**Profit –Volume Chart** The impact of cost and revenue on profit at various levels of activity can be represented in profit volume chart which highlights the loss area at the levels of activity below the break-even volumes and the profit area at levels of activity above the break-even volume. Profit curve cuts the vertical axis below the point at zero profit even when there are no sales the fixed cost must be paid and, consequently, the area below the break-even volume represents loss. The profit-volume chart is shown in figure

The following steps are involved in construction of profit-volume chart:

ILLUSTRATION

Construct a profit-Volume graph and also verify the same with the mathematical analysis.

SOLUTION

**CVP Analysis with change in selling Price **The effects on the potential profits of contemplated selling price changes can be represented on a profit-volume chart as shown in figure

**CVP Analysis in Multi-product Situations** where a company manufacturing more than one product of varying profitability, a change in the profitability of one product will lead to change in the profitability of group as a whole. The profit-volume chart may be used to illustrate the effects of changes in product mix by drawing a product profit path as shown in **FIGURE MULTI-PRODUCT PROFIT PATH CHART** so separate profit lines are drawn for each of the assumed profit mixes as shown in figure for each individual product.

**Curvilinear
Break-even Analysis**

Under marginal costing approach the main assumption is that selling price and variable cost per unit will remain constant at any level of activity. The production and sales can be pushed upto the maximum plant capacity so long as the contribution margin is positive. This assumption is valid only as long as it is not necessary to reduce the selling price per unit to increase sales volume or the variable cost per unit do not increase because of the operation of the factors of diminishing return after a certain stage of production has been reached. The basic assumption behind this is that cost-volume profit relationship is linear. In practical business scene, increased sales volume may be obtained price concessions are offered to the customer. In economic theory, initially total cost will increase at a declining rate per unit, then at a constant rate per unit in the second stage and finally at an increased rate per unit.

This behaviour of costs is due to the fact that due to economies of scale, gains in operating efficiencies have the effect to reduce costs per unit to a certain point beyond which with further increase in level of production, operating inefficiencies tend to bring diminishing returns. The effect of the decreasing price per unit with increase in demand and the increasing cost per unit, due to diminishing returns is to have a profit figure that increases upto a point and then decrease until it is converted into a loss. The break-even chart, therefore become curvilinear instead of linear model which we have discussed earlier. In the curvilinear model, the optimum production level is where the total revenue line, exceeds the total cost line by the largest amount.

Figure shows the total sales revenue and cost curves with the optimum output level. It will also be seen that there are two break-even points one at lower capacity level and the other at a higher capacity level. It should be noted the two bread-even points are different in the sense that while increasing production and sales beyond the first bread-even point will increase profit but increase in volume beyond the second break – even point will result in loss. The economists’ model is valid over a wide range of activity and it allows quantities, unit price and cost inputs to vary. The accountants’ model on the other hand is valid only for a short relevant range of activity and only the quantity varies, limit price and cost structure remaining constant. Beyond this range, however, the price and costs would also undergo change. If we analyse the two and the accountant’s model is consistent with the economic theory. The accountant’s linear model is meant for short term decision to maximize the total contribution margin. If long-term decision model is required, there are two options open; one is to develop the curvilinear model of the economist for the wide range of activity. The other is to reassess the entre revenue and cost patterns which would be valid for a new relevant range of activity and develop the accountant’s linear model with this new set of data.

**CVP
Analysis under Conditions of Uncertainty**

The following techniques are useful in CVP analysis under conditions of uncertainty:

- Our ability to carry out a particular task will be better at each successive attempt. It is a mathematical portrayal of the decreasing rate of increase in costs whilst experience is being gained in a new task. The applicability of learning curve is more important in cases where the labour input in activity is large and the activity is complex.
- Sensitivity analysis seeks to test the responsiveness of outcomes from decision models to different input values and constraints as a basis for appraising the relative risk of alternative courses of action.
- The use of simulation technique makes the cost-volume-profit relationship more realistic. Parameters are specified for each factor and within the parameter, factor values are selected randomly. The technique use random numbers and is used to solve problems which involves conditions of uncertainty. Under varying conditions, the costs vary based on various combinations of factors. Thus the manager depending on varying conditions should be able to predict sale quantity, sale value, variable cost, fixed cost with certain perception.

- The normal distribution also called the ‘normal probability distribution’ is the most useful theoretical distribution for continuous variables. Many statistical data concerning business and economic problems like CVP analysis are displayed in the form of normal distribution. The main property in the normal distribution is , as the size of the sample is increased the sample means will tend to be normally distributed. This characteristic makes it possible to determine the minimum and maximum limits within which the population values lie. For example, within a range of population mean 3, 99.73% or almost all the items are covered.

In the theoretical distribution, many problems can be solved only under the assumption of a normal population. As the sample size becomes large, the normal distribution serves as a good approximation of many discrete distributions whenever the exact discrete probability is laborious to obtain or impossible to calculate accurately. The limiting frequency curve is obtained as ‘n’ becomes large it is called the normal distribution curve or simply normal curve as shown in figure the normal curve is ‘bell – shaped’ and symmetrical in its appearance. If the curves were folded along its vertical axis, the two halves would coincide. The number of cases below the mean in a normal distribution is equal to the number of cases above the mean, which makes the mean and median coincide. The height of the curve for a positive deviation of 3 units (i.e., +3) is the same as the height of the curve for negative deviation of 3 units (i.e., – 3). The height of the normal curve is at its maximum at the mean.

The area under the normal curve is distributed as follows:

The normal distribution curve can be drawn as shown in figure

**Practical
Application of Linear Programming Technique**

The usefulness of the linear programming (LP) technique is given below:

**Allocation of scarce resources**– when scarce or limited raw material, labour hours or machine hours are to be allocated to various products, the problem is to determine the best allocation that would maximise profit.**Product – mix problems –**an industrial concern was available a certain production capacity on various manufacturing processes and has the opportunity to utilise this capacity to manufacture various products. Typically, different, products will have different selling prices, will require different amounts of production capacity at the several processes and therefore, will have different unit’s profits; there may also be stipulations on maximum and / or minimum production levels. The problem is to determine the optimal mix so that the total profit is maximised.**Determination of join products profitability –**in case of products involving joint costs where one or more of the joint products may be processed further. LP technique may be applied to determine the profitability of further processing.**Cost – volume – profit analysis –**LP technique can be applied to multi product cost –volume profit analysis. Problem of short – term capacity utilisation where a number of products are involved may be conveniently formulated into LP models. LP may be viewed as extension of the conventional cost – volume analysis where there is no uncertainty and where the costs and revenues are assumed to be linear functions.

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