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Law of Variable Proportions | Law of Diminishing Returns

 The law of variable proportion is one of the fundamental laws in economics. This law deals with the short-run production function. In short run, input and output relations are studied by keeping at least some factors / inputs of production constant. The law of variable proportion states that when more and more unit of a variable input are applied with the given number / size of
number / size of the fixed inputs, the total output may initially increase at an increasing rate up to some extent, then it increases at decreasing rate, it reaches to the maximum level and ultimately it starts to decrease. This tendency of production in the short run is termed as the ‘Law of Variable Proportion’ or the Law of Diminishing Return’. This law of production exhibits the direction of the change of total product and the marginal product of the variable factor combined with fixed factors.

Different economists have offered definitions of the Law of Variable Proportion or the Law of Diminishing Return in their own way.

Alfred Marshall discussed the law of diminishing returns in relation to agriculture. He defines the law in these words “An increase in the capital and labor applied in the cultivation of land causes in general a less than proportionate increase in the amount of produce raised, unless it happens to coincide with an improvement in the arts of agriculture.”

Similarly, according to Prof. Leftwitch, “The law states that if the input of one resource is increased by equal increments per unit of time while the inputs of other resources are held constant, TP will increase, but beyond some point the resulting TP will increases will become smaller and smaller.”

From the definitions presented above, it is clear that the law of variable proportions or the law of diminishing returns refers to the behavior of output as the quantity of one factor is increased keeping the quantity of other factors fixed, and further it states that the marginal and average product will eventually decline.

The law of variable proportions has the following assumptions:
1. Constant Technology: The law of variable proportion assumes that the state of technology is constant. The reason is that if the state of technology changes, the marginal and average productivity of variable may rise instead of diminishing because of technological improvement.
2. Short-run: This law specially operates in the short-run because in the short run, some factors are fixed and the proportion of others has to be varied. It assumes that labor (L) is variable factor while capital (K) is the fixed factor.
3. Homogeneous Factors: This law is based on the assumption that the variable factor (labor) is applied unit by unit and each factor unit is homogeneous / identical in amount and quality.
4. Changeable Input Ratio: The law supposes that it is possible to produce output by changing the ratio of factor inputs; the ratio of fixed and variable factor is changeable; there is no fixed proportion of production function.

In the short period of time, capital is held constant in manufacturing while in agriculture land is held constant and other inputs are used in varying number. The law of variable proportions or the law of diminishing returns is illustrated with the help of hypothetical data given in table. In the illustration, labor service is supposed as a variable factor.

 No. of Labor Total Product Marginal Product Average Product Stage of Returns (L) (TPL) APL=TPL/L MPL=∆TPL/∆L 1 2 3 4 5 6 7 8 9 10 11 8 22 39 52 60 66 70 72 72 70 66 8 11 13 13 12 11 10 9 8 7 6 8 12 17 13 8 6 4 2 0 -2 -4 1st Stage 2nd Stage 3rd Stage

The data in table show that production changes due to change in variable factor (labor). As the number of labor is increased, initially, both marginal and average product of labor increases. After employing more of labor, both APL and MPL falls more rapidly. Falling in the APL and MPL will continue as more labor is put in the production. Hiring of 9th unit of labor adds only nominal amount of output on iron industry. After then, the additional unit of labor i.e., 10th unit, the marginal product becomes negative. Here after, production becomes less.

The operation of the law of variable proportion can be explained in the following figure. Three Stages of Variable Proportions

The vertical axis of the both figure shows the total, average and marginal product of the variable factor and the horizontal axis shows the units employed of the variable factor. The quantity of variable factor is increased relative to the fixed factors, there may arise three different stages.

First Stage

This stage covers the production ranges OL2 units of labor. In the first stage, total product (TP) increases at increasing rate up to the point A (i.e., called point of inflexion) after then TP increases at diminishing rate. At that movement of operation, MPL increases till the use of OL1 unit of labor and begins to decline whereas APL continuously increases till OL2 unit of labor. In this stage, MPL is greater than APL. MPL and APL become equal at OL2 units of labor employed, as shown by the point B. Point B is the end of this stage and the second stage starts.

