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Linear Programming in Finance, Accounting and Economics

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Linear Programming in Finance, Accounting and Economics Sijia Lu 7289928683 Abstract This article is literatures review about five articles, which apply linear programming to Finance, accounting and economics. The mathematical method is found of crucial importance in those fields. The paper shows how theoretical inference in linear programming throws light upon realistic practice, and how empirical evidence supports those theories. Keywords: finance; accounting; economics; linear programming; investment analysis Linear Programming in Finance Application of Linear Programming to Financial Budgeting and the Costing of Funds” explored how to allocate funds in an enterprise by applying linear programming. As Charnes, Cooper and Miller analyzed, at least three problems are to be considered to solve the allocation problem: 1) Plans for production, purchases, and sales under certain structure of the firm’s assets, in order to maximize its profit or reach other objectives. 2) The change of the firm’s profit per unit change in the structure of the assets. 3) Opportunity cost of the firm’s funds.

The article starts with a simple example with one commodity and one warehouse. Let B be the fixed warehouse capacity, A be the initial stock of inventory in the warehouse, xj be the amount to be sold in period j, yj be the amount to be sold in period j, pj be the sales price per unit in period j, and cj be the purchase price per unit in period j, then we have due to the cumulative sales constraint; due to the warehouse capacity constraint; due to the buying constraint; due to the selling constraint; and with our goal of maximizing The dual problem is also obvious.

It is to minimize subject to and to where As we learned, “dual theorem of linear programming” says that the two optimal values of the original problem and the dual problem should be equal. Using this theorem, the authors then reached a new method of evaluating assets. Because , we have in which the two sides must have the same units of measure. So it is now obvious that t*k represents the value per unit of net warehouse capacity and u*k represents the value per unit of initial inventory in the warehouse. Similarly, consider the financial problem, which has liquidity constraints as here j-? represents payments and j-r represents receipts, M0 is the initial cash available and M is the balance the firm desired the maintain. By examining the dual problem of this, we can find corresponding dual variables for the problem called, say, vk. Again, from the equality we found before, we can learn that the two sides of the equation have the same units of measure. It is then seen that the v’s should be dollars per unit time per dollar invested. The valuation of assets or investments is of crucial importance to any business.

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So far, by simply applying the dual theorem, Charnes, Cooper and Miller have created a new method of evaluating assets or investments. This method of evaluating is also easy to find out answers. It is intelligent to examine the units of measure rather than try to solve the specific problems. The interesting thing is that in realistic problems, we can find true meanings of theoretical dual variables. Then the authors mixed the two former problems together to see a more realistic case – a warehouse problem with financial constraints.

So the following new constraints are added: Now if we define We’ll get the new dual problem: Here, V1 is the incremental cumulative internal yield rate. Or it is the opportunity cost the capital invested – “it shows the net amount to which an additional dollar invested in the firm will accumulate if left to mature to the end of the planning horizon. ” This is also easy to understand in terms of economics, maximizing profit can be the same as minimizing the opportunity costs. The article then went through several practical problems using the dual variable evaluating method.

It is also interesting to find out that all the commodities are directly linked to the funds-flow while the goods-flow can be avoided in the warehouse problem with multiple commodities. An Example “A linear programming model for budgeting and financial planning” created an accounting experiment in which the dual variables introduced earlier were calculated which can also be considered as a sensitivity analysis. This can be seen as application and verification of Charnes, Cooper and Miller’s earlier theory. In the linear programming problem listed below, (1) represents the interests earned with a rate of 0. 29%; (2) holds because firm’s sale of securities will not be more than the beginning balance of this amount; (3) represents the maximum collection of receivables will not exceed the beginning balance of account receivable; (4) means the initial cash balance constraints the purchase of securities; (5) indicates “contribution” on a unit sale per unit deduction from the ending goods inventory, with prevailing selling price being $9. 996 and cost of production $2. 10; (6) holds because of the cost structure: in the $2. 10 cost, $1. is the material cost and $1. 1 is the conversion cost (direct labor cost and direct overhead); (7) represents the production capacity limits by limiting the value of raw materials; (8) holds because conversion is also limited to raw materials at the beginning of the period; (9) means market limit to the sales by constraint on the standard cost; (10) means sales are also limited because it can not be more than the beginning balance of completed goods; (11) represents the repayment of loans will not exceed the beginning balance of outstanding loans. 12) indicates the limit of accounts payable; (13) is the depreciation charge equation with a rate of 0. 833; (14) indicates the structure of costs to be incurred in the current period, including fixed expenses ($2,675,000), variable cost, effective interest penalty for discounts not taken on accounts payable (at a rate of 3. 09%), and interest on loans (at a periodic rate of 0. 91%); (15) represents income tax is accrued at 52% of net profit and the dividend equals to $83,000 plus(minus) 5% of the excess(shortage) of the expected profit, $1,800,000; (16) is the limit of minimum cash balance required by the company policy; (17) holds because an expected price rise in the next period leads the company to decide the ending inventory should be at least the minimum sales expected in the next period; (18) means ending materials must be sufficient for the production of next period; (19) is the payment limits: all income taxes payable and dividends must be paid by the end of current period.

