginal costs rise again from a certain amount. This is the case when production reaches its capacity. For example, when the leather supplier cannot deliver more than 40,000 units of leather. So a new supplier would have to be found. Suppose the company wants to produce 30,000 footballs due to the increased demand. The old leather supplier can cover 25,000 of the soccer balls, whereas a new leather supplier is used for 5,000 soccer balls with a price per unit of $ q_2 (new) = 1 € $.

The cost function for the 5,000 soccer balls looks like this:

$ K (x) = € 2.5 + $ 50,000

$ K´ (x) = 2.5 $

The Marginal cost curve so is as follows:

Less than 20,000 pieces = € 2 (old supplier without discount)

20,000 - 25,000 pieces = 1.5 € (old supplier with discount)

More footballs = € 2.5 (new supplier)

Average cost

The average cost (or unit cost) is the average cost per unit. These are calculated by dividing the costs by the quantity produced.

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Average cost: $ k (x) = \ frac {K (x)} {x} $

We take the example from above again.

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First, let's consider the average cost to produce 20,000 footballs. The cost for 1 unit of rubber is $ q_1 = € 0.5 $ and for 1 unit of leather $ q_2 = € 0.75. The fixed costs are € 50,000. 1 unit of rubber and 2 units of leather are required to produce a soccer ball. the cost function is thus:

$ K (x) = 2x + $ 50,000

with $ x = $ 20,000, the total costs are € 90,000. The costs per piece are then calculated:

$ k (x) = \ frac {90,000 €} {20,000 pieces} = 4.5 € / piece $.

The more that is produced, the lower the average cost. The reason for this is the distribution of the fixed costs over a larger amount (fixed cost degression). However, the average costs can also rise again. If, for example, production reaches its limits and, for example, the input factors have to be obtained from a new, more expensive supplier because the old one cannot produce the required quantity. Then the variable costs increase.

It can of course also be the case that the production of such quantities pushes the company itself to its capacity limits. For example, a new warehouse, a new machine and / or new employees have to be purchased so that production is even possible. Then the fixed costs increase.

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Suppose the company now wants to produce 21,000 soccer balls. The leather supplier and also the rubber supplier can continue to deliver at the above prices. However, the previous warehouse is no longer sufficient. A new warehouse has to be rented. The fixed costs increase by € 10,000:

$ K (x) = 2x + $ 60,000

The total costs are:

$ K (21,000) = 2 \ cdot 21,000 + 60,000 = € 102,000.

The costs per piece are then:

$ k (21,000) = \ frac {102,000} {21,000} = € 4.86 $.

The average costs have increased due to the increase in fixed costs due to the new warehouse.

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How much would the company have to produce now to achieve average costs of € 4.5 / pce again?

$ k (x) = \ frac {K (x)} {x} $

With

$ k (x) = € 4.5 $

$ K (x) = 2x + $ 60,000

Insertion gives:

$ 4.5 € = \ frac {2x + 60,000} {x} $

$ 4.5 € = 2 + \ frac {60,000} {x} $ | $ -2 $

$ 2.5 € = \ frac {60,000} {x} $ | $ \ cdot x $

$ 2.5 € \ cdot x = $ 60,000 | $: 2.5 € $

$ x = 24,000 pieces $

The company has to produce 24,000 pieces to bring the average cost back to € 4.5.