Causes of returns in stage first
1. Increase in efficiency of fixed factor: In the initial stage, the quantity of fixed factor is abundant in comparison to the quantity of variable factor. As more units of variable factors are added to the constant quantity of fixed factor than the fixed is more intensively and effectively utilized i.e., the efficiency of fixed factors are added to it.
2. Increase in efficiency: In the initial stage, we get increasing returns because as more units of the variable factors are employed, the efficiency of the variable factors itself increases. The reason of efficiency is that with sufficient quantity of variable factor, introduction of division of labor and specialization becomes possible which result in higher productivity.

Second Stage

This stage covers the production ranges between OL2 and OL3 units of labor. In this stage, both MPL and APL decline but positive. However, MPL declines at the faster rate. It is important to note that, at OL3 unit of labor, TP becomes maximum as shown by the point M in figure (A) and MPL becomes zero level at point OL3.

Causes of returns in stage second
1. Scarcity of fixed factor: In the short-run, the quantity of fixed factor cannot be varied. For this reason, the further increases in the variable factor will cause marginal and average product to decline because the fixed factor then becomes inadequate relative to the quantity of variable factor.
2. Indivisibility of fixed factor: If the fixed factors are perfectly divisible, there no change in proportion or increasing and decreasingly returns. In short-run, the size of plant is being unchanged, so the fixed factors are indivisible. Therefore, the excess variable factor are used in combination with indivisible fixed factor, the average product of variable factor diminishes.
3. Imperfect substitutability of the factors: The operation of law of diminishing returns is the imperfect substitutability of one factor for another. The perfect substitute of the scarce fixed factor been available, then the capacity of the scarce fixed factor during the 2nd stage would have been made up by increasing the supply of the perfect substitute with the result that output could be expended without diminishing returns.

Third Stage

Third stage begins with the decline in TPL. As the figure shows, the use of labor in excess of OL3 cause decline in TPL and negative MPL because of overcrowding of labor. However, APL will be positive i.e., tends to zero but not becomes zero.

Causes of returns in third stage
1. Too excessive amount of fixed factor: As the amount of the variable factor continue to be increased to constant quantity of the other; a stage is reached when the total product declines and marginal product become negative. This is due to the fact that the quantity of variable factor becomes too excessive relative to the fixed factor so that they affect each other’s efficiency.

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Production of maximum output in the firm | Production of a given output at minimum cost | Production of maximum output with a given level of cost

Production of maximum output in the firm is possible in the given level of expenditure which can be studied with the help of Iso-cost curve and Iso-quant curve.

It is clear that any desired level of output can be produced by a number of different combinations of inputs. But the manager of the firm most make major decisions such as which input combination to be used, what the input, combination will be optimal.

 The firm can choose from among different combinations of capital (K) and Labor (L) to produce a given level of output or faced with specified input prices, it can choose from among many combinations of K and L that would lead to fixed level of cost i.e. expenditure. Thus, the firm has to make either of two input choice decision.
1. Choose the input combination that yields the maximum level of output possible with a fixed output (i.e. output maximization subject to cost constraint).
2. Choose the input combination that leads to the lowest cost of producing a fixed level of output (i.e. cost minimization subject to output constraint).

The solution to any constrained maximization or minimization problem is obtained by choosing the level of each activity whereby the marginal benefits from each activity. Per dollar (\$) spend are equal. Here the profit maximizing firm has to choose the input combination for which the marginal product divided by input price is the same for all inputs used. The implication is that for two input cases, a firm attains the highest level of output when,

MP/ PL = MP/ PK or MPw = MPr

Where, w and r are respectively the prices of labor (PL) and capital (PK). Thus, the MRTS = MP/ MPK equals the factor price ratio (wr).

Input Prices and Iso-costs

The Iso-quant shows the desire of the producer. Usually, a firm is supposed to have a fixed amount of money to buy resources. The Iso-cost line is the producer’s budget line. In determining the optimal input combination, a profit maximizing producing unit firm or producer has to pay attention to relative input prices, it is to minimize the cost of producing a given output or maximizing output for a given level of cost. Input prices are determined by the market forces.