And because we can considers our goal as maximizing net additions to retained earnings, we have substitute the K’s with figures of balance sheet, which is showed below, we can calculate the X’s As we learned before, a dual evaluator indicates the change in net addition to retained earnings if the constraints corresponding to the given evaluator were relaxed by one dollar. For example, the dual evaluator of (7) is $3. 594936. This means that if production capacity ere increased in case that exactly one additional dollar’s raw material is used, the retain earnings will increase $3. 94936. To see this case in detail, table 5 shows what happens after altering the firm’s raw material processing capacity by one unit. Additional cash can be obtained in 3 ways: a) selling securities; b) borrow from a bank; c) delay payment on account payable. But the cheapest way is a). Thus we can calculate the opportunity cost per dollar by: the firm loses interest income of $0. 00229 of every dollar of securities sold while savings from taxes and dividends can relieve this loss, calculate the periodic loss, it is $0. 00104424. Evaluate this loss from an aspect of infinite periods:

Apply this to the last step of deduction, we get $3. 594936, again. Our former inference is thus confirmed. Not only from the mathematical aspect but also from the accounting aspect. In this case, linear programming offers a highly flexible instrument. As in the case, “all sensitivity changes within any specific part of the model are evaluated in terms of their effect on the entire model. ” It is also highlighted, as we mentioned above, this kind of evaluation can be done without actually solving the entire problem. Thus this method is not only reasonable but also convenient.

Linear Programming in Economics So far we have seen the application of linear programming in the field of finance and accounting. Now let’s see an interesting example which apply linear programming to economics. A linear program can approximate product substitution effects in demand. In general, the demand function may be written as (1) where p is an N * 1 vector of prices, q is an N * 1 vector of quantities, a is an N x 1 vector of constants, and B is an N x N negative semidefinite matrix of demand coefficients. And the objective function for the competitive case can be written as maximize 2) where c(q) is an N * 1 vector of total cost functions, q >= 0, AND Substitute (1) into (2) We have the new objective function Maximize (3) In economics, we know that the total welfare of transactions can be separated into two parts: consumers’ surplus and producers’ profit. In mathematics, these two parts can be written as We also represent the resource scarcity by adding constraints (4) The Kuhn-Tucker conditions, which are necessary (but not su? cient) for a point to be a maximum are: Thus the Kuhn-Tucker necessary conditions for the original problem are equation (4) plus

For monopoly market, the object function is a little different, it is to Maximize (5) while the Kuhn-Tucker necessary conditions are equation (4) plus From the competitive market objective function (3) and the monopoly market objective function (5), we can see that both involve a quadratic form in p. In order to set up the LP tableau, define a function representing the area under the demand curve as (6) And the total expenditure function as (7) Then we can derive the following figure for (6) and (7): The representation of the piecewise linear approximation in LP is shown for the two-good, separable-demands case, in table 1. here costs for the ith product in the jth activity producing it are represented by cij; unit outputs of the ith product in the jth activity producing it are given by yij; The quantities sold of the ith product corresponding to the endpoint of the jth segment are defined as qj; Values of W for the ith commodity corresponding to the amount sold, qj, are given by wij; Values of R for the ith commodity corresponding to the amount sold, qj, are represented by rij; The target level of producer's income is denoted by Y*.

Note that the LP problem has its certain properties. In table 1, no more than two adjacent activities from the set of selling will enter the optimal basis at positive levels. And also, by use of the function R in the constraint set, the model includes a measure of income at endogenous prices. The article then looks into a more complicated case where there exists substitution of demands. That is, one good’s demand can be substituted by the other one’s.

An assumption, as the basis of the approximation procedure developed for this situation, is that commodities can be classified into groups, which allow the marginal rate of substitution (MRS) to be zero between all groups but nonzero and constant within each group. Then consider a group consisting of C commodities. We can create table 3 for the situation: The authors pointed out that “each of the blocks of activities [W's R's -Q's 1] constitutes a set of ‘mixing’ activities for one segment of the composite demand function for the commodity group”. i. e. [W's R's -Q's 1]T=

