James R. Martin, Ph.D., CMA
Professor Emeritus, University of South Florida
Citation: Martin, J. R. Not dated. Chapter 12 Appendix: Absorption Costing Profit Functions When the Fixed Overhead Rate Changes. Management Accounting: Concepts, Techniques & Controversial Issues. Management And Accounting Web. https://maaw.info /Chapter12Appendix.htm
This Appendix shows how a fixed overhead rate change effect can offset an inventory change effect to cause exceptions to the usual direct versus absorption costing generalizations related to net income.
If the fixed overhead rate changes, the applicable profit function depends on the cost flow assumption chosen. Two separate profit functions are provided below to illustrate the concepts involved. First, an equation is developed for the FIFO assumption, and then two equations are developed for the LIFO assumption.
When the fixed overhead rate changes and FIFO is used, two rates must be included in the equation, i.e., the old rate and the new rate. In addition, the units in the beginning inventory must be kept separate from the other units that are sold during the period. The absorption costing profit function, based on a FIFO cost flow, may be stated in the following manner.
 NIA = -TFC + (P-V)(XS) - F0(BFG) - F1(XS-BFG) + F1(XP)
F0 = Previous period fixed overhead rate
F1 = Current period fixed overhead rate
BFG = Beginning finished goods in units.
If XS<BFG, substitute XS in each place BFG appears in the equation.
Observe that the adjustment for the inventory change is stated in three parts in Equation 10. The term we used in the previous equations, i.e., -F(XS-XP) is replaced by -F0(BFG) - F1(XS-BFG) + F1(XP). The term -F0(BFG) represents the prior period fixed overhead costs included in the beginning inventory that are charged to cost of goods sold. The next term, -F1(XS-BFG), represents the amount of the current period's fixed overhead costs that is charged to cost of goods sold. The last term represents the current period fixed overhead costs that were charged to the inventory. Equation 10 may be simplified by combining the last three terms and substituting NID for -TFC + (P-V)(XS). The result is Equation 11 which is easier to work with.
 NIA = NID - (F0-F1)(BFG) - F1(XS-XP)
The term - (F0-F1)(BFG) represents the effect of a fixed overhead rate change on absorption costing net income. The last term, -F1(XS-XP), represents the effect of an inventory change on absorption costing net income. Equation 11 shows that if there is no change in the fixed overhead rate, then F0 is equal to F1 and the term (F0-F1)(BFG) drops out of the equation. Then Equation 11 would provide the same results as Equations 2 and 3 presented earlier in this chapter. However, if the fixed overhead rate changes, the units in the beginning inventory will be charged into cost of goods sold at the previous period overhead rate. The usual direct/absorption costing generalizations are valid as long as the absolute amount of the first term (F0-F1)(BFG) is less than the absolute amount of the second term F1(XS-XP). The generalizations are not valid when the absolute amount of the first term is equal to, or greater than, the absolute amount of the second term. What happens is that the effect of the fixed overhead rate change offsets the effect of the inventory change. These situations will be illustrated with an example after we examine the profit functions needed for the LIFO cost flow assumption.
Before moving on to the next section, one additional point should be considered. Equations 10 and 11 are based on the assumption that the number of units sold (XS) are greater than the number of units in beginning finished goods (BFG). However, if XS<BFG, then both equations need to be altered by replacing BFG with XS.
When the LIFO cost flow assumption is used, two separate profit functions are needed. One function is applicable when unit sales are equal to, or less than units produced. Another profit function is applicable where unit sales are greater than units produced. If unit sales are equal to, or less than, units produced, then the Equation may be written as follows:
 NIA* = NID - F1(XS-XP)*
Where XS<XPThe term F1(XS-XP) drops out of the equation when XS=XP and represents the current period fixed overhead costs that are capitalized in the inventory when XS<XP. Equation 12 is equivalent to Equation 3, since only the current fixed overhead rate is involved.When unit sales are greater than units produced, the appropriate equation must include the prior period rate, or if more than one inventory layer is involved, a weighted average prior period rate. The applicable equation is:
 NIA** = NID - FA(XS-XP)**
Where XS>XPFA = Weighted average prior period fixed overhead rate based onthe number of units sold from each inventory layer involved.When the number of units sold exceeds the number of units produced, some of the units produced in a prior period must be sold in the current period. If the extra units (XP-XS) are equal to, or less than, the number of units in the most recent prior period inventory layer, then FA would equal F0. However, if the number of extra units exceeds the number of units in the most recent prior period inventory layer, then FA represents a weighted average fixed overhead rate based on the number of units sold from each of the prior period inventory layers involved. Thus, the adjustment FA(XS-XP) represents the amount of prior period fixed overhead costs that is charged to cost of goods sold. For all practical purposes, the direct/absorption costing generalizations are always valid under the LIFO cost flow assumption. An exception could only occur if the fixed overhead costs were equal to zero.