The equation of total cost is C = rK + wL where all the terms have their usual meaning. Total cost is simply the sum of the cost of K units of capital at r \$ per unit of L units of labor at w \$ per units.

Suppose, capital costs \$100 per month per unit (r = \$100) and labor receives a wage of \$200 per unit (w = S200). Then the firm’s total cost function is

C = 100 K + 200 L ……………………. (i)

Now, suppose that the firm decides to spend \$2000 per month for capital and labor. Thus, equation becomes \$2000 = \$100 K + 200 L.

The process of solving this equation is

2000 = 100 K + 200 L
or, K = 200 – 2 L

In a general situation, if a fixed amount Ḹ is to be spent, the firm can choose among the combinations given by

K = Ḹ/r – w/r . L

Production of a given Output at Minimum Cost

Whatever output a firm chooses to produce, the production manager is desirous of producing it at the lowest possible cost. To achieve this objective, the production process must not only to be technically efficient but economically efficient too. So, the production process has to organize in the most efficient manner.

For example, suppose that at given input prices r and w, a firm wishes to produce the output indicated by Iso-quant Q0 as shown in the figure.

In the figure, KL1, KL2 and KL3 are the three Iso-cost lines from which the producer can choose at the given factor prices. The firm will choose the lowest level of expenditure that enables output level Q0 = 200 to be produced. As shown in the figure, output level Q0 will be produced at the cost level Q0 will be produced at the cost represented by Iso-cost line KL1.

Any cost outlay below that such as represented by KL is not feasible since it is impossible to produce output Q0 with these factor combinations. Any factor combinations above that represented by K1L1 are not considered because the firm seeks to produce the desired output at least cost. If combination A or B is chosen at the cost outlay represented by K2L2, the producer can reduce costs by moving along Q0 to point E. Point E shows the optimal resource combination, K0 units of capital and L0 units of labor. This is known as the level of cost combination of inputs.

This Iso-quant shows the desired rate of factor substitution and the Iso-cost is the actual rate of factor substitution. A firm reaches equilibrium and thus minimizes cost when the Iso-quant is tangent to the lowest possible Iso-cost line. Thus, equilibrium is reached when the Iso-quant representing the chosen output is just tangent to an Iso-cost line. At this tangent the slopes of the two curves are equal, production at least cost requires that the MRTS (Marginal Rate of Technical Substitution) of capital for labor be equal to the ratio of the price of labor to the price of capital.

MPL PMK = w / r

or, MPL / w = MPK / r

where, MPL = Marginal production of labor
MPK = Marginal productivity of capital
r = Price of capital
w = Price of labor

Production of Maximum Output with a given Level of Cost

It is an alternative technique but more preferable way of presenting the optimization problem. It is to assume that the firm chooses a level of output and then select the factor combination that permits production of that output at least cost. This approach seems to be more practical than the previous one. It is assumed that the firm can spend only a fixed amount of money to spend and it seeks to attain the highest level of output consistent with that amount of outlay. It can be explained with the help of figure below:

The Iso-cost line KL shows all possible combination of the two inputs that can be purchased with a fixed market prices. Three Iso-quants are shown in the figure. At the given level of cost, output level Q2 is unattainable. Neither output level Q1 nor level Q2 would be chosen, since higher levels of output can be produced with the fixed cost outlay. The highest possible output with the given level of cost is produced by using OL0 amount of labor and OK0 amount of capital. At point E, the highest attainable Iso-quant (i.e. Q) is just tangent to the given Iso-cost (KL). Thus, in the case of constrained output maximization, the MRTS of capital for labor equal the input-price ratio (the price of labor to the price of capital).

