The Impact of Yield-Dependent Trading Costs on Pricing and Production Planning under Supply and Demand Uncertainty Burak Kazaz and Scott Webster Whitman School of Management, Syracuse University, Syracuse, NY 13244
[email protected],
[email protected] January 30, 2010 This paper studies the role of the yield-dependent cost structure influencing the optimal choice of the selling price and production quantity for a firm that operates under supply and demand uncertainty. The problem is commonly observed in the agricultural industry. We consider a firm that initially leases farm space in order to grow fruit. The realized amount of fruit supply fluctuates due to weather conditions, diseases, etc. At the end of the growing season, the firm has three options: convert its crop supply to the final product, purchase additional supplies from other growers, and sell some (or all) of its crop supply in the open market without converting to the finished product. Earlier literature considers the unit purchasing cost and the unit revenue from selling the fruit supply in the open market as static cost parameters. However in agricultural settings, these two trading cost parameters change with the realized yield. Considering a price-setting agricultural firm, we investigate the impact of the yield-dependent cost structure present in the secondary options of purchasing and selling fruit in the open market. Our work makes four sets of contributions: 1) It determines the optimal selling price, production, purchasing, and selling quantity decisions under supply and demand uncertainty; 2) It shows that the use of a static cost exaggerates the initial investment in the farm space and the expected profit significantly, and the actual value gained from a secondary (emergency) option for an agricultural firm is lower under the yield-dependent cost structure; 3) It proves that the risk neutral firm does not benefit from fruit futures, which may help explain the lack of such markets for certain crops (e.g., olives, citrus, grapes); 4) It concludes that due to the yield-dependent nature of the cost parameters, supply uncertainty presents a higher risk for agricultural firms than businesses that operate under static cost parameters. Keywords: supply uncertainty, yield-dependent purchasing cost, pricing.
1. Introduction On December 2, 2006, The New York Times reported in the article “Nature Getting the Blame for Costly Orange Juice” that the average price of a gallon of orange juice had increased by 14%, going from $4.45 a year earlier to $5.09. The article states that the two largest orange juice producers in the US, Tropicana of Pepsi Co. and Minute Maid of Coca-Cola Company, warned consumers of more price increases because of what is expected to be a lackluster orange crop. The orange supply in 2007 turned out to be the smallest in 17 years due to a summer where four hurricanes (Charley, Frances, Jeanne, and Wilma) crisscrossed the state of Florida, where more than 90% of the orange crop is used for making orange juice, satisfying more than 80% of the country’s orange juice demand. On top of Florida’s low orange supply, the state had to battle with citrus canker and citrus greening, two highly contagious bacterial diseases that destroy citrus crops. On January 16, 2007, The Associated Press reported that four consecutive nights of freezing temperatures ruined approximately $1 billion worth of California’s citrus, and further increased the concerns for the country’s orange supply and its selling price. Recent news shows that the same pattern continues in 2009. On the morning of October 9, 2009, The Associated Press reported that the US
1
Department of Agriculture forecasts Florida’s orange crop to be at 8.25 million tons, which is 18% less than that of 2007. On the evening of October 9, The Associated Press announced that the orange juice futures went up by 10.1% based on the earlier report, warning consumers for an even higher price. These reports exemplify the impact of fruit supply fluctuations on the selling price of an end product, such as the orange juice. The fluctuations in crop supply have a significant effect on the purchasing costs and revenues that can be generated from selling the fruit in the open market. Specifically, lower realizations of fruit supply imply higher prices for the fruit in the open market. Due to high fluctuations in crop supply, agribusinesses typically perceive the price of fruit in the open market as random. The Ayvalık Chamber of Commerce in Turkey, the region that provides more than 70% of the country’s production of olive oil, reports how influential the crop yield fluctuations are in the purchasing cost of olives in the Turkish olive oil industry. Using the behavior of olive prices, Kazaz (2004) describes the relationship between crop yield and the purchasing cost of olives, and defines the unit purchasing cost as “yield-dependent.” Similar observations can be made in other agricultural industries, such as the wine industry. The labeling requirements with references to the growing region (e.g. Sonoma County, Napa Valley) and the year of the vintage force this industry to operate under a yield-dependent purchasing cost, where the unit cost is strongly related to the amount of the crop collected in the region in a specific year. In the case of Australian wine grapes, the purchasing cost of grapes increased more than five times from a year before based on the realized supply. Due to an oversupply in 2006, the purchasing price of wine grapes was reported to be averaging approximately $100 per ton, going below the cost of growing the fruit. 