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 THE ROLE OF COST IN PRICING JOINT PRODUCTS: 
 A CASE OF PRODUCTION IN FIXED PROPORTIONS 
 
 Daniel L. Jensen 
 
 #133 
 
 College of Commerce and Business Administration 
 
 University of Illinois at Urbana-Champaign 
 

FACULTY WORKING PAPERS 
 College of Commerce and Business Administration 
 University of Illinois at Urbana-Champaign 
 September 11, 1973 
 
 THE ROLE OF COST IN PRICING JOINT PRODUCTS: 
 A CASE OF PRODUCTION IN FIXED PROPORTIONS 
 
 Daniel L. Jensen 
 
 #133 
 
September, 1973 
 
 THE ROLE OF COST III PRICING JOINT PRODUCTS: 
 A CASE OP PRODUCTION IN FIXED PROPORTIONS 
 
 Daniel L. Jensen 
 University of Illinois at Urbana-Champaign 
 
 Introduction 
 
 The purpose of this paper is twofold: first, to examine 
 the information requirements for the pricing of joint products 
 arising in fixed proportions, and second, to identify the 
 accounting procedures that will meet those requirements . The 
 analysis is based on a deterministic model of the short-run 
 price-output decision for two complementary joint products 
 produced in fixed proportions. Consequently, the results of 
 the analysis do not extend to cases of variable proportions 
 or to other cases in which such a model is invalid. Moreover, 
 the analysis focuses on the price-output decision and does 
 not extend to problems of inventory valuation or income 
 determination. 
 
 When cost is joint in relation to two outputs produced 
 in fixed proportions, economists argue that any allocation 
 of such cost between the products must be arbitrary unless 
 determined by reference to demand functions. In other words, 
 
2 
 
 an allocation lacks economic justification unless it reflects 
 marginal revenues under the optimal price policy. Moreover, 
 since the demand-based allocation of the joint cost derives 
 from the optimal price policy, it cannot be used in reaching 
 the optimal price policy. One might be tempted to conclude 
 that accountants serve no economic purpose by assigning any 
 joint cost to products produced in fixed proportions. Such 
 a conclusion may be warranted when profit is maximized by 
 selling the entire output after separate processing; in. this 
 case, the demand-based allocation must be derived from the 
 optimal price policy and, hence, cannot be uned in reaching 
 that policy. But when profit maximization requires the sale 
 or disposition of some production at split-off, then the 
 demand-based allocation assigns the entire joint cost to the 
 other product, all of which is separately processed. In this 
 case the decision process may benefit by an accounting policy 
 that resists assigning any joint cost to such production. 
 More specifically, v/hen demand information is incomplete such 
 that split-off sales are known to be necessary but the optimal 
 prices are unknov/n, then such an accounting policy may be 
 useful in guiding experimentation with price-output possibilities 
 in an effort to maximize profit. In this way, the economic 
 analysis provides a limited justification for the by-product 
 method of accounting for joint production. 
 
 When a joint product is marketed partly at split-off, the 
 marginal cost of the separately processed portion is required 
 
3 
 
 in order to determine its optimal price and sales level. 
 Consequently, an accounting policy that makes that marginal 
 cost available will facilitate profit maximization with 
 respect to such production. Although economists and account- 
 ants have recognized that the marginal cost of separately 
 processed joint production (part of which is sold at split- 
 off) is required in order to determine its profit maximizing 
 price and sales level, the form of the required marginal cost 
 has not been clearly specified for all cases of interest. 
 The analysis below demonstrates that the appropriate form of 
 the required marginal cost depends on the assumptions made 
 concerning separate processing and disposition at split-off. 
 
 Formulation of the Maximization Problem 
 
 If two products are produced under fixed proportions in 
 quantities x-, and Xp, then the ratio of the outputs must be 
 constant for all levels of production, that is, x., = ^' x o» 
 If the production process is subject to an upper limit on 
 capacity, K, then two constraints are required to represent 
 the relations of production. Let us suppose that x., may not 
 exceed (a. *K) and that Xp may not exceed (ap«K) where a-, and 
 ap give the production of the two products per unit of 
 
 capacity. The upper limits on production of the two joint 
 
 p 
 
 products may be written as follows: 
 
 x l = a l * K 
 U) *2 = a 2 ' E 
 
In a typical cost accounting problem, we consider batch pro- 
 duction in this way, K units of material are introduced to 
 begin processing the batch and ^ach unit of material input 
 yields a, units of the first product and a 2 units of the second. 
 