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Firm making decision with regard to the Optimum employment of two inputs

A profit maximizing firm (business unit) tries to minimize its cost for a given output or to maximize its output for a given total cost. The logic of iso-quant tells that a given output can be produced with different input combinations. Given the input prices, only one of the input combinations confirms to the least-cost criteria. We may now combine the iso-quants and iso-cost to determine the optimal input employment or the least cost combination of inputs. There are two conditions for the least-cost combination of inputs (a) the firm order condition, (b) the second order condition. The first order condition of the least-cost input condition of the least-cost input combinations can be expressed in both the physical and value terms.

Given the two inputs K and L, the first order condition in physical terms required that marginal rate of exchange (MRE) between K and L must equal the ratio of their marginal physical product (MPP) i.e.,

MPMPK = ∆K /∆L ……………… (i)

 Where ∆K / ∆L is the marginal rate of exchange (MRE) between K and L, and MPL / MPK is the ratio of marginal physical productivities of L and K i.e., on input combination at which factor exchange ratios equals the ratios of their marginal productivities is the least-cost input combination.

The first order criterion of the least-cost input combination is also expressed as the MRP ratio of K and L must equal their price ratio, i.e.,

MPMPL = P/ PK or, MP/ PL = MP/ PK ………………….. (ii)

Where, MPL and MPK are marginal products of labor and capital respectively, and PL and PK are prices of labor and capital respectively.

In equation (ii), (∆K / ∆L) slope of the iso-quant, and MP/ MPK = slope of the iso-quants. It refers that the least cost combination exists at a point where iso-quant is tangent to the iso-cost. The least cost combination of K and L is graphically presented in the following figure.

The iso-quant Iq2 is targeted to iso-quant K2L2 at point E. At this, combination of K and L equals OG of K plus OH of L. This combination of K and L is optimal as it satisfies the least-cost criterion, i.e.,

∆K / ∆L = MP/ MPK

In case of second order condition, the second order combination requires that the first order condition be fulfilled at the highest possible iso-quant. The first order condition is satisfied also on points A and F, the points of intersection between Iq1 and iso-cost K2L2 in above figure. Since at these intersection points ∆K / ∆L = MP/ MPK. But, points A, F and E are on the same iso-cost, point E is on an upper iso-quant while E is associated with on output of 400 units, points A and F being on a lower iso-quant, are associated with an output of 200 units. It means that the total cost, a firm can produce 200 units. It means that given the total cost, a firm can produce 200 units as well as 400 units. Therefore, a profit maximizing or cost minimizing firm chooses input combinations at point E rather than A or F.

In case of value terms, the following condition has to be operated for optimal employment of two points.

P/ PK = MRP/ MRPK or MRP/ PL = MRP/ PK ………………… (iii)

Where, MRP = marginal revenue productivity of the factor.

It may be inferred from equation (iii) that least-cost or optimal input combination requires that the MRP ratios of inputs should be equal to their price ratios. In other way, the MRP and factor price ratios of all the inputs must be equal.

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Determination of Optimal Employment of an Input

 To see how the economic productivity of an input, as defined by its marginal revenue product, is related to the factor for productive purposes, one needs only consider the simple question. If the price of input X in the production system is \$100, how many units of X would a firm use? If the marginal revenue products exceed the related cost, that level of such input number could be employed.

The relationship between resource productivity as measured by the marginal revenue product and optimal employment or factors use can be maximized. If marginal revenue exceeds marginal cost, profits most increase. In the context of production decisions, this means that if the marginal revenue product of an input, i.e., the marginal revenue generated by its employment in a production system exceeds its marginal cost, then profits are increased as input employment increases.

Similarly, when the marginal revenue product is less than the cost of the factor, marginal profit is negative, so the firm would reduce employment of that factor.

This concept of optimal resource employment can be clarified by examining a very simple production system in which a single variable input labor (L), is used to produce a single product, Q. Profit maximization requires that production be at a level such that marginal revenue equals marginal cost. Since the only available factor in the system is input L, the marginal cost of production can be expressed as:

MC = ∆C / ∆Q = P/ MPL …………….. (i)

That is dividing PL, the price of a marginal unit of L, by MPL, the number of units output gained by the employment of an added unit of L, provides a measure of the marginal cost of producing each additional unit of the product.