1 The New York Times reported in the article “A Drought in Australia” on April 17, 2008 that a long drought in 2007, however, cut the country’s wine grape supply by more than 50%, causing the price of wine grapes to rise significantly, ranging between $500 to $3,000 per ton. The impact of crop yield fluctuations can be seen easily in the citrus fruit prices in the US. As mentioned earlier, the increased number of hurricanes since 2004 has devastated the grapefruit crops in Florida, which serves as the primary source for the production of grapefruit juice in the country. As a result of a busy hurricane season in 2004, the amount of grapefruit crop decreased by 48.47% from 2003 to 2004. In one year in the same period, the price of grapefruit increased by 156.88%. Figure 1 shows the annual grapefruit crop collected in the US in the last decade and the average price of the fresh grapefruit (primarily used for the making of grapefruit juice). 2 The figure demonstrates the impact of crop yield on grapefruit fruit prices in the open market. 1
ABC News Online, http://www.abc.net.au/news/newsitems/200604/s1615457.htm. Citrus juice producers claim that grapefruit prices are the best representative of the yield-dependent purchasing costs due to the fact that a smaller percentage of grapefruit is sold directly to consumers than oranges. The data for high-quality orange prices in the US Department of Agriculture are clouded by direct sales to consumers. 2
2
US Grapefruit Production and Prices
Fresh Grapefruit Production and Prices in the US 30 25 20 15
Grapefruit Prices US$
10
Grapefruit Production US (100K tons)
5 0 2000 2001 2002 2003 2004 2005 2006 2007 2008 2009
Figure 1. Fresh Grapefruit Production and Prices in the US (Source: National Agricultural Statistics Services of the US Department of Agriculture, Annual Reports 2000 through 2009). The fluctuations in the citrus, olive, and grape supplies exemplify how important it is for agricultural businesses to incorporate the yield-dependent cost and revenue structure (associated with supply uncertainty) into pricing and production planning decisions. This paper responds to this need by examining a firm’s combined decisions of price-setting and production-planning while operating under yielddependent trading cost and revenue (generated from selling the fruit in the open market). The paper shows that the yield-dependent nature of cost and revenue streams from the fruit supply has a significant influence on the profitability of agricultural businesses. By comparing to the traditional modeling approaches with static cost parameters, the study demonstrates that the agribusinesses operate in an increased level of risk with lower expected profits due to the yield-dependent nature of the cost parameters. Firms in the agricultural industry counter the uncertainty on such cost parameters and supply fluctuations by employing three methods. It is important to note that there are no financial instruments that mitigate the risk of uncertain fruit costs. While there are futures for the finished product in the case of olive oil, orange juice and some higher quality wine, there is no futures market for the fruit of olives, oranges and grapes. As a result, the three methods used in practice play a critical role in mitigating the supply risk. The first method corresponds to leasing farm space in order to grow fruit in anticipation of reducing the future purchasing costs. When compared with the expected purchasing cost over supply uncertainty, the expected cost of growing the fruit is typically less expensive. Leasing farm space is a common practice among citrus juice producers in the US, olive oil producers in Turkey, and many wine producers throughout the world. The lease is determined by the number of trees (or vines), and the producer incurs the cost of growing the fruit and maintaining the farm space (which includes pruning, stem cutting, weed control, and insect and disease management). However, this strategy forces the producer to deal with the additional risk of supply uncertainty. The production of olive oil, in particular, is a risky investment because olive
3
trees bear fruit only once in two years. The realized supply at the end of the two-year-long growing season can fluctuate significantly due to weather conditions, diseases, etc. Leasing farm space is also widely practiced in the wine industry. An executive in one of the largest wine producers in the US has described the benefit of leasing farm space as reducing the risk associated with a lower return on equity, and therefore, avoiding the potential decrease in the perceived value of the firm in financial markets. The second method of battling supply uncertainty involves the second chance opportunity decisions. The firm can purchase additional fruit from other growers when the realized supply is insufficient, or alternatively, can sell some (or all) of its fruit crop in the open market without converting to the final product. As described above, the unit purchasing cost and the unit spot-selling revenue of the fruit (e.g., citrus, olives, and grapes) change with the amount of realized