 A distinction is made between (i) production of a joint 
 product that is sold (or scrapped) at the split-off point 
 and (ii) production of a joint product that is sold after 
 further processing. If y 1 and y 2 represent separately 
 processed joint production and u-, and Up represent production 
 sold (or scrapped) at the split-off point, then the capacity 
 constraints may be written as follows? 
 
 *1 + U l = a l K 
 (2) 
 
 y 2 + u 2 = a 2 K 
 
 Provided that capacity^ K, can be readily adjusted 9 only one 
 of the two products will be sold at the split-off point; this 
 result will be demonstrated below. A diagrammatic represent- 
 ation of these relationships and definitions is given in Figure 1. 
 Unprocessed joint production is assumed to be sold in a perfectly 
 competitive market at a constant unit contribution, s, , given 
 
 by the difference between a constant unit selling price and a 
 
 3 
 
 constant unit selling cost. The total contribution from sales 
 
 of unprocessed production is given by: 
 (5) s^ + s 2 u 2 
 
 where s-, and s 2 are the constant unit contributions and u« 
 and u 2 are the quantities of unprocessed joint production. 
 
(b-K) 
 
 (B 1 .U 1 ) 
 
 (s 2 -u 2 ) 
 
 ■^ y* 
 
 y x + U]L = Xl = a x K 
 
 y ? + u 2 = x 2 = a^K 
 
 2 
 
 u, 
 
 FIGURE 1. —Joint Production Problem 
 
On the other hand, separately processed production is 
 assumed to be sold in an imperfectly competitive market in 
 which the quantity sold is inversely related to the unit 
 price. Since the objective of this analysis is to determine 
 a profit-maximizing price policy, revenue from the separately 
 
 processed joint production is written as a function of price 
 
 5 
 as follows: 
 
 (4) p l' D l^ p l^ + p 2 t:D 2^ p 2^ 
 
 where p. is the price of the separately processed ith product 
 and D. (p.) is its demand function* In order to simplify the 
 notation, the demand function will hereafter be written D. . 
 
 The cost function recognizes two kinds of unit cost: 
 (i) unit costs of processing the separate products beyond 
 the split-off point, c-, and Cp? and (ii) a unit cost of 
 batch capacity, b. Since revenue is given as a function of 
 price, cost is also given as the following function of price: 
 
 (5) c^ % + c 2 D 2 ^ ^ K 
 
 The first two terms are interpreted as the cost of processing 
 the two products beyond the split-off point. The final term, 
 bK, gives the cost which varies with the size of the batch, 
 
 the capacity cost exemplified here by the cost of the jointly 
 
 7 
 processed material input. 
 
By combining the revenue and cost functions given by 
 equations (3), (4), and (5) and the joint production constraints 
 given by equations (2), a general programming problem may be 
 formulated as follows: 
 
 MAXIMIZE ^ = D + p D _ e A - o^ + s^ ♦ a 2 » 2 - *K 
 vPl»P2» 1* 2' ' 
 
 SUBJECT TO: ^l + u l = a l K 
 
 $2 + u 2 ~ a 2 K 
 
 p-p P 2 » u it u 2 , K > 
 
 The problem is to obtain the prices ( ?1 and p 2 ), the quantities 
 of unprocessed joint production (^ and u g ) , and the batch 
 size (K), that maximize profit on a production process issuing 
 two joint products in fixed proportions where each product may 
 be processed beyond the split-off point. Production subject to 
 separate processing is sold in an imperfectly competitive market. 8 
 Production subject to disposition at the split-off point is sold 
 in a perfectly competitive market to yield a unit contribution 
 of s. dollars for the ith product. 
 
 In order to derive necessary conditions for a solution to the 
 maximization problem above, we write the Lagrangian form as 
 
 follows: 
 
 (6) L = Tf - ^ 1 (B 1 + u x - a x K) - ^0 2 + * 2 - a 2 K) 
 
 where Lisa function of p lf p 2 , i^, u 2 , and K. The dual 
 
variables, $-, and ?L* ma y ^ e interpreted as the marginal 
 opportunity costs of the two products. Moreover, the dual 
 variables imply an assignment of joint cost between the 
 products as will be shown below. 
 