Since MR must equal MC at the profit maximizing output level, MR can be substituted for MC in equation (i), resulting in the expression,

MR = P/ MPL ……………………. (ii)

Equation (ii) must hold for profit maximization since it was demonstrated immediately above that the right-handed side of equation (ii) is just another expression for MC. Solving equation (ii) for PL results in PL = MR x MPL which is defined as the marginal revenue product of L.

PL = MRPL …………………….. (iii)

Equation (iii) states the general result that a profit maximizing firm will always employs an input up to the point where its MRP is equal to its cost. If the MRP exceeds the cost of the input, profits are increased by employing additional units of the factor.

Similarly, when the resources’ price is greater than its MRP, profit is increased by using less of the factors.

Only at the level of usage where MRP = P are profits maximized than the costs incurred (PL). Only at L where PL = MRPL, will total profits be maximized. If PL were higher, the quantity of L demanded would be reduced. Similarly, if PL were lower the quantity of L purchased would be greater.

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Production Function | Linear Production Function | Quadratic Production Function | Cubic Production Function | Power Production Function

Production involves the transformation of inputs into outputs. Production involves the transformation of inputs into physical output. The output is thus, a function of factors which are also called inputs. The term ‘production function’ refers to the relationship between the inputs and outputs produced by them. The functional relationship between physical inputs and physical output of a firm is known as production function. Algebraically, production function can be expressed as,

Q = f ( Ld, L, K, M…. )
Where,
Q stands for the quantity of output, Ld, L, K and M stand for the land, labor, capital and management respectively.

 The above equation shows that the quantity (Q) of output produced depends upon the quantities of the factors used. Simply, production function expresses the relationship between the quantity of output and the quantities of the various inputs used for the production. More precisely, the production function states the maximum quantity of output that can be produced with by given quantities of various inputs.

In economic theory, there are two types of production functions on time basis. The production function when the quantities of some inputs such as capital and labor are kept constant and the quantity of one input such as labor is varied. This kind of production function [Q = f (K, L)] is known as short-run production function. The study of short-run production function is the subject-matter of the law of diminishing returns which is also called the law of variable proportion.

Secondly, we study production function (input – output relation) by varying all inputs, and this is called long-run production function and can be expressed as Q = f (Ld, L, K, M ). This form is the subject-matter of the law of returns to scale. Generally, the terms constant and increasing returns are used with reference to constant and increasing returns to scale.

Besides these, there are other production function, such as

1) Linear Production Function

A linear production function would take the form:

Total production Y = a + bx

From this, function for the managerial production will be,

= Y / X.a / x + b

The equation for the marginal product will be,

∆Y / ∆X = b

2) Power Function

A power function expresses output Y, as a function of input in the form:

y = AXα

It contains certain properties:
a) The exponents are the elasticity of production. Thus, in the above function, the exponent α represents the elasticity of production.
b) The equation is linear in the logarithms, i.e., it can be expressed as,

log Y = logA + αlogX

when the power function is expressed in logarithmic form as above, the coefficient α represents the elasticity of production.

c) If one input is increased while all others are held constant, marginal product will be decline.

The production function may be quadratic, taking the following form:

Y = a + bx – cx2

Where, the dependent variable Y shows total output and the independent variable X represents input. The small letters are parameters; their probable values are determined by a statistical analysis of the data.

a) The minus sign in the last term denotes diminishing marginal returns.
b) The equation allows for decreasing marginal product but not for both increasing and decreasing marginal products.
c) The elasticity of production is not constant at all points along the curve as in a power function, but declines with input magnitude.
d) The equation never allows for an increasing marginal product.

4) Cubic Production Function

The cubic production function takes the following form:

Y = a + bx + cx2 – dx3

Some important special properties of a cubic production function are:
a) It allows for both increasing and decreasing marginal productivity.
b) The elasticity of production varies at each point of the curve.
c) Marginal productivity decreases at an increasing rate in the later stages.