supply. This paper demonstrates that the yielddependent unit purchasing cost has a significant impact on the optimal decisions regarding the sale price and the production quantity. It should also be emphasized that, unlike the practices observed in the manufacturing industry, agricultural businesses cannot use inventories strategically to battle supply uncertainty. For example, the olive oil producer needs to press its olives within 48 hours of collection in order to achieve the highest quality of its final product. As a result, the oil producer cannot hold inventories of olives for future production of olive oil. Furthermore, olive oil that is stored longer than two years is considered to develop an acidic taste. Similar observations can be made for the production of fresh orange juice and wine. Freshly squeezed orange juice has a superior taste than the juice obtained from oranges stored in inventory. Therefore, high quality orange juice is obtained by pressing the oranges immediately after the collection. In wine production, grapes are also pressed immediately after their collection. Blackburn and Scudder (2008) provide other examples of deteriorating product quality in fresh produce (e.g. melons and sweet corn). As a result, like in the case of most perishable goods, the problem for the agricultural business can be modeled in a single-period context. The third method for battling uncertainties deals with influencing the demand by appropriately selecting the price of the final product. As exemplified by the orange juice prices, setting prices can be an important tool for managing supply uncertainty. Considering the operating environment for the agricultural businesses, we consider the price-setting behavior of the firm after crop yield is realized. The model is a two-stage stochastic program with recourse, where the growing season of the fruit corresponds to the first stage of the model. In this stage, the producer determines the amount of farm space to be leased that maximizes the expected profit under supply uncertainty. The amount of farm space leased can be perceived as the maximum amount of production attained under the ideal growing conditions. At the end of the growing season, the producer observes only a proportion of this amount. The selling season corresponds to the second stage of the model. In this stage, the producer makes four decisions: 1) the amount of end product to be made from internally grown fruit through the leased farm space, 2) the amount of additional fruit to
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be purchased from other growers for the production of the final product, 3) the amount of fruit to be sold in the open market without converting to the end product, and 4) the selling price of the finished product. There are several distinguishing factors that separate our work from earlier studies. Earlier research that examined production planning problems under supply uncertainty (as yield uncertainty) has considered prices to be exogenous. The body of literature that incorporates the price-setting behavior of a monopolistic firm into the decision making process under supply uncertainty is limited. This paper advances these works by highlighting the impact of the yield-dependent purchasing cost on pricing and production decisions under supply uncertainty. The paper is organized in the following order: A review of the literature is provided in Section 2. The model is introduced in Section 3 and the impact of supply uncertainty is demonstrated under a deterministic but price-dependent demand. Section 3 also provides numerical illustrations using data from the Turkish olive oil producers. The analysis of the problem under demand uncertainty is presented in Section 4. Section 5 includes the conclusions, managerial insights, and future research directions. All proofs and derivations are presented in the Appendix.
2. Literature Review The problem of production planning under supply uncertainty has received considerable attention. Price is commonly assumed to be exogenous. Yano and Lee (1995) provide an extensive review of production and inventory problems under yield uncertainty, which is the foundation for supply uncertainty. Models that consider production capabilities in a single and multiple period(s) can be found in Gerchak et al. (1988), Gerchak (1992), Henig and Gerchak (1990), Henig and Levin (1992) and Shih (1980). Gerchak et al. (1988) and Henig and Gerchak (1990) prove that the optimal production quantity does not depend on the yield distribution in a periodic review inventory problem. Henig and Levin (1992) determine the optimal order quantity with the choice of the vendor and the quantity to be delivered to customers. Hsu and Bassok (1999) and Bitran and Silgert (1994) identify the amount of optimal production with the availability of downward substitution. Bollapragada and Morton (1999) describe efficient myopic heuristics for periodic review inventory problems. The problem of production planning under random yield is also considered in assembly lines (see Gerchak et al. 1994) and in N-stage serial systems (see Lee and Yano 1988). Grosfeld-Nir and Gerchak (2004) provide an extensive review of the literature that studies multiple lot sizing decisions under random supply and demand in make-to-order systems. Rajaram and Karmarkar (2002) consider the issue of supply uncertainty in the process industry. Galbreth and Blackburn (2006) extend this work in the context of remanufacturing. The opportunity to obtain additional fruit resembles the setting in Jones et al. (2001). In their paper, the hybrid seed corn producer gets a second chance of production in a different region of the world and experiences supply uncertainty. Our problem differs from the one presented in Jones et al. (2001) in two
