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Constrained Maximization in Managerial Economics

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Constrained Maximization in Managerial Economics

SOLUTION:
Maximization is an economics theory, that refers to individuals or societies gaining the maximum amount out of the resources they have available to them.[KOTACK,2005]
Constrained Maximisation is a term in economics used to refer to and is concerned with the restrictions imposed on the availabilty of resources and other requirements.( ) it tries to explain using prescribed forumlae such as the langarian method how firms can solve issues to do with constrained maximisation. In this context however we are more interested in the maximisation of profits in firms, we are interested in the contraints imposed on managers that limit their options when making decisions.
For instance in profit making organnisations the primary objective is to make profits so it endeavours to explain what firms have to contend with in their objective. In this context profit maximization is the process by which a firm determines the price and output level that returns the greatest profit, and in doing so the company may have constraints on the bugdet, human resourse, inputs in terms of raw materials , capital expenditure etc.
Contrained Maximisation shows the relationship between inputs such as the ones mentioned and how they ultimatly affect the output. In solving any constrained maximisation problem the objective is to see how other variables can be manipulated to achieve the highest output, to do this Managerial economists use the constraint equation for one of the decision variables , then susbstitute for that variable in the objective function.
A typical example of constrained maximisation can be shown by examining a study done by M.T Maloney on a Bee Keeper Steve Cheung to analyse the problem of keeping bees as an integrated endeavour to explain contrained maximisation.
Be keepers sometimes paid and sometimes received pay depeneding on the marginal value of their pollination services relative to the value of honey they collected. For instance a farmer that uses bees to pollinate apple trees and to make honey from nector . the farmers outputs are apples and honey . Assuming there is a trade off between the two outputs i.e. the nector gathered from apple trees does not produce as much honey as that found elsewhere. When the farmer is interested in apple pollination he places the hives close to the apple trees. This reduces the amount of honey while increasing the production of apples. Conversly, when the farmer increases honey production , apple production is sacrificed.
This relationship can be explained using derivatives, for instances if we let the production function be characterised by y=f(X1, X2), where y is the number of bees , X1 is honey , and X2 is apples. This is an inverse production function since the input is the left hand side variable ; the partial derivative of input with respect to one of the outputs f1, tells us how much the input must increase for one unit change in the output holding the other output constant.
If we go further, we can also make the asumption that the farmer has a fixed number of bees , the farmer maximises profits by allocating the bees based on the prices of honey and the prices of apples. The problem is one of Maximising profit by keeping the number of bees constant. The farmer has has only one choice . Variable , that is , the placement of the fixed number of hives. However, we are interested in examining his behaviour in terms of the amount of honey and apples that he brings to the market. We analyse his behaviour in terms of these two variables , knowing the choice of one implies the choise of the other . This relationship is identified by setting the optimisation problem up using the method of lagrange.(M.T.Maloney.p11,1973)
The above example describes a sitaution which is typical in most organisations, it basically describes a Managerial decision dilema that of increasing out put while keeping all other inpu variables in balance. Most managers are usauly faced with similar problems of how they would allocate their resources which may range from capital expenditure, raw materials and other inputs to achieve the highest profits.
By using the lagrangian formula, we can determine how behaviours change as paramters change, in this context behaviour is defined in terms of choices and how they vary and ultimatley affect the outcome. Similarly in the farmers case if he gets more bees, the profits will increase; if a firm increases capital expenditure or improves the quality of raw materials their profits are likely to increase.
Marginal cost and revenue, depending on whether the calculus approach is taken or not, are defined as either the change in cost or revenue as each additional unit is produced, or the derivative of cost or revenue with respect to quantity output. It may also be defined as the addition to total cost as output increase by a single unit. For instance, taking the first definition, if it costs a firm 400 USD to produce 5 units and 480 USD to produce 6, the marginal cost of the sixth unit is approximately 80 dollars, although this is more accurately stated as the marginal cost of the 5.5th unit due to linear interpolation. Calculus is capable of providing more accurate answers if regression equations can be provided.

Total Cost-Total Revenue Method


Pofit Maximization - The Totals Approach
To obtain the profit maximizing output quantity, we start by recognizing that profit is equal to total revenue (TR) minus total cost (TC). Given a table of costs and revenues at each quantity, we can either compute equations or plot the data directly on a graph. Finding the profit-maximizing output is as simple as finding the output at which profit reaches its maximum. That is represented by output Q in the diagram.
There are two graphical ways of determining that Q is optimal. Firstly, we see that the profit curve is at its maximum at this point (A). Secondly, we see that at the point (B) that the tangent on the total cost curve (TC) is parallel to the total revenue curve (TR), the surplus of revenue net of costs (B,C) is the greatest. Because total revenue minus total costs is equal to profit, the line segment C,B is equal in length to the line segment A,Q.
Computing the price at which to sell the product requires knowledge of the firm's demand curve. The price at which quantity demanded equals profit-maximizing output is the optimum price to sell the product.

Marginal Cost-Marginal Revenue Method


Profit Maximization - The Marginal Approach
If total revenue and total cost figures are difficult to procure, this method may also be used. For each unit sold, marginal profit equals marginal revenue minus marginal cost. Then, if marginal revenue is greater than marginal cost, marginal profit is positive, and if marginal revenue is less than marginal cost, marginal profit is negative. When marginal revenue equals marginal cost, marginal profit is zero. Since total profit increases when marginal profit is positive and total profit decreases when marginal profit is negative, it must reach a maximum where marginal profit is zero - or where marginal cost equals marginal revenue. This is because the producer has collected positive profit up until the intersection of MR and MC (where zero profit is collected and any further production will result in negative marginal profit, because MC will be larger than MR). The intersection of marginal revenue (MR) with marginal cost (MC) is shown in the next diagram as point A. If the industry is competitive (as is assumed in the diagram), the firm faces a demand curve (D) that is identical to its Marginal revenue curve (MR), and this is a horizontal line at a price determined by industry supply and demand. Average total costs are represented by curve ATC. Total economic profits are represented by area P,A,B,C. The optimum quantity (Q) is the same as the optimum quantity (Q) in the first diagram.
If the firm is operating in a non-competitive market, minor changes would have to be made to the diagrams.

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