5) Power Production Function (Cobb-Douglas Function)

Power functions have been employed in a large number of empirical production studies, particularly since Charles W. Cobb and Paul H. Douglass’s pioneering work in the late 1920s. The impact of this work was so great that power production functions are now frequently referred to as Cobb-Douglas production functions.

Cobb-Douglas production function can be expressed as,

Q = AKα Lβ
Where, Q = total output
L = index of employment of labor in manufacturing
K = index of fixed capital in manufacturing

The exponents α and β are the elasticities of production i.e., α and β measure the percentage response of output to percentage changes in labor and capital respectively.

Properties of Power Function:
a) Power functions allow the marginal productivity of a given input to depend upon the levels of all inputs employed a condition that often holds in actual production systems.
b) They are linear in logarithms and thus can be easily analyzed using linear regression analysis.

log Q = log A + α log K + β log L

The least squares technique can be used to estimate the coefficients of equation and thereby the parameters of equation.

c) Power functions facilitate returns to scale estimation. Returns to scale are easily calculated by summing the exponents of the power function. If the sum of the exponent is less than one, (α + β < 1) diminishing returns are included. A sum greater than one (α + β > 1) indicates increasing returns. Finally, if the sum of the exponent is exactly one (α + β = 1), returns to scale are constant.

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Investment Decision under Uncertainty

Uncertainty refers to a situation in which a decision is expected to yield more than one outcome and the probability of none of the possible outcomes is known. Therefore, decisions taken under uncertainty are necessarily subjective. However, analysis have devised some decision rules to impart some objectively to the subjective decisions, provided decision-makers are able to identify the possible ‘states of nature’ and can estimate the outcome of each strategy. Some such important decision rules are discussed below:

1. Wald’s maximum decision criterion

Wald’s maximum decision criterion says that the decision-makers should first specify the worst possible outcome of each strategy and accept a strategy that gives best out of the worst outcomes. It gives a conservative decision rule for risk avoidance. However, this decision rule can be applied by those investors who fall in the category of risk averters. This investment rule can also be applied by firms whose very survival depends on avoiding losses.

2. Minimax regret criterion

 Minimax regret criterion is another decision rule under uncertainty. This criterion suggests that the decision-makers should select a strategy that minimizes the maximum regret of a wrong decision. What is regret? “Regret is measured by the difference between the pay-off of a given strategy and the pay-off of the best strategy under the same state of nature. Thus, regret is the opportunity cost of a decision.

3. Hurwicz decision criterion

Hurwicz has suggested another criterion for investment decision under uncertainty. In his opinion, full realization of optimistic pay-off or full realization of most pessimistic pay-off is a rare phenomenon. The actual pay-off of a strategy lies somewhere between the two extreme situations. According to Hurwicz criterion, therefore, the decision-makers need to construct a decision index of most optimistic and most pessimistic pay-offs of each alternative strategy. The decision index is, in fact, a weighted average of maximum possible and minimum possible pay-offs, weight being their subjective probability such that sum of probabilities of maximum (Max) and minimum (Min) pay-offs equals one.

4. Laplace decision criterion

The Laplace criterion uses the Bayesian rule to calculate the expected value of each strategy. As mentioned earlier, Bayesian rule says that where meaningful estimate of probabilities is not available, the outcome of each strategy under each state of nature must be assigned the same probability and that the sum of probabilities of outcome of each strategy must add up to one. For this reason, the Laplace criterion is also called the ‘Bayesian criterion’. By assuming equal probability for all events, the environment of ‘uncertainty’ is converted into an environment of ‘risk’.

Once this decision rule is accepted, then decision-makers can apply the decision criteria that are applied under the condition of risk. The most common method used for the purpose is to calculate the ‘expected value’ as defined in the case of pay-off matrix in section. Once expected value of each strategy is worked out, then the strategy with the highest expected value is selected.

This decision rule avoids the problem that arises due to subjectivity in assuming a probability of pay-off. This criterion is, therefore, regarded as the criterion of rationality because it is free from a decision-makers attitude towards risk.

