SOME APPLICATIONS OF FORECASTING Prof. Thomas B. Fomby Department of Economics Southern Methodist University May 2008
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1 SOME APPLCATONS OF FORECASTNG Prof. Thoas B. Foby Departent of Econoics Southern Methodist University May 8 To deonstrate the usefulness of forecasting ethods this note discusses four applications of forecasting in the world of business and personal investents: Supply Chain Manageent, nventory Control, Budgeting in Governent, and Personal nvestents.. Supply Chain Manageent The supply chain anager of a anufacturing copany is in charge of aking sure that the necessary productive resources (capital, labor, coponent parts and the like) are always available for producing the required output deanded of the copany by its custoers. So the supply chain anager ust deterine a good forecast of the deand for the anufacturing copany s output and fro that estiate deterine the necessary resources to produce the forecasted aount of output. One way to view this need for supply chain forecasts is to envision the anufacturing copany producing output according to a Fixed Proportions production function. Let Q denote the units of output produced by the anufacturing copany and capital (K) and labor (L) being the input to the copanies production process. A Fixed Proportions production function is of the for K L Q in, () K L where ( K, L ) is the fixed proportion of capital and labor required to produce one unit of output. For illustration, let ( K, L ) (, ). Then the Fixed Proportion production function with the iplied activity ray and isoquants is depicted in the figure below:
2 Fixed Proportion Production Function K Activity Ray 4 Q= Q= L (K, L ) = (, ) The activity ray has a slope of two and represents the inial cobinations of capital and labor (in a two-to-one ratio) that will produce the various levels of output. The perpendicular isoquants for Q = and Q = are drawn in the figure and depict the fixed proportions relationship iplied by the forulation (). Assue you want to produce Q =, then you should use the input vector (,) (4,) that is K = 4 and L =. Using the cobination K = 5 and L = will still produce units of output because of the fixed proportion requireent. Notice that the Fixed Proportions production function exhibits constant returns to scale. f you double the inputs (in proportion of course), you double the output. For exaple, if then K L Q in, () K L K L Q Q in,. () K L Therefore, in this siple case (which ay be approxiately true for any copanies production processes in the short run), we see that a forecast of the deand for output Q, say, Qˆ, will iediately give rise to forecasts of K and L, say Kˆ and Lˆ, such that
3 ˆ ˆ ˆ K L Q in, (4) K L where Kˆ and Lˆ are inially sufficient to produce Qˆ. This situation is then ideal for the supply chain anager. She first forecasts the deand for the anufacturer s product Qˆ and then, using the Fixed Production function s relationship (4), she backs out the input forecasts Kˆ and Lˆ required to produce the forecasted output. Thus, she anages the supply chain. The greater the accuracy of her forecasts of Q, K, and L, the greater the profit of the copany. Of course, this discussion can be extended to ultiple inputs k-diensional Fixed Proportions production function,,, K and the K Q in,, (5) () () K () where,,, ) denotes the input proportion of the k inputs that gives rise to ( () () K () one unit of output. Again, given a forecast of Q, corresponding forecasts of the k inputs can be produced. Such is the supply chain proble in its crudest for.. nventory Control f instead of anaging production, you are an owner of a retail store, you are likely interested in anaging your inventory efficiently as part of your plans to axiize your profits. During the onth you wish to have plenty of your product lines on hand for custoers to consider and buy but, on the other hand, you don t want to have too uch inventory on hand because you then would have too uch of your oney (capital) tied up in idle inventory that otherwise could be drawing interest in a bank account soewhere. This is called the inventory control proble and you are an inventory controller. Suppose that for your copany you deterine that your optial inventory of your products, y, y,, y, given deand for the products of y, y,, y is deterined by the following optial inventory-to-sales ratios: ( ) y SR opt ( ) y, SR opt,, y y SR ( opt) y. () y Then given that you perceive your product deand for the next sales period to be
4 ,,,, (7) Your estiated optial inventory for your products, say, (8), ˆ,, y can be deterined fro the optial sales ratios as in ( ) SR opt ( opt) or SR ˆ y ( ) SR opt ( opt) or SR ˆ y () SR ( opt) ( opt) or SR. That is, we have established an optial inventory level for the next sales period for our products, naely, ˆ,,, y, based on our product-line forecasts,,,,. Obviously, the ore accurate our product line forecasts are, the ore efficient our inventory levels well be thus increasing the botto line profit of the copany. (opt) Of course, one ight say, How are the optial inventory-to-sales ratios, SR i, deterined? One way to deterine these ratios is by a trial and error syste which sees what inventory-to-sales ratios for each of the product lines leads to a iniu of leftover unsold product at the end of the sales period.. The City Manager Proble But forecasting is not only useful in anufacturing operations (the Fixed Proportions production proble) and optial inventory control in retail stores (inventoryto-sales ratio proble) but also for public sector jobs like those of city anagers anaging the budgets of cities. Consider the City Manager proble. The fiscal budgets of city governents are often deterined by the aount of tax revenue that the city governent hopes to garner in the coing year. Consider the following graph of Plano s sales tax revenues fro February to Noveber 5. 4
5 By Mo n t h Ye a r How is one to forecast the total of sales tax revenue expected for Plano in the following year? One a forecast of the next-year s revenues is produced, the city anager then oves to establish the budgets of the various departents of the city governent possibly using a proportional assignent of additional venues iplied by the forecasts. Again, the ore accurate the forecast of the City s next year revenues, the ore accurate the budget setting process will be and thus the less the possible isallocation of resources in the City governent over the coing year. V. Personal nvestents: The Bond Market Now let s consider a personal investent proble. As we know fro beginning finance courses, the prices of bonds vary inversely with the level of interest rates. n fact the price of a bond that pays $R dollars per year in perpetuity at a rate of the annual interest rate r is priced at P $R. () r Consider the following plot of annual averages of onth t-bill rates fro through 7 obtained fro the 8 Econoic Report of the President. f one can identify the peaks of the interest rate cycles and invest long in bonds and, when troughs of interest rate cycles coe about, invest short in interest sensitive assets that don t do well in rising interest rate regies, then one s investent perforance will likely iprove as a result. 5
6 Annual -Mo. T-bill Rates Ye a r
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