 The necessary conditions^ for a solution to the maximization 
 problem are as follows t 
 ( 7 ) pj d£ + DJ ~ c ± dj - i x d* 4 
 
 (8) 
 
 pj ( P * d* + DJ - c x dj - ^ dj) - 
 
 
 
 ( 9) 
 
 Pf >, 
 
 
 (10) 
 
 p| d* + D* - c 2 dg - ^ d* ^ 
 
 
 (11) 
 
 p 2 ^2 d l * D 2 ~ ^2 d 2 "" ^2 d 2^ ~ 
 
 
 
 (12) 
 
 P| > o 
 
 
 (15) 
 
 s l "" **1 ^ ° 
 
 
 (14) 
 
 uj ( s 1 - ^) = 
 
 
 (15) 
 
 u* ^ 
 
 
 (16) 
 
 S « *~ © o c*» KJ 
 
 
 (17) 
 
 . u£ ( s 2 - jzJp) - 
 
 
 (18) 
 
 u* > 
 
 
 (19) 
 
 ~ b + a l ^1 * a 2 ^2 ^ ° 
 
 
 (20) 
 
 K* ( - b + a, *$., + a ? jO = 
 
 
 (21) 
 
 K* \ 
 
 
8 
 
 Variables in the conditions above carrying a superscript 
 asterisk (*) are optimal values, that is, the prices, disposal 
 quantities and capacity that maximize profit* The symbol d. is 
 interpreted as the derivative of demand with respect to price, 
 that is 9 
 
 d. =s dD i / dp., 
 
 and d* is that derivative evaluated at the optimal price, pf. 
 
 Optimal Price 
 
 Our primary purpose is to obtain a characterization of 
 the optimal prices in terms of cost and demand parameters to 
 enable recommendations for the assignment of costs. To this 
 end, let us assume that price is strictly greater than zero; 
 then conditions ( 8) and (11 ) yield the following expression 
 for the optimal price: 
 (22) pf = -D* / d* + c ± + i v i - 1,2. 
 
 This equation says that the optimal price must be the sura 
 of the separate processing cost, c., and the marginal 
 opportunity cost of the ith product, 0., plus the term (-D|/d|) 
 which derives from the demand function. Since D* is positive 
 and d* is negative, the term (-D*/df ) is positive, In other 
 words, the optimal price is the sum of a positive demand 
 component and the marginal cost (c. + t/) m 
 
9. 
 If we denote the price elasticity of the ith product by 
 
 N. where N. is defined as -d.p./B. (which is greater than 
 
 1 1 J. 1 -L 
 
 zero provided d s is negative) and if we let Nf he the elasticity 
 evaluated at the optimal price, then equation {22} can be 
 rewritten as follows: 
 
 Inspection of equation (23) shows (i) that (c, + ^. ) is 
 
 positive if and only if price elasticity, NJ, is strictly 
 
 greater than unity; (ii) that (c i + i*) is zero if and only 
 
 if price elasticity equals unity; and (iii) that (c. + £/) 
 
 is negative if and only if price elasticity is strictly less 
 than unity. This relationship between price elasticity and 
 
 marginal cost Is used to characterize the optimal output policy 
 
 later in the paper* 
 
 In addition, marginal cost must equal marginal revenue 
 
 at the optimal price. Prom the theory of demand, we know that 
 
 price (p), marginal revenue (MR), and price elasticity (N) are 
 
 related by the equation: 
 
 (24) (N - 1) p = N (MR) 
 
 v/here p is any price and demand is diff erentiable and has an 
 
 inverse. Since optiraality requires that both (23) and (24) 
 
 be satisfied, the optimal price, p|, occurs at the equality 
 
 of marginal revenue and marginal cost, that is, MR = (cj + £*)• 
 
10 
 
 As noted above, the marginal opportunity costs, £* and 
 ^p, imply an allocation of joint cost. This may be demonstrated 
 by reference to equation (20). If batch capacity, K, is greater 
 than zero, then condition (20) requires that the marginal Capacity 
 cost, b, equal the sum of the marginal opportunity costs weighted 
 by the input-output coefficients, a., and a OJ that is, 
 
 (25) b ~ a l ^1 + a 2 ^2 
 
 This means that the unit capacity cost, b s must be completely 
 allocated between the two joint products. In other words, $>. 
 may be interpreted as the share of b assigned to the ith product 
 under the optimal pricing policy. 
 
 Further discussion of the implied allocation is facilitated 
 by a distinction between two characterizations of the optimal 
 solution which we shall call Case I and Case II. In Case I, 
 the optimal price-output policy requires disposition or sale 
 
 of production both at the split-off point and after separate 
 
 11 
 processing. In Case I±, the optimal policy requires that the 
 
 entire production of both products be marketed after separate 
 
 12 
 processing. 
 