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ways: 1) the sale price is endogenous to the model, therefore the producer uses the sale price as a mechanism to hedge against fluctuating supply, 2) when the producer purchases additional fruit from other growers she does not experience another supply uncertainty. A similar setting with one unreliable and one reliable supply source is also examined in Kazaz (2004), Tomlin (2009), Tomlin and Snyder (2006); and the scenario with multiple unreliable suppliers is investigated in Tomlin and Wang (2005), Dada et al. (2006), and Federgruen and Nang (2008). However, these papers consider only exogenous selling price, and are not concerned with the insights that come as a result of the price-setting behavior. Price-setting behavior has been widely studied under demand uncertainty (see Petruzzi and Dada 1999, 2001, Federgruen and Heching 1999, 2002, Boyacı et al. 2006, Kocabıyıkoğlu and Popescu 2007), however, it has not been examined extensively with regard to supply uncertainty. Li and Zheng (2006) is the first study to consider endogenous pricing for inventory replenishment under supply and demand uncertainty. Their model differs from ours in three ways. First, in their model, supply and demand uncertainty are realized at the same time, whereas in our model supply uncertainty is revealed first, giving the opportunity to the firm to adjust its pricing and production-related decisions before demand is realized. Secondly, their model allows for backordering, and therefore, the firm always satisfies the demand in its entirety. This feature enables them to obtain unique optimal solutions without enforcing a restriction on the probability density function (pdf) of supply or demand uncertainty. Our model, however, features lost sales, and therefore, it becomes necessary to enforce restrictions on the pdf of demand uncertainty in order to warrant the uniqueness of the optimal policy. Third, their model provides the second-chance opportunity to purchase additional products after supply and demand are realized, but their purchasing cost is static (i.e., independent of the yield), whereas our purchasing cost changes with the realized supply. Both technical and managerial results depart significantly in the presence of the yield-dependent purchasing cost. Tang and Yin (2007) also study pricing under supply uncertainty. Their demand function is linear in price and ignores the influence of demand uncertainty. Supply uncertainty in their paper is restricted to a discrete uniform distribution, and therefore, their results are not generalizable. This paper extends their work by: 1) incorporating continuous and arbitrary distributions that define supply uncertainty, 2) generalizing the price-dependent demand function, 3) incorporating demand uncertainty, 4) considering the firm’s opportunity to sell the crop in the open market, and once again, 5) featuring the yield-dependent nature of the purchasing cost and revenue. As will be demonstrated later, the influence of the yielddependent trading costs is significant in the initial investment and the profitability of the firm. Tomlin and Wang (2006) consider price-setting behavior under supply uncertainty for a firm that produces multiple products with downward substitution. However, their prices are market-clearing prices
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as uncertainty is resolved before setting prices in their model; thus, the insight stemming from the pricesetting behavior is limited. The challenges in agricultural businesses, different than a repetitive manufacturing environment, are also receiving attention in recent literature. While the problem settings and the decisions investigated are distinctly different than our model, the interested reader is referred to Burer et al. (2009) and Huh and Lall (2008) for other challenging problems in the agricultural industry.
3. The Model This section presents the modeling approach used for the pricing and production planning problem of an agricultural business that leases farm space and experiences supply uncertainty. The model is a two-stage stochastic program, where the first stage corresponds to the growing season of the fruit, and after the production takes place, the second stage is the selling season of the final product. At the beginning of a growing season, the firm determines the optimal amount of farm space to be leased, denoted Q, at a unit cost of leasing cl . While all structural results are derived using linear leasing costs (i.e., clQ), the results continue to hold when the leasing cost is convex increasing in Q. Randomness in supply is represented with a stochastically proportional yield (or, multiplicative random error term), denoted with u% , where u is a realization, g ( u ) is the pdf defined on a support [ul , uh ] , and G ( u ) is the cumulative distribution function (cdf) with a mean u = E [u% ] and a variance σ 2 .