WHEN DO EXCEPTIONS TO THE GENERALIZATIONS OCCUR?
The purpose of this section is to make you aware that exceptions to the usual direct versus absorption costing generalizations occur under certain circumstances. There are three variables that determine whether the usual direct versus absorption costing generalizations are applicable in a given situation.
First, for exceptions to occur, the fixed overhead rate must change from one period to the next. The rate will change when either the numerator or the denominator of the rate calculation changes. The numerator changes when the amount of budgeted fixed costs increases. The denominator changes when additional capacity is added or when planned production is used as the denominator.
The second requirement for an exception to occur is that either the average cost or FIFO cost flow assumptions must be chosen. We will restrict our discussion to FIFO, but similar results occur when the average cost assumption is used.
The third requirement involves the rate of growth or decline in unit sales and production. Assuming production is based on sales, (i.e., units sales + desired ending finished goods - beginning finished goods) the exceptions occur when unit sales increase or decrease at either a constant rate or a decreasing rate. The following example illustrates what happens when unit sales increase at a decreasing rate.
Assume that a manufacturer produces a single product and the following budgeted data are applicable to the next five years. Although it is not realistic to assume that prices will be constant for a five year period, this allows us to isolate the effects of the inventory changes and the fixed overhead rate changes.
|Data for Example 12-3||$|
|Variable manufacturing cost per unit||1.00|
|Variable selling and administrative expenses per unit||1.00|
|Total fixed manufacturing costs per period||120,000|
|Total fixed selling and administrative costs per period||30,000|
Overhead rates are based on planned production. A summary of budgeted unit sales and production, as well as the fixed overhead rates, are provided below.
To Be Produced* XP
Based on desired ending inventories of 20% of next year's sales.
The percentage increases in unit sales are 150, 100, 50, 32.2, and 10 for the six year period. Net income comparisons are provided below assuming the firm uses planned production as the denominator for overhead rate calculations.
Assumption not relevant
According to the usual generalizations, absorption costing net income should exceed direct costing net income since the units produced are greater than the units sold for each year. Comparing absorption costing net income when LIFO is used, with direct costing net income indicates that the generalizations are valid for every year in the illustration. However, comparing absorption costing net income when FIFO is used, with direct costing net income reveals that exceptions occur in years 2 through 5. Equation 11 can be used to illustrate why the exceptions occurred. First, examine the equation for year 1.
NIA = NID - (F0-F1)(BFG) - F1(XS-XP)
= -120,000 - (0-8)(0) - 8(10,000-15,000)
= -120,000 + 40,000
The generalization that absorption costing net income will be greater than direct costing net income is valid since there is only one fixed overhead rate involved. This is because there is no beginning inventory in the first year. The difference in net income is solely a result of the inventory increase. This caused $40,000 of the current period fixed overhead costs to remain in ending finished goods under absorption costing. Now, examine the equation for year 2.
NIA = -75,000 - (8-4)(5,000) - 4(25,000-30,000)
= -75,000 - 20,000 + 20,000
The effect of the fixed overhead rate change exactly offset the effect of the inventory change. The rate change caused the 5,000 units in BFG at $8 per unit to be replaced by 5,000 units at $4 per unit. Thus, using FIFO, $20,000 of prior period fixed overhead costs are charged into cost of goods sold. However, since the inventory increased, $20,000 of the current period fixed overhead costs remains in the ending finished goods inventory. Now, examine the equation for year 3.
NIA = 0 - (4-2.1818)(10,000) - 2.1818(50,000-55,000)
= 0 - 18,182 + 10,909
= - 7,273
The effect of the fixed overhead rate change (-18,182) more than offset the effect of the inventory change (10,909). The decrease in the fixed overhead rate caused the 10,000 units in BFG at $4 fixed overhead per unit to be replaced by 10,000 units at $2.1818 per unit. Thus, $18,182 of prior period fixed costs were charged into cost of goods sold. This effect was partially offset by the inventory increase which caused $10,090 of the current period fixed overhead cost to remain in the ending finished goods inventory.The net income differences for years 4 and 5 can be reconciled in the same manner. The requirements for Problem 12-7 include a reconciliation for these two years.
1 This appendix is based on the following article: Martin, J. R. 1992. How the effects of company growth can reverse the LIFO/FIFO decision: A possible explanation for why many firms continue to use FIFO. Advances In Management Accounting (1): 207-232.