 In general the marginal opportunity costs are found by 
 
 solving the maximization problem. If Case II characterizes 
 
 the optimal solution, then the marginal opportunity costs will 
 
 be given in terms of both cost and demand parameters. But if 
 
 Case I characterizes the solution, then the marginal opportunity 
 
 costs are given in terms of cost parameters alone. In particular, 
 
11 
 
 the entire joint process cost is assigned to the product 
 whose whole production is separately processed, Weil argues 
 that the marginal opportunity costs lead to a proper and use- 
 ful allocation of joint processing cost." it how can they "be 
 useful in reaching the optimal price-output policy if they 
 emerge simultaneously with it? Only if we know that Case I 
 obtains but do not know the optimal prices will the marginal 
 opportunity costs be useful in reaching the optimum* But, 
 
 as Weil points out, the marginal opportunity costs may assist 
 
 14 
 in reaching other decisions. 
 
 Optimal Prices Under Case _I 
 
 Pricing a Join t ?r o d uct Marketed Bo th B e i c re an d Aft er 
 Seoarr-t e Processings If some of the first product is sold 
 at the split-off point, a characterization of its marginal 
 
 opportunity cost, £ , may be obtained from equation (14). 
 
 j. 
 
 If uj > 0, then jzL = s,, In words, when some of the 
 first product is sola at sp >ff , the marginal opportunity 
 cost is equal to the marginal contribution of sales at the 
 split-off point. This means that expansion of the separately 
 processed quantity reduces the quantity sold at the split-off 
 point and thereby sacrifices contribution at a rate of s^ dollars 
 per unit. 
 
12 
 
 Substituting s n for ^-. in equation (22) yields the 
 
 -L J. 
 
 following expression for the optimal price of the separately 
 processed production: 
 
 (26) uf > implies that - -Dj/d^ + c 1 + s, 
 
 that is, the price of separately processed production must 
 provide for the recovery of "both the unit cost of separate 
 processing and the unit contribution foregone on sales at the 
 split-off point. 
 
 The optimal price may be interpreted in terms of equation 
 (23) to yield the relationship between the marginal cost (c-, + s,) 
 and the demand elasticity at which the separately processed 
 production should be marketed* If the marginal cost is positive, 
 then the separately processed nroduction should be marketed at 
 demand elasticity greater than unity. If, on the other hand, 
 the unit contribution on sales at the split-off point is 
 negative and sufficiently large in absolute value to make the 
 marginal cost negative, then the separately processed production 
 should be marketed at demand elasticity less than unity. In 
 this case, the marginal cost is the cost of separately processing 
 an additional unit less the disposition cost (plus the negative 
 contribution) avoided by doing so. In the special case that 
 the marginal cost equals zero, the separately processed production 
 is marketed at the point of unitary demand elasticity* 
 
13 
 
 These conclusions differ from those reached on the MS model, 
 
 but they are not inconsistent with them, Manes and Smith conclude 
 that "disposal of units of a product cannot yield maximum profit 
 unless the product is marketed at the point of unitary demand 
 elasticity, i.e., where marginal revenue = (or where marginal 
 revenue = marginal cost of processing and selling this item 
 alone) # "l^ The conclusion that sales of separately processed 
 production should be expanded until marginal cost equals marginal 
 revenue is correct. But marginal cost does not always equal 
 zero and equality does not always occur at unitary demand elasticity. 
 Consequently, revenue from separately processed production may be 
 falling, rising or stable at the optimal price. ^ 
 
 In summary, when some production of a joint product is 
 sold at the split-off point, the price of production sold 
 after separate processing is given by the sum of (i) a positive 
 demand component (-Bf/dj), (ii) the unit cost of separate pro- 
 cessing (c i ), and (iii) the unit contribution of sales at the 
 split-off point (s.). The separately processed production may 
 be marketed at demand elasticity greater than, equal to, or 
 less than unity depending on whether (c- + s.) is positive, 
 zero, or negative. 
 
 Pricing a Joint Product Marketed Only After .Separate 
 Processing , We turn now to the price of the second product, 
 the entire production of which is sold after separate processing. 
 Prom equation (25) v/e know that the unit capacity cost, b, must 
 
14 
 
 equal the sum of a^ and a 2 ^ 2 , where a. ± is the number of units 
 of the ith product produced by one unit of batch capacity and 
 where f*. is the marginal opportunity cost of the ith product. 
 Recall that if u* is positive, then ^ = s^ . Substituting this 
 equality in equation ( 25 ) yields: 
 
 (27) i 2 * d/ a 2 )^ b " a i s i ) 
 
 This means that the marginal capacity cost of the second 
 product equals the entire marginal capacity cost minus the 
 marginal contribution on unprocessed production of the first 
 product with respect to production of the second, This 
 finding may be interpreted in terms of the price of the 
 second product by substituting equation (2?) in equation 
 (22) a s follows: 
 
 (28) Pj = - D ?/ d 2 + c 2 r ^ a 2 "" ( a i/ a 2^ s i 
 The optimal price of the second product provides for the 
 recovery of not only its marginal processing cost "but also 
 the entire capacity cost. In addition, the optimal price is 
 reduced for a share of the contribution on unprocessed pro- 
 duction of the first product. If the marginal cost of the 
 second product (c 2 + b/a 2 - a-,s-./ap) is positive, as is 
 likely, then the product should be marketed at demand 
 elasticity greater than unity. 
 
