BY DR LOIZOS CHRISTOU

OPTIMIZATION

Optimization Techniques

Optimization: What consumer chooses

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Optimization: What the consumer chooses

Goal is to understand how a consumer make choices.

We have the two pieces necessary for the analysis: 1: the consumer budget constraint. (how much

he can afford to spend) 2: the consumer preferences (what he want to spend it on

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Optimizaton

Maximization and minimization of some functions subject to cost and output (Q) is optimization.

Best allocation of resources as resources are limited, consumer will purchase optimal combination of goods.

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Optimization Techniques and New Management Tools      

The first step in presenting optimisation techniques is to examine ways to express economic relationships. Economic relationship can be expressed in the form of equation, tables, or graphs. When the relationship is simple, a table and/ or graph may be sufficient. However, if the relationship is complex, expressing the relationship in equational form may be necessary.

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Optimization Techniques and New Management Tools      

Expressing an economic relationship in equational form is also useful because it allows us to use the powerful techniques of differential calculus in determining the optimal solution of the problem.

More importantly, in many cases calculus can be used to solve such problems more easily and with greater insight into the economic principles underlying the solution. This is the most efficient way for the firm or other organization to achieve its objectives or reach its goal.

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Example 1

Suppose that the relationship between the total revenue (TR) of a firm and the quantity (Q) of the good and services that firm sells over a given period of time, say, one year, is given by

TR= 100Q-10Q2

(Recall: TR= The price per unit of commodity times the quantity sold; TR=f(Q), total revenue is a function of units sold; or TR= P x Q).

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Example 1

By substituting into equation 1 various hypothetical values for the quantity sold, we generate the total revenue schedule of the firm, shown in Table 1. Plotting the TR schedule of table 1, we get the TR curve as in graph 1. In this graph, note that the TR curve rises up to Q=5 and declines thereafter.

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0

50

100

150

200

250

300

0 1 2 3 4 5 6 7

Q

TR

Example 1

Equation1: TR = 100Q - 10Q2

Table1:

Graph1:

Q 0 1 2 3 4 5 6TR 0 90 160 210 240 250 240

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Example 2

Suppose that we have a specific relationship between units sold and total revenue is precisely stated by the function: TR= $ 1.50 x Q. The relevant data are given in Table 2 and price is constant at $ 1.50 regardless of the quantity sold. This framework can be illustrated in graph 2.

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Example 2 

Unit Sold TR Price

1 1.5 1.5

2 3  

3 4.5  

4 6  

5 7.5  

6 9  Table2:

Graph of the relationship between total revenue and units sold

01.5

34.5

67.5

9

1 2 3 4 5 6

Unit sold for time period

Reve

nue

per

time

peri

od

Graph2:

Total, average and marginal cost

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The relationship between total, average, and marginal concepts and measures is crucial in optimisation analysis. The definitions of totals and averages are too well known to warrant restating, but it is perhaps appropriate to define the term marginal.

Total, Average, and Marginal Cost

A marginal relationship is defined as the change in the dependent variable of a function associated with a unitary change in one of the independent variables.

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Total, Average, and Marginal Cost

Q TC AC MC0 20 - -1 140 140 1202 160 80 203 180 60 204 240 60 605 480 96 240

AC = TC/Q

MC = TC/Q

(Recall: Total cost: total fixed cost plus total variable costs; Marginal cost: the change in total costs or in total variable costs per unit change in output).

Table3:

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Total, Average, and Marginal Cost

The first two columns of Table 3 present a hypothetical total cost schedule of a firm, from which the average and marginal cost schedules are derived in columns 3 and 4 of the same table. Note that the total cost (TC) of the firm is $ 20 when output (Q) is zero and rises as output increases (see graph 3 to for the graphical presentation of TC). Average cost (AC) equals total cost divided by output. That is AC=TC/Q. Thus, at Q=1, AC=TC/1= $140/1= $140. At Q=2, AC=TC/Q =160/2= £80 and so on. Note that AC first falls and then rises.

