## Sunday, March 15, 2020

### How to Calculate the 7 Cost Measures

How to Calculate the 7 Cost Measures There are many definitions relating to cost, including the following seven terms: Marginal costTotal costFixed costTotal variable costAverage total costAverage fixed costAverage variable cost The data you needÃ‚  toÃ‚  compute these seven figuresÃ‚  probably will come in one of three forms: A table that provides data on total cost and quantity producedA linear equation relating total cost (TC) and quantity produced (Q)A nonlinear equation relating total cost (TC) and quantity produced (Q) Following are definitions of the terms and explanations of how the three situations should be dealt with. Defining Terms of Cost Marginal costÃ‚  is the cost a company incurs when producing one more good. Suppose its producing two goods, and company officials would like to know how much costs would increase if production was increased to three goods. The difference is the marginal cost of going from two to three. It can be calculated thus: Marginal Cost (from 2 to 3) Total Cost of Producing 3 Ã¢â‚¬â€œ Total Cost of Producing 2 For example, if it costs \$600 to produce three goods and \$390 to produce two goods, the difference is 210, so thats the marginal cost. Total costÃ‚  is simply all the costs incurred in producing a certain number of goods. Fixed costsÃ‚  are the costs that are independent of the number of goods produced, orÃ‚  the costs incurred when no goods are produced. Total variable cost is the opposite of fixed costs. These are the costs that change when more is produced. For instance, the total variable cost of producing four units is calculated thus: Total Variable Cost of Producing 4 units Total Cost of Producing 4 Units Ã¢â‚¬â€œ Total Cost of Producing 0 units In this case, letÃ¢â‚¬â„¢s say it costs \$840 to produce four units and \$130 to produce none. Total variable costs when four units are producedÃ‚  is \$710 since 840-130710.Ã‚   Average total costÃ‚  is the totalÃ‚  cost over the number of units produced. So if the company produces five units, the formula is: Average Total Cost of Producing 5 units Total Cost of Producing 5 units / Number of Units If the total cost of producing five units is \$1200, average total cost is \$1200/5 \$240. Average fixed costÃ‚  is fixed costs over the number of units produced, given by the formula: Average Fixed Cost Total Fixed Costs / Number of Units The formula for average variable costs is: Average Variable Cost Total Variable Costs / Number of Units Table of Given Data Sometimes a table or chart will give you the marginal cost, and youll need to figure the total cost. You can figure the total cost of producing two goods by using the equation: Total Cost of Producing 2 Total Cost of Producing 1 Marginal Cost (1 to 2) A chart will typically provide information regarding the cost of producing one good, the marginal costÃ‚  ,and fixed costs. Lets say the cost of producing one good is \$250, and the marginal cost of producing another good is \$140. The total cost would be \$250 \$140 \$390. So the total cost of producing two goods is \$390. Linear Equations Lets say you want to calculate marginal cost, total cost, fixed cost, total variable cost, average total cost, average fixed cost, andÃ‚  average variable costÃ‚  when given a linear equation regarding total cost and quantity. Linear equations are equations without logarithms. As an example, letÃ¢â‚¬â„¢s use the equation TC 50 6Q. That means the total cost goes up by 6 whenever an additional good is added, as shown by the coefficient in front of the Q. This means there is a constant marginal cost of \$6 per unit produced. Total cost is represented by TC. Thus, if we want to calculate the total cost for a specific quantity, all we need to do is substitute the quantity for Q. So the total cost of producing 10 units is 50 6 X 10 110. Remember that fixed cost is the cost we incur when no units are produced. So to find