Understanding Economics 53 4- Cost Curves
Understanding Economics 54 4- Cost Curves
K = Machine hours FC = Fixed Cost
dL MPdTP
L = Labour hours VC = Variable Cost
L APTP
TP = Total Product i.e total output TC = Total Cost TC = FC + VC MP= Marginal Product MC = Marginal Cost
dTP MCdTC
AP= Average Product AFC = Average Fixed
Cost TP
AFCFC
w = wage rate/hour AVC = Average
Variable Cost TP
AVCVC
r = interest rate i.e. cost of
capital/hour AC = Average Cost AVC AFC
TP
ACTC
Relationship between Marginal Product (MP) and Total Product (TP)
Marginal Product (MP) represents the slope of the Total Product function (the slope of every total is its marginal!) i.e. the ratio of change in TP and change in quantity of labour.
MPL
dL dTP dX
dY
Product Total
of Slope
Relationship between Marginal Product (MP) and Average Product (AP)
Consider your class, where the average age is say, 18 years. Supposing that a new student decides to join your class, the average age of the class increases if his age exceeds 18 years and decreases if it is less than that. The same rule applies to Average Product (AP) i.e. the ratio of Total Product (TP) and quantity of labour (L) and Marginal Product (MP). Observing columns 4 and 5 simultaneously we notice that AP rises as long as MP is higher than AP and falls when MP is less than AP, as shown in the diagram below. Direction of the change in MP does not determine the direction of the change in AP. AP increases as long as MP is above AP, irrespective of the direction of the change in MP.
Diagram 4.1
MP AP MP
AP
L1 L2 L3 L
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Understanding Economics 55 4- Cost Curves
The diagram shows that AP rises as long as MP is above AP and falls when MP is below AP. MP cuts AP from above and at its maximum point. Thus, AP is maximized when MP equals AP.
Marginal Product (MP) is maximized when L1 labour hours are employed and diminishing returns set in beyond this point so that total product rises but at a falling rate. As table 4.1 shows, diminishing returns set in between the 4th and 5th labour hour.
Average Product (AP) is maximized when L2 labour hours are hired, beyond which AP begins to fall as MP lies below AP.
Total Product (TP) is maximized when L3 labour hours are hired, beyond which MP becomes negative and Total Product falls.
Thus along a Total Product curve, which is steep to start with but becomes increasingly flatter later on (rising MP i.e. increasing returns followed by falling MP i.e. decreasing returns), MP is the first to decrease, followed by AP and then Total Product.
(try J/07/3/04)
In order to examine the effects of law of variable proportion on a firm‟s costs, we need to define the nature of costs. In the short run, total cost can be divided into two categories: fixed cost and variable cost.
Fixed cost
Fixed cost is that portion of total cost which does not vary with the number of units made. It may change due to other factors like inflation but not the units of output produced. Examples include rent of building, depreciation, interest payments made to bank, royalties and salaries.
Our initial example (table 4.1) assumes price of labour and capital to be £10 per hour (see columns 6 & 7). Column 8 shows Fixed Cost (FC), the product of amount of capital (K) and the
price of 1 machine hour (or r). Fixed Cost curve is a straight horizontal line as shown in diagram 4.2.
Since Fixed Cost does not change with output, Average Fixed Cost (AFC) decreases with each additional unit produced as greater output spreads fixed costs over a larger volume- hence Average Fixed Cost is downward sloping, as shown in diagram 4.3 (try J/08/3/07 and N/02/3/07).
Diagram 4.2 Diagram 4.3
Costs
AFC Output FC
Cost
Output
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Understanding Economics 56 4- Cost Curves
Variable Cost
Variable cost is that portion of total cost which varies directly with the number of units produced e.g. the cost of raw materials and piece rate wages.
Variable Cost (VC) (see column 9) is the product of the quantity of variable input i.e. labour and the price of one labour hour (or w). Variable Cost very obviously increases with additional units of output; however Average Variable Cost (AVC), the ratio of Variable Cost and Total Product may increase, decrease or stay the same. In the given example, it initially decreases but eventually increases with output (see column 13).
The following example should make the distinction between fixed and variable costs clearer.
Consider the rent of a building that a tenant must pay- it is fixed for the period of lease agreement. The rent must be paid to the landlord through out the lease period, irrespective of the sales revenues or profits made. Thus, rent is a fixed cost. However, rent becomes variable once the lease period expires. It is then upto the tenant to either vacate the building and avoid paying the rent or renew the agreement with the landlord for another lease period. Rent becomes fixed once again when the lease agreement is renewed.
Following is a Short Run Total Cost function derived from a short run production function.
X = f
L,KSRTC = VC + FC SRTC = L.w + K.r In the equation above:
SRTC = Short Run Total Cost (Assuming that capital is fixed) L= Number of labour hours
w = Wage rate per hour K = Number of machine hours r = Cost of capital per hour VC= Total variable cost FC= (total) fixed cost.