To sum up, uncertainty is an important factor in investment decisions but there is no unique method of dealing with uncertainty. There are several ways of making investment decisions under the condition of uncertainty. None of the methods as described above lead to a flawless decision. However, they do add some degree of certainty to decision-making. The choice of method depends on the availability of necessary data and reliability of a method under different conditions.

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Utility and Risk Aversion in Investment Decision

The assumption of risk aversion is basic to many decision models in managerial economics. Because this assumption is too crucial, it is appropriate to examine attitudes toward risk and discuss why risk aversion might hold in general.

1. Possible risk attitudes

In this theory, three possible attitudes towards risk are presented as aversion to risk, indifference to risk, and preference for risk. Risk aversion characteristics individuals who seek to avoid or minimize risk. Risk neutrality characteristics decision makers focus on expected returns and disregard the dispersion returns (risk). Risk seeking characteristics decision makers prefer risk. Given a choice between more risky and less risky investments with identical expected monetary returns, a risk averter select the less risky investment and a risk seeker select the riskier investment. Faced with the same choice, the risk-neutral investor is indifferent between the two investment projects.

Given the importance of attitudes towards risk in economic decision making, it is important to ask what factors are involved in the determination of such attitudes. Managerial economics tends to presume that the majority of economic participants are risk averters, and it makes this presumption on the basis of the principle of diminishing marginal utility of money.

2. Relation between money and its utility

At the heart of risk aversion is the notion of diminishing marginal utility for money. If someone with no money receives \$5000, it can satisfy his or her most immediate needs. If such a person then receives a second \$5000, it will obviously be useful, but the second \$5000 is not quite so necessary at the first \$5000. Thus, the value or utility of the second or marginal \$5000 is less than the utility of the first \$5000 and so on. Consequently, diminishing marginal utility of money implies that the marginal utility of money income or wealth diminishes for additional increments of money.

Now, if this principle holds generally then it has an important implication for attitudes towards a 50/50 risk of gaining or losing a given monetary amount. This is that the extra benefit from making an equally likely gain is less than the loss of benefit from enduring an equality likely loss. For this reason, the diminishing marginal utility of money tends to make for risk aversion.

3. Adjusting the valuation model for risk

 To the extent that diminishing marginal utility leads directly to risk aversion, then this risk aversion can be reflected in the basic valuation model used to determine the worth of a firm. If a managerial decision affects the firm’s risk level, the value of the firm is affected. Two primary methods are used to adjust the basic valuation model to account for decision making under conditions of uncertainty.

Under conditions of risk, the profits shown in the numerator of the valuation model as π equal the expected value of profits during each future period. This expected value is the best available estimate of the amount to be earned during any given period. However, since profits cannot be predicted with absolute precision, some variability is to be anticipated. If the firm must choose between two alternative methods of operation, one with high expected profits and high risk and another with smaller expected profits and lower risks, some technique must be available for making the alternative investments comparable. An appropriate ranking and selection of projects is possible only if each respective investment project can be adjusted for considerations of both time value of money and risk.

The certainty equivalent method is an adjustment to the numerator of the basic valuation model to account for risk. Under the certainty equivalent approach, decision makers specify the certain sum that they are comparable to the expected value of a risky investment alternative. The certainty equivalent of an expected risk amount typically differs in monetary terms but not in terms of the amount of utility provided.

Another way to incorporate risk in managerial decision making is to adjust the discount rate of denominator of the basic valuation model equation (iii). Like certainty equivalent factors, risk-adjusted discount rates are based on the trade off between risk and return for individual investors.

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Measurement of Risk by Probability Distribution

 The probability of an event is the chance, or odds, that the event will occur, if all possible events or outcomes are listed, and if a probability of occurrence can be assigned to each event, the listing is called a probability distribution. For example, suppose a sales manager observes that there is a 70% chance that a given customer will place a  specific   order   within   the  next   two
weeks, versus a 30% chance that the customer will not. This situation is described by the probability distribution shown in table.

