Sum of Continuous Function and Discontinuous Function
Economic Applications of Continuous and Discontinuous Functions
There arc many natural examples of discontinuities from economics, In fact economists often adopt continuous functions to represent economic relationships when the use of discontinuous functions would be a more literal interpretation of reality. It is important to know when the simplifying assumption of continuity can be safely made for the sake of convenience and when it is likely to distort the true relationship between economic variables too much. Our first example illustrates a class of situations in which it is usual to use a model with continuous functions even though this is a distortion of reality in a literal sense. In most such cases the assumption is not a harmful one. However, as many of the other examples illustrate. the idea of discontinuity may be inherent in an economic model itself, with the solution hinging entirely on the existence of some point of discontinuity of the relevant function.
Divisibility and the Production Function
The first step in modeling the decisions of a firm is usually the analysis of die available technology. This relationship between inputs used and outputs generated is generally presumed to be represented by some production function. In the case of one input, call it x, and one output, call it y, we can write y = f (x).
What does it mean to say that this function is continuous on some range of values (usually x > 0. x e R)? In the first place, to assume that /(x) is continuous at a point .v - a implies that /(x) is defined on some open interval of real numbers containing the point a. This means that x must be infinitely divisible. That is. one can choose x to be a value that deviates even by infinitesimal amounts from ,r = a.
An example of an input {and an output) that would not be infinitely divisible would be holts used in the production of an automobile. Since one would not use a fraction like a half of a bolt, it would only make literal sense to treat bolts as integer valued. Therefore it does not make sense to contemplate an open interval of points including some value x = a bolts. However, if a manufacturer producer 20,000 vehicles per year using 1,050 bolts in each vehicle, it seems reasonable tc simply treat bolts and vehicles as infinitely divisible and represent the relationship between them as y = x/1.050. where v is the number of bolts used and y is the number of vehicles produced. (Of course, we have ignored all the other inputs. )
Figure 4.7 illustrates this function while Ihe true relationship would include only the points (1,050. I), (2.100, 2), (3,150. 3), etc. If one uses the continuous function y = (1/1.050).» in the process of solving some decision problem for the firm and discovers the solution involves some value of x that i*. nol a multiple of 1.050, then using Ihe closest value that is a multiple of 1.050 would probably bt reasonably accurate. Thus, even if a commodity is not infinitely div isible, we ma> often assume that it is. without distorting reality very much.
A Salary Schedule with a Bonus Payment
Suppose that a salesperson receives a salary according to a contract that establishes a relationship between pay and the level of sales made by the salesperson. In particular, suppose that the contract stipulates that the salesperson's monthly salaiy will be composed of three parts: (i) a basic amount of $800, (ii) a commission ol 10%. and (iii) a lump-sum bonus of $500 if the salesperson's sales for the month reach or exceed $20,000. From this description, one can see that her salary will jump by $500 if the critical level of $20,000 worth of sales is achieved. This implies a discontinuity in her salary schedule. Letting S represent sales per month and p represent the salesperson's pay for the month, n follows that the function describing her salary-sales relationship is p =
$800 + 0.15. S< $20,000 $1.300+0.1.S. S> $20,000
which is illustrated in figure 4.8.
3,200
2,400
1,600
3,200
2,400
20,000
Figure 4.8 A salary schedule vvitli u bonus payment
The facl thai ihe bonus of $500 is achieved once S reaches ihe critical value of $20,000 leads to the result that the left-hand limit of the salary function at s = $20,000 is $2,800 while the right-hand limit is $3,300. The existence of this discontinuity has interesting economic implications. Consider the following scenario. There are three salespersons, called A. B. ¿ nd C. Their cumulative sales for the month, not including the last day, are $26,000 for A, $18,5<X> for B, and $6,000 for C. The 10% commission on sales will give each a similar incentive to make extra sales on the last day of the month. But will the $500 bonus possibility have a different effect on the three salespersons? Assuming that it is plausible to generate a few thousand dollars worth of sales in a day but virtually impossible to create more than $10,000 worth of sales, one would expect that salesperson B will try harder on the last day to increase sales than will the other two.