Marginal Cost
Marginal cost is the cost incurred on producing an additional unit of output. It is the change in total cost due to a change in the number of units produced. The following formula is used to calculate marginal cost:
output in Change
cost total in Change cost
Marginal dX MCdTC
As capital is fixed and prices of labour and capital i.e. w & r are beyond the control of an ordinary firm, any change in total cost is precisely because of the change in the quantity of variable input i.e. labour hours. Thus,
dTC = w.dL
where dTC is change in TC, w is the wage rater per hour and dL is the change in the quantity of labour.
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Understanding Economics 57 4- Cost Curves
The relationship between Law of Returns and Marginal Cost is very interesting. Increasing returns occur when additional quantities of variable input combined with a fixed input increase output at an increasing rate. In other words, firms experiencing increasing returns face diminishing costs as they have to hire fewer labour hours to increase output at a constant rate. The following table helps understand the relationship between law of returns and law of costs.
Table 4.2
TP L
dL
MPdTP K W R TC = w.L + r.K
MP
dTC
dTP MC dTC
0 0 - 5 10 10 50 -
100 5 20 5 10 10 100 0.5
200 9 25 5 10 10 140 0.4
300 12 33.3 5 10 10 170 0.3
400 15 33.3 5 10 10 200 0.3
500 19 25 5 10 10 240 0.4
600 24 20 5 10 10 290 0.5
700 30 16.67 5 10 10 350 0.6
The production function above initially shows increasing returns and then decreasing returns. 5 labour hours are required to produce the first 100 units but only 4 are needed to make an additional 100. Increasing returns continue till 12 labour hours. Constant returns occur i.e. total product rises at a constant rate between 9 and 15 labour hours. Diminishing returns set in beyond 15 labour hours since increasing output by the same quantity i.e. 100 units requires ever increasing quantities of labour hours. The last column shows Marginal Cost and it is interesting to note that there are diminishing (marginal) costs whenever there are increasing returns. Total Cost starts to increase at an increasing rate i.e. MC begins to rise when diminishing returns set in.
Alternatively,
w MP MP
dTC dL
dTP .
MCdTC
Note: As discussed earlier, dTC equals w.dL, the product of wage rate and change in the quantity of labour.
Thus, it is clear that whenever MP rises (increasing returns to variable input) MC falls (diminishing cost) and there are increasing costs (MC rises) when there are diminishing returns (MP falls).
Table 4.1 also proves this relationship (see column 4 and 11). It is also shown that MC is lowest when MP is at its maximum (try N/06/3/03).
Average Cost
Average Cost (column 14) is the ratio of Total Cost and Total Product (output). Average Cost may also be computed by summing Average Variable Cost (AVC) and Average Fixed Cost (AFC).
TC = FC + VC
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Understanding Economics 58 4- Cost Curves
Dividing both sides by X i.e. output:
X VC X FC X
TC AC = AFC + AVC
The following diagram helps understand the relationship between AC and AVC. Costs are shown along Y axis and output along X axis.
Diagram 4.4
AC AVC AC
AVC
Output
AC and AVC are both U shaped but are not parallel. The vertical distance between AC and AVC i.e. (AC – AVC) equals AFC and since AFC always decreases with output, the vertical distance between AC and AVC always decreases with increases in output. As AFC never becomes zero, AVC always lies below AC.
Marginal Cost, Average Cost and Average Variable Cost Diagram 4.5
AC AVC
AC MC AVC MC
Output
X1 X2 X3
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Understanding Economics 59 4- Cost Curves
The Marginal Cost curve is somewhat hockey shaped as MC falls initially (increasing returns) and then rises (decreasing returns). AC and AVC fall as long as MC is below AC and AVC. MC cuts AVC and AC from below and at their respective minimum points. Diminishing returns set in at output X1. AVC and AC continue to fall till output X2 since MC is still below them. AVC starts to increase at X2 but AC continues to fall till X3. Between X2 and X3, the decrease in AFC outweighs the increase in AVC. AC increases beyond X3 as decreasing AFC no longer outweighs increasing AVC.
Viewing from origin, MC increases first, followed by AVC and AC respectively.
There is no relationship between Marginal Cost and Average Fixed Cost curve. MC can cut AFC at any point.
Total Cost (TC), Variable Cost (VC) and Fixed Cost (FC)
In the panel of output and costs, Fixed Cost shows as a straight horizontal as it does not change with the number of units made. Total Cost and Variable Cost are parallel curves, the constant difference between them measuring Fixed Cost. Total Cost and Fixed Cost share the same vertical intercept as Total Cost equals Fixed Cost at zero output. Put another way, variable costs are zero when no output is generated and TVC therefore begins from the origin.
The slope of both TC and TVC measures Marginal Cost. In the diagram, Total and Variable Cost Curves initially become flatter showing decreasing Marginal Cost and increasing returns to a variable input. Then roughly around output X1, TC and VC start becoming steeper showing increasing Marginal Cost and diminishing returns to variable input.
The relationship between Marginal Cost, Average Variable Cost and Average Cost can also be verified by looking at the following pair of diagrams:
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Understanding Economics 60 4- Cost Curves