Relative prices of commodities in the group are assumed fixed, both within and between segments, and are defined by Also define the quantity index as and price index as where Then we create table 4, which is a simple extension of the single product case. Only the selling activities are shown. in which The price-weighted total quantity is (8) To extend the case of demand in fixed proportions within a group, define matrix A as The elements in matrix Q can now be calculated as (9) substitute (8) into (9), we have The price-weighted total quantity, q*sm, is given by so (9) is equal to hen calculate the elements of W and S Now we are able to calculate the MRS By rearranging we get MRS=-p2/p1, the required result. An Expansion The use of linear programming in the field of economics was continued in the paper “Endogenous Input Prices in Linear Programming Models”. In this paper, the author provides a method for formulating linear programming models in which one or more factors have upward sloping supply schedules, and the prices are endogenous. Instead of examining the demand function, Hazell starts from the function of the producers, whose goal is to maximize their profit here x is a vector of output levels; p and c are vectors of market prices and direct costs, respectively; d is a vector of labor requirements; L is the amount of labor employed at wage w. Now if the buyer of labor is monopoly, or the market is a monopsolistic market,due to economic definition we’ll have Then the problem becomes Again we use Kuhn-Tucker conditions to solve for the optimal solution. L;0, so we have = w+? L Thus, given the optimal amount of labor used (L*), the associated market-clearing wage is w* = a + PL*, and this is smaller than ? by PL*.

This is correct by intuition and empirical evidence. Similarly, if the situation is competitive market , we can derive? =w, which is quite different from the former case. Using the method of Duloy and Norton, Hazell calculate the supply curve of labor, which is actually a stepped function, showed as below: Hazell pointed out that “stepped supply functions arise artificially from using linearization techniques, but they also arise in reality when different sources of labor are identifiable which can be expected to enter the labor market as the wage reaches critical levels. And then he also mentioned another way to find out the supply function of labor. This article is a development and application of the former article. The method for achieving these results utilizes the sum of the producers' and consumers' surplus, and is an extension of existing methods for solving price endogenous models of product markets. Linear Programming in Daily Investing Linear programming is such a useful tool that we can find its advantages in finance, accounting and also economics. But what about in our daily life?

How can linear programming help when we make decisions about our own investing, say, our own financial portfolios in various stocks? In 2004, C. Papahristodoulou and E. DotzauerSource wrote an article about these questions, named “Optimal Portfolios Using Linear Programming Models”. This paper is about three models: The classical quadratic programming (QP) formulation and two new ones -- (i) maximin, and (ii) minimization of mean absolute deviation. The first model is to s. t. where i and j are securities; ?ij is the covariance of these securities; xj is the portfolio allocation of security j.

These are the variables of the problem and should not exceed an upper bound uj; ? is the minimum (expected) return required by a particular investor; and B is the total budget that is invested in portfolio. The second model is established so the minimum return is maximized. Regarding the constraints, one might assume that every period's return will be at least equal to Z. For period t, this constraint can be formulated as: where rjt, is the return for security j over period t. The third model simplifies the Markowitz classic formulation is to use the absolute deviation as a risk measure.

It is proved by Konno and Yamazak that “if the return is multivariate normally distributed, the minimization of the mean absolute deviation (MAD) provides similar results as the classical Markowitz formulation”. And as is known, MAD is defined as We define first all Yt >0 variables,t = 1, ... ,T. These Yt variables can be interpreted as linear mappings of the non-linear Thus, the objective function is to minimize the average absolute deviation and the constraints added are Then the author tested all three models, using monthly returns from 67 shares traded in the Stockholm Stock Exchange (SSE), between January 1997 and December 2000.

As expected, the maximin formulation yields the highest return and risk, while the QP formulation provides the lowest risk and return, which also creates the efficient frontier. The minimization of MAD is close to Markowitz. The results are as follows: All three formulations though, outperform the top equity fund portfolios in Sweden. They also conclude, “When the expected returns are confronted with the true ones at the end of a 6-month period, the maximin portfolios seem to be the most robust of all. ” Conclusion We have seen the crucial importance of linear programming to finance, accounting, economics and also our daily life.

It turns difficult problems into easier ones. By using this mathematic way of solving problem, we can achieve more intelligent choices while wasting less. The study of linear programming is so useful that in the future, it will hopefully find more use in the world of economics and management. References “Application of Linear Programming to Financial Budgeting and the Costing of Funds”, A. Chares, W. W. Coopers, and M. H. Millerss, The Journal of Business, Vol. 32, No. 1, Jan. , 1959 (pp. 20-46) “A Linear Programming Model for Budgeting and Financial Planning”, Y. Ijiri, F. K. Levy, and R. C.

Lyon, Journal of Accounting Research, Vol. 1, No. 2, Autumn, 1963, (pp. 198-212) “Prices and Incomes in Linear Programming Models”, John H. Duloy and Roger D. Norton, American Journal of Agricultural Economics, Vol. 57, No. 4, Nov. , 1975 (pp. 591-600) “Endogenous Input Prices in Linear Programming Models”, Peter B. R. Hazell, American Journal of Agricultural Economics, Vol. 61, No. 3, Aug. , 1979 (pp. 476-481) “Optimal Portfolios Using Linear Programming Models”, C. Papahristodoulou and E. Dotzauer, The Journal of the Operational Research Society, Vol. 55, No. 11, Nov. , 2004 (pp. 1169-1177)

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