15 
 
 Implementation of th e Opt imal Policy 
 
 The preceding sections examined the profit-maximizing 
 price policy with special attention to the case in which 
 some joint production is sold at split-off. The present 
 section considers a graphical analysis of these results 
 as v/ell as results obtained by Manes and Smith and by 
 Colberg under somewhat more restrictive assumptions. 
 In general, the purpose of this section is to show how 
 graphs may be used to obtain the profit-maximizing output 
 policy from fully specified demand and cost functions. 
 
 Three production situations are examined. The first, 
 which was examined by I'anes and S.-r.ith, requires all' joint 
 production to receive separate processing and assumes that 
 excess production receives costless disposition* The 
 second, which was examined by Colberg, also assumes that 
 excess production does not affect profit,, but does not 
 require that such production receive separate processing. 
 The third situation accords with the model presented earlier 
 in this paper and permits sales at split-off in a perfectly 
 competitive market. Thus the third case comes closest to 
 representing the joint cost problem as it is usually 
 represented in cost accounting texts. 
 
 The optimal policy under the MS model may be demonstrated 
 by reference to Figure 2. Since the two products arise in 
 fixed proportions, both revenue and cost may be written as 
 
Pig. 2"b 
 
 K* ' 
 
 FIGURE 2 
 
16 
 functions of K. Profit maximization requires that batch 
 capacity, K, be set such that the appropriate marginal 
 revenue equals total marginal cost. In Figure 2, total 
 marginal revenue, FTRm> is the vertical summation of the 
 marginal revenue lines of the separate products, MR-, and 
 MR«. Since the MS model requires that all joint production 
 receive separate processing, MC«, is a-, c, + a^Cp + h. All 
 that needs be done io to determine the output levels, k^, 
 kp, and k m , at which marginal cost intersects the three 
 marginal revenue lines. The optimal batch size, X*, is 
 then simply the largest k-value. If k™ is the largest 
 k-value, as it is in Figure 2a, then the entire production 
 of both joint products is marketed at prices equal to the 
 separate demand (or average revenue) functions at km. 
 On the other hand, if k ? is the largest k-value, as in 
 Figure 2b, then the entire production of the second product 
 is marketed at a price equal to average revenue at k ? , but 
 only part of the first product's production will be marketed 
 and that at the price corresponding to MR., of zero. The 
 unmarketed production of the first product receives costless 
 disposition. It is important to notice that the optimal 
 policy was achieved without reference to any assignment 
 of cost. 
 
17 
 
 Colberg's model is like the MS model in that excess 
 production of a joint product receives eostlesr disposition, 
 but it is unlike the IIS model in that such production does 
 not receive separate processing. Again, three k-values are 
 determined by reference to marginal cost as follows: 
 (i) k-, is n by the intersection of MR, 
 and MC, = su o. + b; 
 
 J, 1 X 
 
 (ii) k« is given by the intersection of If« 
 
 and MCp = &pCp + b; 
 (iii) k, T is given by the intersection of MR m 
 
 and MC«, - a 1 c, + &pCp + b. 
 As demonstrated in Figure 3, the optimal batch size, K*, 
 is given by the largest k-value. If km is the largest 
 k-value, as in Figure 3a, then the entire production of 
 both joint products in marketed at prices equal to the 
 average revenue functions at k m . On the other hand, if 
 kp is the largest k-value, as in Figure 3b, then the 
 entire production of the secord product is marketed after 
 separate processing at a price equal to average revenue 
 at kp. But only part of the first product's production will 
 be marketed and that at the price corresponding to the equality 
 of MR, and a, c-; . 
 