A Discontinuous Income Support Program
Many welfare programs or "income support programs" offer individuals who are not employed a fixed or lump-sum monthly payment that is made only if the individual does not earn any income. Once an individual earns any income whatsoever. the payment is stopped. Consider the following hypothetical example. Suppose a single parent of two preschool-aged children can collect a monthly welfare payment of $750 provided she doesn't work. However, once she earns any positive amount of income, the welfare payment stops. Assume that she could earn $ 15 per hour at some job for which the number of hours worked per month is entirely flexible. The income of this person. K. as a function of hours worked, h. is given below
The graph depicting this person's income as a function of hours worked is provided in figure 4.9. It is clearly discontinuous at h = 0 hours worked.
This type of discontinuity, which is a property of many "all or nothing" income support programs, has been the subject of a great deal of debate. One can see that a person in such a program would have to work 50 hours per month just to match the income earned from the support payments. Since the person would face childcare and other costs of working, the "all or nothing" property of this program presents a serious deterrent to the incentive to work.
An alternative scheme would be to allow a person in this situation to keep a certain fraction of income earned in addition to the $750 monthly payment. Suppose, for example, that the person were allowed to retain 50% of any earnings, with the other 50% representing a payback of the income support up to the level where the entire $750 is paid back. A person facing a wage rate of $15 per hour will have paid back the full $750 only after working 100 hours or more per month (0.5 x 100 * 15 = 750). For 0 < h < 100, the effective wage rate is 50% of $ 15 (or $7.50). and so net income for this range of hours worked is Y(h) = 7.5/?. After this amount of earnings, the individual would keep any excess. Therefore under this program the person's income schedule would be the following:
The graph for this income schedule is provided in figure 4. H) Notice that i\ is continuous. [Check that lim;,_M,) >'(/') = lim/,_ioo- y[h) = k(100).]
Many economists prefer this second plan because it avoids the discontinuity of the first plan. In the first plan there is effectively a large penalty for working at ¡ill. since income drops from $750 per month to almost zero if the individual chooses only a few hours of work. Under the second plan the person always earns more income by choosing to work more. The result is that the person will be more likely choose some positive hours of work under the second plan making himself/herself better off and also reducing the cost of the program to the government.
Continuous Marginal-Product Functions
The marginal product of an input is the amount by which output increases ^ a result of an additional unit of that input being used, given fixed amounts of other inputs available. This concept is useful in economics when analyzing the decision-making problem of firms in the short run when the level of some input \ can be altered (variable inputs) hut other input levels are fixed (fixed inputs«
A full treatment of the marginal product of an input will be taken up in the following chapter on derivatives. However, there are some interesting problems concerning the continuity of marginal-product functions that are useful to consider here. For example, suppose that the function y = 10L relates the amount of output produced, y. to die amount of labor input employed. L. forgiven fixed levels of other inputs. One can then see that an increase in L of one unit always leads to an increase in output of 10 units. Thus the marginal product of labor function is the constant function y = 10 and so is continuous on the interval |0. oz).
Notice that this marginal-product function has the rather unrealistic property thai more output is generated by using more labor even for very large values of labor. Since the amounts of all other inputs are fixed, one might anticipate that it makes more sense to imagine that as L increases, the added output generated begins to fall and may even become zero or negative. The following discussion shows that this phenomenon may occur in such a way that the marginal-product function should he modeled a.s a discontinuous function.
Marginal-Product Function with a Capacity Constraint
In many production processes the maximum output that can be produced by increasing the amount of a variable input depends on the amount of the other (fixed) inputs available. A good example is a coal-fired electricity generating station. There will always be some absolute maximum amount of power that can be generated from a single station. This maximum is generally referred to as the capacity of the station. For example, if a station has a 1,500-megawatt capacity, this means that no matter how much coal or other inputs are available, the maximum amount of energy that can be generated in a twenty-four hour period (per day) is 36,000 megawatt-hours (i.e.. 24 x 1.500).
Suppose that we want to determine the marginal pioduct of coal for a case in which there is enough of all inputs other than coal to keep a 1,500-megawatt power plant operating at capacity. Assume that it lakes 250 pounds of coal to generate one megawatt-hour of energy, and so one ton of coal will generate eight megawatt-hours ofenergy. Therefore the marginal product (per day) of coal is eight megawatt-hours (per ton) as long as capacity has not been reached. Once capacity has been reached, however, the marginal product of coal drops to zero. Thus, once 4,500 tons of coal have been used in a day to generate electricity, the generating station will have reached capacity (4.500 x 8 = 36.000). If we let x represent tons of coal used per day and y the marginal product of coal in megawatt-hours, then the marginal product of coal is given by the function
18, 0<.r <4.500 V" |o. I >4.500
Continue reading here: Definition of a Tangent Line
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