At the end of the growing season, the firm realizes crop yield in the amount of Qu . Given the realized supply at the beginning of the selling season, corresponding to the second stage of our model, the firm makes four decisions to maximize its expected profit: 1) the selling price, denoted p, 2) the amount of realized supply from the leased farm space (internal growth) to be converted to final product, denoted qi , 3) the amount of fruit to buy from other growers, denoted qb and 4) the amount of fruit to sell in the open market, denoted qs . The sum of the converted supply and the fruit sold in the open market is restricted to be no more than the realized crop yield, i.e., qi + qs ≤ Qu. In citrus juice, olive oil and wine production, the firm presses the fruit in order to obtain the final product, and we let c p denote the unit pressing (processing) cost. The firm incurs a processing cost of c p ( qi + qb ) . Before the selling season begins, the firm may choose to increase its production by purchasing additional fruit from other growers at a unit cost that changes with the realized yield. We denote the yielddependent unit cost of buying additional fruit as b ( u ) , and the firm incurs b ( u ) qb . The firm also has the flexibility to sell its fruit in the open market at a unit revenue that changes with the realized yield. The yield-dependent unit revenue from sales of the fruit is denoted s ( u ) , and the firm earns s ( u ) qs . There is a positive spread, defined as δ(u) = b(u) – s(u) > 0 for all u, between the unit purchasing cost and the unit
7
selling price of the fruit in the open market. This prohibits an arbitrage opportunity, i.e., the firm cannot make profits by simply buying and selling crop at the same time in the open market. It should be emphasized here that we make no assumptions regarding the shape of the yield-dependent costs except that both
b(u) and s(u) are decreasing in yield u. 3 For example, we do not require that expected trading costs are equal to the their costs at the expected yield; specifically, we allow E[b(u% )] ≠ b(u ) and E[ s (u% )] ≠ s (u ) . The conclusions of the study are robust as the results hold under convex, concave, and other forms of yield-dependent fruit trading cost functions. Demand is defined as price-dependent and stochastic, expressed as d(p, ε% ), where d(p) is the expected demand for a given selling price and ε% is a random error term that has a pdf of f(ε), a cdf of F(ε) defined on a support [ε l , ε h ] , and mean E [ε% ] = 0. Demand d(p,ε) is decreasing in price, increasing in realization
ε, and twice differentiable in price and the random error term, and therefore, there exists a unique inverse p(d,ε). Similar to Kocabıyıkoğlu and Popescu (2007), we assume that revenue p(d,ε)d is strictly concave in demand for any ε ∈ [εl, εh], i.e., 2p'(d,ε) + p"(d,ε)d < 0. We define the value of the demand error term that equates the demand to the realized crop supply q = Qu as ε(p, q), i.e., d(p, ε(p,q)) = q = Qu. The model is a two-stage stochastic program with recourse, where the firm chooses the optimal amount of farm space to be leased under supply and demand uncertainty in Stage 1. Given the realized crop, at the beginning of Stage 2, the firm determines the selling price, the amount of final product to be produced from internally grown and externally purchased fruit as well as the amount of fruit to be sold in the open market without converting to the final product. Leftovers of finished product at the end of the selling season are salvaged at a unit revenue of s2 . The model can be expressed as follows: uh
Stage 1: max E ⎡Π ⎣ ( Q ) ⎤⎦ = −cl Q + ∫ P ( Q, u ) g ( u ) du Q≥0
(1)
ul
Stage 2: Given Q and u, P ( Q, u ) =
max
( p , qi , qb , qs ) ≥ 0
π ( p, qi , qb , qs Q, u ) where
⎧ ⎫ ⎪ ⎪ ⎪−c p ( qi + qb ) − b ( u ) qb + s ( u ) qs ⎪ ⎪ ε ( p ,qi + qb ) ⎪ ⎪ ⎪ π ( p, qi , qb , qs Q, u ) = ⎨+ ∫ ⎡⎣ p × d ( p, ε ) − s2 ( ( qi + qb ) − d ( p, ε ) ) ⎤⎦ f ( ε ) d ε ⎬ εl ⎪ ⎪ ⎪ ⎪ εh ⎪+ ⎪ ⎡ p × ( qi + qb ) ⎤⎦ f ( ε ) d ε ⎪ ε p , q∫ + q ⎣ ⎪ ⎩ ( i b) ⎭ s.t. qi + qs ≤ Qu
3
We use the terms increasing/decreasing and positive/negative in their weak sense throughout the paper.