Fig e 3a 
 
 Fig. 3b 
 
 UR2 3 
 
18 
 
 Under the model considered by this paper, the firm 
 has the option of selling a joint product in a perfectly 
 competitive market before it incurs the separate processing 
 cost. Under this model, three k -values again lead to an 
 identification of the optimal batch size, The three k-values 
 are calculated ollov/s: 
 
 (i) k-. is given by the intersection of MR-, 
 and MC-, = a-,c-, - a 2 s 2 + *>; 
 (ii) k« is given by the intersection of T4Rp 
 
 and MC 2 = a 2 c 2 " a l s l + D ' 
 (iii) k™ is given by the intersection of MR m 
 
 and MC m = (c, - s^a-^ + (c 2 - s 2^ a 2 + **• 
 Again, the largest k-value is the optimal batch capacity. 
 If k m = K*, then the entire production of the two joint 
 products is sold after separate processing (Case II )„ 
 On the other hand, if kp is the largest k-value, as in 
 Figure 3b, then the entire production of the second product 
 is sold after separate processing* But only part of the 
 first product's production will be marketed after separate 
 processing; the remainder will be sold at split-off. In 
 other words? an instance of Case I is indicated when either 
 k, or k« exceeds k m . 
 
19 
 
 In summary, the optimal batch size under each model 
 is the largest of three k -values — k, , kp, and k«. k.. is 
 the optimal sales level for the first product when profit 
 maximization requires sale or disposition of some of the 
 second product at split-off* Similarly, kp is the optimal 
 sales level for the second product when profit maximization 
 requires sale or disposition of some of the first product 
 at split-off, nally, km is optimal sales level for 
 both products when profit maximization requires that the 
 entire production of both products be sold after separate 
 processing. While the three k-values have essentially the 
 same interpretation under all three models, different 
 marginal costs enter their determination in each model. 
 But every marginal cost includes the marginal joint pro- 
 cessing cost, b, which demonstrates that implementation 
 of the optimal policy does not require an allocation of 
 such cost under any of the models considered, 
 
20 
 
 Accounting for Bv-Products 
 
 mi., , .1 - - i ii n-- i ■ ■-" ii ■ . ■ ii 
 
 Manes and Smith argue that their analysis justifies 
 the traditional approach to "by-product cost accounting 
 whenever profit maximization results in withholding some 
 joint production from the market. The}' interpret the 
 traditional approach "to allocate no input cost to the 
 by-product and to indicate net revenue of the by-product 
 
 either as a reduction in the cost of goods sold of the major 
 
 17 
 
 product or as supplemental income." The nature of the 
 
 justification requires close scrutiny. Consider the imple- 
 mentation of the optimal policy under the Manes and Smith model, 
 
 Suppose that k ? is the maximum k-value; then part of the 
 first product's output is withheld from the market. If 
 the first product is treated as a by-product, it receives 
 no cost; the entire input cost is assigned to the second 
 product which is called the main or major product. But 
 how does this assignment of cost assist in fixing the 
 optimal prices? He optimal price of the second product 
 is its average revenue at kp* which is calculated without 
 reference to cost* The optimal price of the first product 
 is its average revenue at the sales volume that brings 
 marginal revenue to zero where zero is the marginal cost 
 of expanding sales of the first product. In other words, 
 the marginal cost of the by-product is required in order 
 to fix the optimal price and sales volume of the by-product. 
 While the assignment of cost is not useful in fixing the 
 optimal price of the main product, it is useful in fixing 
 the optimal price of the by-product. 
 
21 
 
 A similar argument can be made for the other models 
 considered in the preceding section, but not without 
 recommending a modi: accounting procedure 
 
 for by-products. The p: dified to assign 
 
 each unit of t parately | d by-product its marginal 
 cost — (c- ) under the Colberg mod nd (c-, + s-,) under the 
 third model. 
 
 Returning to the Manes and Smith model, there is a 
 second './ay in which the marginal costs may be used in reaching 
 the optimal policy. Suppose that demand functions are less 
 than completely known; the firm knows only that some part of 
 the first product's production must be scrapped as in Figure 2b, 
 How then does the firm maximize profit? In this case, it is 
 useful to consider the separate marginal costs. The optimal 
 prices of the two products are given as follows: 
 
 p* = - D*/d* 
 
 p* = - D*/d| + c 2 + (b + a 1 c 1 )/a 2 
 
 Since the positive demand components of the optimal prices 
 are unknown, a reasonable strategy is to adjust the prices 
 upward by small increments from the level of the marginal 
 costs— zero for the first product and c + (b + a,c, )/a„ 
 for the second — until incremental revenue equals incremental 
 cost. 
 
 "! Q 
 
22 
 The firm would commence the process of adjustment by- 
 announcing a price for the second product equal to its 
 
 20 
 
 marginal cost* Having satisfied the resulting demand, 
 
 ♦' 
 
 the firm would repeat the process with an incremented price. 
 The two observations of demand would enable the firm to 
 compute marginal revenue. If the computed marginal revenue 
 exceeds the marginal cost, the firm would announce a further 
 increment in price and continue the process until marginal 
 revenue fell to the level of marginal cost. Experimentation 
 with the price of the first product (some of which is 
 known to require costless disposition) would be conducted 
 simultaneously in a similar way except that marginal revenue 
 would be brought to zero. 
 