Q TC AC MC0 20 - -1 140 140 1202 160 80 203 180 60 204 240 60 605 480 96 240

Table3:

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Total, Average, and Marginal Cost

Marginal cost (MC), on the other hand, equals the change in total cost per unit change in output. That is, MC= TC/Q where the delta () refers to “a change”. Since output increases by 1unit at a time in column 1 of table 3, the MC is obtained by subtracting successive values of TC shown in the second column of the same table. For instance, TC increases from $ 20 to $ 140 when the firm produces the first unit of output. Thus MC= $ 120 and so forth. Note that as for the case of the AC and MC also falls first and then rises (see graph 4 for the graphical presentation of both AC and MC). Also, note that at Q=3.5 MC=AC; this is the lowest AC point. At Q=2; that is the point of inflection whereas the point shows MC at the lowest point.

Q TC AC MC0 20 - -1 140 140 1202 160 80 203 180 60 204 240 60 605 480 96 240

Table3:

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Total, Average, and Marginal Cost

0

60

120

180

240

0 1 2 3 4Q

TC ($)

0

60

120

0 1 2 3 4 Q

AC, MC ($)AC

MC

Graph3:

Graph4:

MC=AC

Total,average and marginal revenue:

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Average Revenue:

Total revenue per unit of output. When all output is sold at the same price, average revenue will be the same as price.

AV=TR/units of out put(Q) average revenue curve (AR),is also the

market demand curve

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Marginal Revenue:

Marginal revenue (R') is the additional revenue that will be generated by increasing product sales by 1 unit.

It indicates the extra revenue received for selling each extra unit

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Total, Average, and Marginal revenue

In the total revenue function, marginal revenue is the change in total revenue associated with a one-unit change in units sold. Generally, we analyse an objective function by changing the various independent variables to see what effect these changes have on the dependent variables. In other words, we examine the marginal effect of changes in the independent variable. The purpose of this analysis is to determine that set of values for the independent or decision variables which optimises the decision maker’s objective function.

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Total, Average, and Marginal revenue

Marginal revenue can be positive, zero or negative When the average revenue (demand) curve is

elastic, marginal revenue is positive and total revenue is increasing.

When the average revenue (demand) curve is inelastic, marginal revenue is negative and total revenue is decreasing.

When average revenue (demand) curve is unit elastic, marginal revenue is zero and total revenue is not changing.

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Profit Maximization

Table 4 indicates the relationship between TR, TC and Profit. In the top panel of graph 5, the TR curve and the TC curve are taken from the previous graphs. Total Profit () is the difference between total revenue and total cost. That is = TR-TC. The top panel of Table 4 and graph 5 shows that at Q=0, TR=0 but TC=$20. Therefore, = 0-$20= -$20. This means that the firm incurs a loss of $20 at zero output. At Q=1, TR=$90 and TC=$ 140. Therefore, = $90-$140= -$50. This is the largest loss. At Q=2, TR=TC=160. Therefore, = 0 and this means that firm breaks even. Between Q=2 and Q=4, TR exceeds TC and the firm earns a profit. The greatest profit is at Q=3 and equals $30.

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Profit Maximization

Q TR TC Profit0 0 20 -201 90 140 -502 160 160 03 210 180 304 240 240 05 250 480 -230

Table 4:

Table 4 indicates the relationship between TR, TC and Profit. In the top panel of graph 5, the TR curve and the TC curve are taken from the previous graphs. Total Profit () is the difference between total revenue and total cost. That is = TR-TC. The top panel of Table 4 and graph 5 shows that at Q=0, TR=0 but TC=$20. Therefore, = 0-$20= -$20. This means that the firm incurs a loss of $20 at zero output. At Q=1, TR=$90 and TC=$ 140. Therefore, = $90-$140= -$50. This is the largest loss. At Q=2, TR=TC=160. Therefore, = 0 and this means that firm breaks even. Between Q=2 and Q=4, TR exceeds TC and the firm earns a profit. The greatest profit is at Q=3 and equals $30.

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Profit Maximization

0

60

120

180

240

300

0 1 2 3 4 5Q

($)

MC

MR

TC

TR

-60

-30

0

30

60

Profit

Graph5:

vertical difference between TR and TC is profit

Profit maximization:

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Two conditions for profit maximization:

1: MC must be equal to MR (this is a necessary condition)

2: slope of MC should be greater than MR(this is sufficient condition)

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The EndThe End

Thanks