Linear Programming in Finance, Accounting and Economics

Linear Programming in Finance, Accounting and Economics Sijia Lu 7289928683 Abstract This article is literatures review ab let let out five articles, which apply bilinear schedule to Finance, enumerateing and political economy. The mathematical method acting is found of crucial grandness in those scene of actions. The composition shows how abstractive inference in linear programme throws light upon realistic practice, and how empirical evidence supports those theories. Keywords finance be political economy linear schedule investment analysis Linear Programming in Finance Application of Linear Programming to Financial Bud astounding and the Costing of bullion explored how to anyocate funds in an disgraceprise by applying linear programming. As Charnes, Cooper and milling machine analyzed, at least three line of works argon to be considered to solve the allocation problem 1) Plans for production, purchases, and sales under certain coordinate of the inviolables assets, in order to maximize its profit or reach other objects. 2) The change of the firms profit per social unit change in the structure of the assets. 3) Opportunity cost of the firms funds.The article starts with a unprejudiced congressman with champion goodness and matchless store. Let B be the unflinching w arhouse qualification, A be the initial stock of inventory in the wargonhouse, xj be the sum of money to be exchange in achievement j, yj be the amount to be sold in period j, pj be the sales cost per unit in period j, and cj be the purchase set per unit in period j, t hen we have cod(p) to the cumulative sales constraint due to the w arehouse cleverness constraint due to the buying constraint due to the selling constraint and with our goal of maximizing The threefold problem is overly unadorned.It is to minimize subject to and to where As we l earn, triple theorem of linear programming says that the both optimal values of the original problem and the dual probl em should be equal. Using this theorem, the causalitys then reached a new method of evaluating assets. Because , we have in which the cardinal sides must have the said(prenominal) 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 salarys and j-r represents receipts, M0 is the initial change available and M is the balance the firm desired the maintain. By examining the dual problem of this, we pot distinguish equal dual variables for the problem called, say, vk. Again, from the equating we found before, we tin fucking learn that the two sides of the equality have the same units of measure. It is then dealn that the vs should be sawhorses per unit time per dollar invested. The valuation of assets or investments is of crucial importance to any bus iness.So far, by undefiledly 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 ingenious to examine the units of measure rather than try to solve the specific problems. The kindle thing is that in realistic problems, we nookie find true meanings of theoretical dual variables. then(prenominal) the authors mixed the two former problems together to see a more(prenominal) realistic slip of paper a warehouse problem with financial constraints.So the following new constraints are added no(prenominal) if we check Well get the new dual problem Here, V1 is the incremental cumulative internal give back rate. Or it is the probability cost the capital invested it shows the net amount to which an special dollar invested in the firm pass oning accumulate if left to get to the depot of the planning horizon. This is also easy to understand in harm of economics, maximizing profit lot 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 pertaining to find out that all the commodities are aspirely linked to the funds-flow while the goods-flow kindle be avoided in the warehouse problem with multiple commodities. An Example A linear programming mildew for budgeting and financial planning created an accounting experiment in which the dual variables introduced before were numberd which can also be considered as a sensitivity analysis. This can be seen as coating and verification of Charnes, Cooper and Millers earlier theory. In the linear programming problem listed below, (1) represents the interests earned with a rate of 0. 29% (2) holds because firms sale of securities will non be more than the beginning balance of this amount (3) represents the upper circumscribe collection of receivables will not exceed the beginn ing balance of account receivable (4) substance the initial cash balance constraints the purchase of securities (5) indicates region on a unit sale per unit deduction from the closing 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 ram cost and direct overhead) (7) represents the production capacity terminations by limiting the value of raw materials (8) holds because conversion is also control to raw materials at the beginning of the period (9) core 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 disparagement charge equation with a rate o f 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 punishment 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 accumulated 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 lower limit cash balance required by the confederation insurance policy (17) holds because an expected price rise in the next period leads the company to decide the shutdown 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 kep t up(p) sugar, we have patronage the Ks with figures of balance sheet, which is showed below, we can guide the Xs As we learned before, a dual judge indicates the change in net addition to retained earnings if the constraints equivalent 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 result that exactly one additional dollars 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 firms raw material treat capacity by one unit. Additional cash can be obtained in 3 styles a) selling securities b) borrow from a beach c) delay payment on account payable. But the cheapest way is a). hencely we can suppose 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, cal culate the periodic loss, it is $0. 00104424. Evaluate this loss from an aspect of endless periodsApply this to the last step of deduction, we get $3. 594936, again. Our former inference is thus confirmed. Not just 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 separate of the perplex 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 through with(p) without actually solving the entire problem. and so 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 lets see an interesting example which apply linear programming to economics. A linear program can approximate product substitution effects in gather up. In ge neral, the guide business office 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 pass on for for the competitive case can be written as maximize 2) where c(q) is an N * 1 vector of total cost blend ins, q = 0, AND Substitute (1) into (2) We have the new objective function Maximize (3) In economics, we know that the total welfare of legal proceeding 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) plusFor 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 equation 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 do the following figure for (6) and (7) The representation of the piecewise linear idea 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 section are define 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 producers 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 goods demand can be substituted by the other ones.An assumption, as the basis of the approximation procedure developed for this situation, is that commodities can be classified into convocations, which allow the borderline rate of substitution (MRS) to be zero between all crowds 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 Ws Rs -Qs 1 constitutes a set of mixing activities for one segment of the composite plant demand function for the commodity group. i. e. Ws Rs -Qs 1T=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 endogenetic 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 comprehend requirements L is the amount of labor employed at rent w. Now if the buyer of labor is monopoly, or the market is a monopsolistic market,due to economic definition well have Then the problem becomes Again we use Kuhn-Tucker conditions to solve for the optimal solution. L0, 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 tame 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 come 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 investment 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 mak e 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) homework and two new ones (i) maximin, and (ii) minimisation 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) run required by a particular investor and B is the total budget that is invested in portfolio. The scrap model is established so the minimum return is maximized. Regarding the constraints, one major power assume that every periods return will be at least equal to Z. For period t, this constraint can be formulated as where rjt, is the retu rn 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 minimisation 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 Mar kowitz. The results are as follows All three formulations though, scale 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 or so 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 waste 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 diary of Business, Vol. 32, No. 1, Jan. , 1959 (pp. 20-46) A Linear Programming Model for Budgeting and Financial planning, Y. Ijiri, F. K. Lev y, and R. C.Lyon, Journal of Accounting enquiry, Vol. 1, No. 2, Autumn, 1963, (pp. 198-212) Prices and Incomes in Linear Programming Models, backside 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)