Each possible outcome is listed in column 1, and the probabilities of each outcome, expressed as decimals and percentages, appear in column 2. Notice that the probabilities sum to 1.0 or 100%, as they must if the probability distribution is to be complete (i.e. represent all possible outcomes). In this very simple example, risk can be read from the probability distribution as the 30% chance of neither the firm nor receiving the order. For most managerial decisions, the relative desirability of alternative events or outcomes is not so easily computed. A more general measure of the relation between risk and the probability distribution is measure of the relation to incorporate risk considerations adequately into the decision-making process. The need for a more general measure of risk can be illustrated by the following example.

Suppose that a firm is able to choose only one of two investment projects, each calling for an outlay of \$10,000. Assume also that profits earned from the two projects are related to the general level of economic activity during the coming year, as shown in table. This table is known as a payoff matrix since it illustrates the monetary outcomes associated with each possible state of profits from project B very much more as a result the state of the economy than do those from Project A. In a normal economy, both projects return \$5,000 in profit. Should the economy be in a recession next year? Project B will produce nothing, whereas Project A still provides a \$ 4,000 profit. If the economy is booming next year, Project B’s profit will increase to \$ 12,000, but profit for Project A will increase only moderately to \$6,000.

Outcomes and Probabilities for Receiving an Order

Event                        Probability of Occurrence
(1)                                        (2)
Do not received order                                        0.3 = 30%
Total                                                                  1.0 = 100%

Project A is clearly more desirable if the economy is in recession, whereas Project B is superior in a boom. In a normal economy, the projects offer the same profit potential, and both are equally desirable. To choose the best project, one needs to know the likelihood of a boom, a recession or normal economic conditions. If such probabilities are available, the expected profits and variability of profits for each project can be determined. These measures make it possible to evaluate each project in terms of anticipated or expected returns, and to measure the risk of such returns in terms of the difference between and expected values.

Payoff Matrix for Project A and B

Profits
State of the Economy         Project A             Project B
Recession                             4,000                     0
Normal                                 5000                  5,000
Boom                                   6,000                 12,000

The expected value is the anticipated receipts from a given payoff matrix with a specified probability distribution. It is the weighted average receipt when the weights are defined by the appropriate probability distribution.

To continue with the previous example, assume that forecasts based on the current trend in economic indicators suggest a 2-in-10 chance of recession, a 6-in-10 chance of normal economy, and a 2-in-10 chance of a boom. As probabilities, the probability of recession is 0.2, or 20%, the probability of normal economic activity is 0.6 or 60% and the probability of a boom is 0.2, or 20%. These probabilities add up to 1.0 (0.2 + 0.6 + 0.2 = 1.0), or 100%, and thereby from a complete probability distribution, as shown in table.

Calculation of Expected Values

 State of the Economy Probability of this State Occurring Profit Outcome if this State Occurs Expected Profit Outcomes (\$) (1) (2) (3) (4) = (2) x (3) Project A Recession 0.2 4,000 800 Normal 0.6 5,000 3,000 Boom 0.2 6,000 1,200 1.0 Expected Profit A Project B Recession 0.2 0 0 Normal 0.6 5,000 3,000 Boom 0.2 12,000 2,400 1.0 Expected Profit B

If each possible outcome is multiplied by its probability of occurrence, and the answers are summed, the weighted average outcomes are obtained. In this calculation, the weights are the probabilities of occurrence, and the weighted average is called the expected value. The above mentioned table illustrates the calculation of expected profits for Project A and B. Each possible profit level in column 3 is multiplied by its probability of occurrence from column 2 to obtain weighted values of the possible profits. Summing column 4 of the table for each project gives a weighted average of profits under various states of the economy. This weighted average is the expected profit from the project.

Risk is a complex concept, and some controversy surrounds attempts to define and measure it. Common risk measures that are satisfactory for most purposes are based on the observation that right probability distributions imply low risk.

The standard deviation is a popular and useful measure of absolute risk. Absolute risk as measured by the standard deviation is the overall dispersion of possible payoff values. The smaller is the standard deviation, the tighter is the probability distribution and therefore the lever is the risk in absolute terms.