8
(2)
(3)
From ∂π/∂qs = s(u) > 0, it follows that inequality (3) is always binding (i.e., qi + qs = Qu). Thus, the second-stage objective function can be expressed as follows:
π ( p, qi , qb Q, u ) = ( p − c p − s ( u ) ) ( qi + qb ) − ( b ( u ) − s ( u ) ) qb + s ( u ) Qu − ( p − s2 )
ε ( p , qi + qb )
∫
F ( ε ) d ε . (4)
εl
While demand in practice can be random, supply uncertainty is a significantly more critical concern for agricultural businesses. This is evidenced in many reports. For example, following a public anger and a protest to boycott French products in 2002, French wines were predicted to go through a significant demand uncertainty in the United States. However, as shown in the book “Judgment of Paris” by G. Taber (pp. 284-285), the demand for French wines in the US turned out to be steady as in the past. The same observation can be made for fine wine, and is supported by the reports of the popular wine investor’s site www.liquidasset.com. In a recent article published in The New York Times (“Wine Market Struggles to Adjust in New Era,” March 31, 2009), it is reported that the overall demand for French wines has been steady despite the economic fluctuations. As a result, we proceed with the analysis of the deterministic (but price-sensitive) demand case in Section 3, and present the analysis of the stochastic demand case in Section 4. 3.1. The Case of No Trading (Buying or Selling) of the Fruit
We first analyze the variant of the problem with deterministic supply and demand by replacing the supply random variable u% with its mean u and the demand random variable ε% with its mean. The firm leases Q units of farm space, and realizes crop yield of Qu . In the second stage, due to the lack of a trading option (buying or selling) of the fruit (i.e., qb = qs = 0 ), the firm is restricted to convert the entire crop to the final product. In this setting, the selling price clears the production, and therefore, d ( p ) = qi = Qu . The first-stage objective function can be expressed as follows:
(
)
Ψ ( Q ) = p ( Qu ) − ( ( cl / u ) + c p ) Qu . Remark 1. a) The optimal amount of farm space to be leased, denoted by Q 0 , under deterministic
supply satisfies p (Q 0u )u + p '(Q 0 u )Q 0 (u ) 2 = cl + c p u ;
(5)
b) the optimal deterministic profit, denoted by Ψ (Q 0 ) , is Ψ ( Q 0 ) = − p '(Q 0u )(Q 0u ) 2 .
(6)
We next analyze the firm’s objective function under supply uncertainty: uh
uh
ul
ul
E ⎡Π ⎣ ( Q ) ⎤⎦ = − ( cl + c p u ) Q + ∫ p ( Qu ) Qug ( u ) du = Ψ (Q) − ∫ [ p ( Qu ) − p(Qu )]Qug ( u ) du .
9
(7)
Proposition 1. a) The first-stage objective function is concave in Q, and the optimal amount of farm
space to be leased satisfies uh
∫ ⎡⎣ p ( Qu ) u + p ' ( Qu ) Qu
2
⎤⎦ g ( u ) du = cl + c p u ;
(8)
ul
b) the optimal profit is uh
E ⎡⎣Π ( Q* ) ⎤⎦ = − ∫ p '(Q*u )(Q*u ) 2 g ( u ) du ,
(9)
ul
and is less than its deterministic equivalent; c) if 3p"(d) + p"'(d)d ≤ 0, then the optimal amount of farm space to be leased is less than that of the deterministic supply, i.e., Q* < Q 0 . The above proposition provides general results regarding the behavior of the optimal amount of farm space to be leased and the optimal profit expression under deterministic and stochastic supply. Because the demand function is not described by a specific function, a closed-form expression is not provided for the selling price. Under the linear demand function d(p) = a – bp, the optimal amount of farm space to be leased and the optimal profit expressions under deterministic and stochastic supply can be expressed as follows (derivations provided in the Appendix): 0 ⎡ a − b ( +c p ) ⎤ u ⎡ a − b ( (cl / u ) + c p ) ⎤ ⎦ =⎣ ⎦ = Q < Q0 , Q* = ⎣ 1 + cv 2 2 (u 2 + σ 2 ) 2u (1 + cv 2 ) 2 ⎡ a − b ( (cl / u ) + c p ) ⎤ u 2 Ψ (Q0 ) 2 ⎛ 1 + cv ⎞ ⎦ * ⎡ ⎤ = = < Ψ (Q0 ) E ⎡⎣Π ( Q ) ⎤⎦ = ⎣ Q u ⎜ ⎟ ⎣ ⎦ 2 2 2 4b ( u + σ ) ⎝ b ⎠ (1 + cv ) 2
*
where cv = σ / u is the coefficient of variation of supply uncertainty. It should be highlighted that the above expressions do not depend on the form of the pdf of supply uncertainty. Moreover, both the mean and the variance terms are influencing the optimal amount of farm space to be leased and the optimal profit. Preserving the mean supply, the above inequality shows that both the optimal amount of farm space to be leased and the optimal expected profit are decreasing in cv. 3.2. Incorporating the Trading Option (Buying, qb ≥ 0, and Selling, qs ≥ 0)
We next incorporate the flexibility to trade the fruit crop in the open market. The firm now has the option of buying additional fruit to be converted to final product (i.e., qb ≥ 0) and the option of selling its fruit supply without converting to the final product (i.e., qs ≥ 0). Considering the (deterministic) demand, the second-stage objective function in (4) can be revised as follows:
π ( p, qi , qb Q, u ) = ( p − c p − s ( u ) ) ( qi + qb ) − ( b ( u ) − s ( u ) ) qb + s ( u ) Qu . qi + qb ≤ d ( p ) qi ≤ Qu
10
(10)
We first present the analysis regarding the buying and selling options independently. Let us begin with the selling option by setting qb = 0 in (10). Recall that s(u) is the yield-dependent unit sales revenue from selling the fruit in the open market. s(u) resembles the salvage revenue in traditional modeling approaches, however, its value decreases with higher crop realization. The firm would consider the selling option of the fruit when the realized crop supply is high. When the yield realization is low, the pricing flexibility enables the firm to set higher prices to compensate for the lack of production. As a result, the firm would set a threshold for a maximum production level for each realization of u, where the remaining crop would be sold in the open market. Let us denote this threshold that determines the firm’s switching decision from utilizing the fruit for production to the option of selling in the open market by TS(u). This threshold can be determined by ignoring the effect of the realized supply, i.e., by setting Qu = 0. This leaves (10) as a maximization over a single variable qi, and the optimal value of qi provides TS(u). Proposition 2. The threshold for the amount of fruit to be purchased from the open market is:
TS(u) = – [p* – cp – s(u)] d′(p*)
(11)
and is increasing in u.