 The adjustment process for the other models considered 
 in the preceding section would be identical except that 
 the marginal costs would be calculated differently. In 
 summary, when the firm's only knowledge of demand is that 
 k-, or kp will be the maximum k-value, then the marginal 
 costs of the separate products may be useful in reaching 
 the optimal price-output polic such cases, there is 
 justification for an accounting procedure that assigns 
 ;}oint products their marginal cost. 
 
23 
 
 Summary 
 The purpose of this paper is to examine the information 
 requirements of the pricing decision for joint production in 
 fixed proportions, "./hen some production requires disposition 
 or sale at split-off, the marginal opportunity costs, v/hich 
 imply an allocation of the joint processing cost, are given 
 in terms of cost parameters alone. In that case, the joint 
 product whose entire production is separately processed is 
 allocated the entire joint cost. Such an allocation is 
 useful in that the decision maker is not lead to consider 
 the joint processing cost in setting the price and level of 
 separate processing; for the other product. The marginal cost 
 of this product takes different forms depending on the 
 conditions under v. r ' ich its sale at split-off and its separate 
 processing take place; consequently, the accountant must 
 exercise care in assigning the non-joint cost for purposes 
 of the pricing decision. Furthermore, id Ls known 
 to obtain but demand information is incomplete (such that 
 the optimal price-output policy is unknown), then the cost- 
 based portion of the 03 price be useful in exper- 
 imentation with price a utput in an effort to reach the 
 optimum. 
 
FOOTNOTES 
 
 ~^See, for example: Marshal R. Colberg, "Monopoly Prices 
 Under Joint Costs: Fixed Proportions," Journal of Political 
 Pconc-y, IL (1941), pp. 103-110; R. P. Panes and Vernon L. 
 Smith, : int Cost Theory and Accounting; Practice," 
 
 The ;.cc 1:^1!:- ?.evie\; t PL (January, 1965), pp. 31-51; and 
 Roman L, . oil, Jr. , "Alloc g Joint Costs," American Pconomic 
 Review , LVIII (December* 1968)7 pp. 1342-1345. 
 
 2 
 The analysis ' : that capacity, K, and production, 
 
 x-} and Xp, are perfectly divisible, 
 
 3 
 
 "This assumption is less restrictive than those employed 
 
 "by Colberg (1941', Panes and Smith (1965)* Manes and 
 Smith assume that all production is either sold after separate 
 processing or scrapped without additional cost and that all 
 production is separately processed. Colberg 5 s most general 
 model does not require that all production be separately processed 
 but does require costless disposition at split-off. The model 
 considered here is somev.'hat more general in that all production 
 need not he separately processed and in that unprocer ed production 
 may generate a nonzero contribution at split-off. All three 
 models, hov;ever, are single-period analyses under which production 
 for inventory is precluded: all production is marketed or scrap- 
 ped v/ithin the production period. 
 
 This characterizes both the Manes and Smith model and 
 the models considered by Colberg, 
 
 c 
 
 Revenue from separately processed production of the ith 
 joint product, R>, may be written in two ways. In general, 
 
 revenue is the product of price and quantity sold, but that 
 product may be given either as a function of price, p., or 
 
 as a function of quantity sold, v., that is, 
 
 R i = V?i 
 
 = Pi*V p P 
 
 i u i' y L 
 
 where y. = D«(p.-)» which is the demand function, and o,. = Z.(y. ), 
 
 which is the inverse demand function. If the inverce exists, 
 then the two formulations of revenue are equivalent, and an 
 optimal price is equivalent to an optimal sales level. The 
 equations above assume that the demands for the two products 
 are independent of one a*. other; hence, the demand for each 
 product is a function of its price alone and not the price of 
 the other product. Similarly, the price of each product is a 
 function of its sales alone and not the sales of the other orodui 
 
g 
 
 If the solution to the maximization problem were sought 
 in terms of outputs rather than prices, then the cost function 
 would have the following form: 
 
 r 
 
 n 
 
 A production process may generate a joint product (e.g., 
 scrap) t is not susceptible of further processing. If 
 such production is sold in a perfectly competitive market, 
 then its contribution per unit of capacity merely reduces the 
 unit capacity cost, b. This accords with the recommended 
 procedure of crediting the net realisable value of scrap to 
 the cost of joint production. ; type of joint product is 
 distinguished from unprocessed joint production that is 
 susceptible of further processing. 
 