the fixed cost, substitute in Q 0 to the equation. The result is 50 6 X 0 50. So our fixed cost is \$50. Recall that total variable costs are the non-fixed costs incurred when Q units are produced. So total variable costs can be calculated with the equation: Total Variable Costs Total Costs Ã¢â‚¬â€œ Fixed Costs Total cost is 50 6Q and, as just explained, fixed cost is \$50 in this example. Therefore, total variable cost is (50 6Q) Ã¢â‚¬â€œ 50, or 6Q. Now we can calculate total variable cost at a given point by substituting for Q. To find the average total cost (AC), you need to average total costs over the number of units produced. Take the total cost formula of TC 50 6Q and divide the right side to get average total costs. This looks like AC (50 6Q)/Q 50/Q 6. To get average total cost at a specific point, substitute for the Q. For example, average total cost of producing 5 units is 50/5 6 10 6 16. Similarly, divide fixed costs by the number of units produced to find average fixed costs. Since our fixed costs are 50, our average fixed costs are 50/Q. To calculate average variable costs, divide variable costs by Q. Since variable costs are 6Q, average variable costs are 6. Notice that average variable cost does not depend on quantity produced and is the same as marginal cost. This is one of the special features of the linear model, but it wont hold with a nonlinear formulation. Nonlinear Equations Nonlinear total cost equations are total cost equations that tend to be more complicated than the linear case, particularly in the case of marginal cost where calculus is used in the analysis. For this exercise, letÃ¢â‚¬â„¢s consider the following two equations: TC 34Q3Ã‚  Ã¢â‚¬â€œ 24Q 9TC Q log(Q2) The most accurate way of calculating the marginal cost is with calculus. Marginal cost is essentially the rate of change of total cost, so it is the first derivative of total cost. So using theÃ‚  two given equations for total cost, take the first derivate of total cost to find the expressions for marginal cost: TC 34Q3Ã‚  Ã¢â‚¬â€œ 24Q 9TCÃ¢â‚¬â„¢ MC 102Q2Ã‚  Ã¢â‚¬â€œ 24TC Q log(Q2)TCÃ¢â‚¬â„¢ MC 1 1/(Q2) So when total cost is 34Q3Ã‚  Ã¢â‚¬â€œ 24Q 9, marginal cost is 102Q2Ã‚  Ã¢â‚¬â€œ 24, and when total cost is Q log(Q2), marginal cost is 1 1/(Q2). To find the marginal cost for a given quantity, just substitute the value for Q into each expression. For total cost, the formulas are given. Fixed cost is found when Q 0. When total costs are 34Q3Ã‚  Ã¢â‚¬â€œ 24Q 9, fixed costs are 34 X 0 Ã¢â‚¬â€œ 24 X 0 9 9. This is the same answer you get if you eliminate all the Q terms, but this will not always be the case. When total costs are Q log(Q2), fixed costs are 0 log(0 2) log(2) 0.30. So although all the terms in our equation have a Q in them, our fixed cost is 0.30, not 0. Remember that total variable costÃ‚  is found by: Total Variable Cost Total Cost Ã¢â‚¬â€œ Fixed Cost Using the first equation, total costs are 34Q3Ã‚  Ã¢â‚¬â€œ 24Q 9 and fixed costÃ‚  is 9, so total variable costs are 34Q3Ã‚  Ã¢â‚¬â€œ 24Q. Using the second total cost equation, total costs are Q log(Q2) and fixed cost is log(2), so total variable costs are Q log(Q2) Ã¢â‚¬â€œ 2. To get the average total cost, take the total cost equations and divide them by Q. So for the first equation with a total cost of 34Q3Ã‚  Ã¢â‚¬â€œ 24Q 9, the average total cost is 34Q2Ã‚  Ã¢â‚¬â€œ 24 (9/Q). When total costs are Q log(Q2), average total costs are 1 log(Q2)/Q. Similarly, divide fixed costs by the number of units produced to get average fixed costs. So when fixed costs are 9, average fixed costs are 9/Q. And when fixed costs are log(2), average fixed costs are log(2)/9. To calculate average variable costs, divide variable costs by Q. In the first given equation, total variable cost is 34Q3Ã‚  Ã¢â‚¬â€œ 24Q, so average variable cost is 34Q2Ã‚  Ã¢â‚¬â€œ 24. In the second equation, total variable cost is Q log(Q2) Ã¢â‚¬â€œ 2, so average variable cost is 1 log(Q2)/Q Ã¢â‚¬â€œ 2/Q.