The first consequence of yield-dependent revenue of selling the fruit becomes clear in the firm’s desired level of production, namely TS(u). In the traditional modeling approaches where the salvage revenue is static and does not change with the realized yield, the firm’s desired level of production would be constant for all realizations of the yield parameter u. Proposition 2 shows that the desired level of production changes with the supply realization, and specifically, increases in the yield fraction. The next proposition shows that when the realized supply is below the desired level of production, the firm produces all of its crops and charges a market-clearing price, and when the realized supply is above the desired level of production, it produces up to the threshold and the remaining fruit is sold in the open market. Proposition 3. For a given realized supply of Qu and the revenue of s(u), the optimal selling price and
the production quantities that maximize π ( p, qi , qs Q, u ) in the second stage are ⎧⎪ p ( Qu ) if Qu ≤ TS ( u ) ⎧⎪( Qu ,0 ) if Qu ≤ TS ( u ) and ( qi* , qs* ) = ⎨ p* = ⎨ ⎪⎩ p (TS (u ) ) if Qu ≥ TS ( u ) ⎪⎩(TS (u ), Qu − TS (u ) ) if Qu ≥ TS ( u )
(12)
and the optimal second-stage profit is
(
)
π ( p* , qi* , qs* Q, u ) = ⎡ p min {TS ( u ) , Qu} − c p − s ( u ) ⎤ min {TS ( u ) , Qu} + s ( u ) Qu . ⎣
⎦
(13)
We next present the analysis of the buying option by setting qs = 0 in (10). Recall that b(u) is the yield-dependent unit purchasing cost of the fruit from the open market. The firm would consider the buying option when the realized crop supply is low due to the pricing flexibility that compensates for the expensive purchasing option. The threshold for the buying option, denoted TB(u), can be determined by ig-
11
noring the internal production option, i.e., qi = 0. This is equivalent to the scenario when the firm does not lease any farm space and acquires all fruit on the open market. Once again, (10) becomes a maximization over a single variable qb, and the optimal value of qb provides TB(u). Proposition 4. The threshold for the amount of fruit to be sold the open market is:
TB(u) = – [p* – cp – b(u)] d’(p*)
(14)
and is increasing in u. The above proposition demonstrates the influence of yield-dependent unit purchasing cost on the amount of fruit acquired from the open market. The threshold for the amount of fruit to be purchased from the open market increases in the yield fraction u. In the traditional modeling approaches that consider static purchasing costs from a secondary source, this threshold would be a constant; however, it changes according to the realization of the yield parameter for the agricultural businesses. The next proposition shows that when the realized supply is below the threshold for purchasing, the firm acquires additional crops from the open market and increases its level of production beyond the realized supply. In this case, the selling price reflects the influence of the purchasing cost. When the realized supply is above the threshold, however, the firm does not engage in additional purchasing, and converts only the internally grown fruit to the final product. In this event, the firm charges a market-clearing price. Proposition 5. For a given realized yield of Qu and the unit purchasing cost of b(u), the optimal selling