 °If all production is sold in perfectly competitive 
 markets, then the problem reduces to a linear programming 
 problem in which nrices are known constants. A thorough 
 treatment of problems of this type is given by Ronald V„ 
 Hartley, "Decision Making When Joint Products Are Involved," 
 The Accounting Review , X1VT (October, 1971), 746-755. 
 
 ^The necessary conditions for a saddle-point solution 
 are also sufficient provided that profit is a concave function 
 of the prices, disposal quantities and capacity. Profit is 
 concave if and only if D-. and D ? are concave when p, - c^ ^ 
 
 and p~ - c~ >. 0, which in turn is true if and only if the 
 
 *d cl 2 2 
 
 second derivative of demands with respect to price, cL and d^t 
 
 are zero or negative. In short e are willing to assume 
 that the second derivative c h demand functions will not 
 assume positive values, then the conditions above are both 
 necessary and sufficient for p. when p. - c. ; ^0 e Since the 
 
 second derivative of a linear demand function is zero, the 
 requirement is fulfilled for our example. See G. Hadley, 
 Nonlinear and Dynamic P rogramming (Reading, Massachusetts: 
 Addison-V/esley Publishing Company, Inc., 1964). 
 
 See C. E. Ferguson, Microeconomic Theory (Homewood, 
 Illinois: Richard D. Irwin, Inc., 1969), pp. 100-102. 
 
If part of. one product's production is sold at the 
 split-off point, then the entire production of the other 
 product will almost always be marketed after separate 
 processing. This result assumes that K, the batch capacity, 
 is sufficiently adjustable to be set at the optimal level. 
 If uj is positive, then ^, = s~„ Prom (20), jjL = b/a^ - (a,/a 2 )s., 
 
 provided k is positive. Substituting in (17), we have the 
 requirement that 
 
 u| (sg - b/a 2 + a i s 2' /a 2^ ~ ° 
 Since the term i- heses rare iuals zero, u£ 
 
 nearly always equals zero, ' uj equals zero, then the 
 
 entire production of the second product is sold after separate 
 processing. 
 
 12 
 
 A third case might be identified in which the entire 
 
 production is sold at split-off. In this case, no price 
 
 policy is necessary because the markets for production at 
 
 split-off are perfectly competitive. 
 
 13 V/eil (1968), pp. 1342-1343. 
 
 14 Ibid ., p. 1343. 
 
 15 Manes and Smith (1965), p. 33. 
 
 ^-^The difference in conclusions is a consequence of their 
 assumption that marginal cost of units scrapped at split-off 
 equals the marginal cost of units sold after separate processing 
 or, equivalently, that excess production receives costless 
 disposition after incurring the separate processing cost. This 
 assumption enables them to write the optimal price in terms of 
 demand parameters alone; from this and the assumed positive 
 price, it follows that demand elasticit ist be unity. This 
 result can be demonstrated under the present model* If we 
 let s^ = - c-j^ when u? > 0, then ^ = - c-. and the right-hand 
 
 side of equation (23) is zero. Since price is positive, this 
 means that N* must equal unity. Moreover, from" equation (22)» 
 
 price is given by the demand component alone. 
 
17 Manes and Smith (1965), p. 33. 
 
 1 ft 
 
 x It is important to notice that Case I does not 
 
 coincide v/ith the by-product care as usually defined. 
 Cost accounting texts define a by-product as a joint 
 product ,r hose importance or value is small relative 
 to other joint products from the same process. Although 
 the definition is not I is clear that it may 
 
 be interpr to include instances of Case II and to 
 exclude instances of Care I. Moreover, the definition 
 is not restrictec to fi: ropo: s. Consequently, 
 when we s~y that the a is of Case I justifies by- 
 product accounting we refer only to the coincidence of 
 the definition Case I, 
 
 1 9 
 
 In the absence of complete specification of demand 
 
 functions, the separate marginal costs are known when either 
 k, or k« is known to be the largest k-value 9 But the 
 
 marginal costs of the separate products are not known when 
 k^ is expected to be the largest k-value unless the demand 
 
 functions are also known. Consequently, if demand functions 
 are unknown, except to the extent that L is expected to be 
 
 the largest k-value, separate marginal costs cannot be used 
 to pursue the optimal policy, 
 
 20 Careful judgment must be exercised in this process 
 of adjustment. If, for example, the firm is unable to 
 satisfy demand in any period, demand in the following period 
 may be affected and some adjustment would be in order. 
 

 7-9*