price and the production quantities that maximize π ( p, qi , qb Q, u ) in the second stage are:
⎧⎪ p ( Qu ) if Qu ≤ TB ( u ) ⎧⎪( Qu , TB (u ) − Qu ) if Qu ≤ TB ( u ) . and ( qi* , qb* ) = ⎨ p* = ⎨ if Qu ≥ TB ( u ) ⎪⎩ p (TS (u ) ) if Qu ≥ TB ( u ) ⎪⎩( Qu ,0 )
(15)
and the optimal second-stage profit is
(
)
π ( p* , qi* , qb* Q, u ) = ⎡ p max {TB ( u ) , Qu} − c p − b ( u ) ⎤ max {TB ( u ) , Qu} + b ( u ) Qu . ⎣
⎦
(16)
We next combine the buying and selling conditions. Note that TB(u) < TS(u), which is a consequence of b(u) > s(u). Proposition 6. For a given realized yield of Qu, the optimal decisions for the selling price, the amount of
crop yield to be converted to finished product, the amount to purchase from other growers and the amount to sell in the open market are: ⎧ p (TB (u ) ) if Qu ≤ TB ( u ) ⎪ if TB ( u ) ≤ Qu ≤ TS ( u ) p = ⎨ p (Qu ) ⎪ ⎩ p (TS (u ) ) if Qu ≥ TS ( u ) *
(17)
12
⎧( Qu , TB ( u ) − Qu ,0 ) if Qu ≤ TB ( u ) ⎪ if TB ( u ) ≤ Qu ≤ TS ( u ) ( qi* , qb* , qs* ) = ⎪⎨( Qu,0,0 ) ⎪ ⎪⎩(TS (u ),0, Qu − TS (u ) ) if Qu ≥ TS ( u )
(18)
and the optimal second-stage profit is ⎧ ⎡ p (TB ( u ) ) − c p − b ( u ) ⎤ TB ( u ) + b ( u ) Qu if Qu ≤ TB ( u ) ⎦ ⎪⎣ ⎪ * * * * if TB ( u ) ≤ Qu ≤ TS ( u ) (19) π ( p , qi , qb , qs Q, u ) = ⎨ ⎡⎣ p ( Qu ) − c p ⎤⎦ Qu ⎪ ⎪⎩ ⎡⎣ p (TS ( u ) ) − c p − s ( u ) ⎤⎦ TS ( u ) + s ( u ) Qu if Qu ≥ TS ( u ) We next consider the first-stage objective function. We define B(Q) = {u : TB ( u ) > Qu , u ∈ [ul , uh ]}
(20)
N(Q) = {u : TB ( u ) ≤ Qu ≤ TS ( u ) , u ∈ [ul , uh ]}
(21)
S(Q) = {u : TS ( u ) < Qu , u ∈ [ul , uh ]}
(22)
Note that B(Q), N(Q), and S(Q) is a partition of [ul, uh], i.e., B(Q) ∪ N(Q) ∪ S(Q) = [ul, uh] and B(Q) ∩ N(Q) = N(Q) ∩ S(Q) = B(Q) ∩ S(Q) = ∅. The set B(Q) contains values of u where it is optimal to buy fruit in the open market, N(Q) contains values of u where it is optimal to not trade fruit in the open market, and S(Q) contains values of u where it is optimal to sell fruit in the open market. Incorporating the optimal second-stage decisions into the first-stage objective function, (7) can be rewritten as follows:
⎧ ⎡ ⎡ p TB (u ) ) − c p − b(u ) ⎤ TB (u ) + b(u )Qu ⎤ g ( u ) du ⎦ ⎦ ⎪ ∫B ( Q ) ⎣ ⎣ ( ⎪ E[Π (Q )] = −cl Q + ⎨ ∫ ⎡⎣( p (Qu ) − c p )Qu ⎤⎦ g ( u ) du + N (Q ) ⎪ ⎪ ⎡ ⎡ p (TS (u ) ) − c p − s (u ) ⎤ TS (u ) + s (u )Qu ⎤ g ( u ) du ⎦ ⎦ ⎩ ∫S ( Q ) ⎣ ⎣
(23)
Proposition 7. The first-stage objective function in (23) is concave in Q, and therefore there exists a
unique optimal solution for Q*. The expected profit in (23) can be interpreted as a convex combination of the expected profits from three policies, buy additional fruit from the open market, do not trade fruit, and sell fruit in the open market. A unique optimal solution is guaranteed due to the concavity in the objective function without enforcing any limitations on the pdf of supply uncertainty. We emphasize that the optimal price is decreasing in the production quantity (and the optimal production quantity is decreasing in the price). This is a consequence of deterministic demand, where the demand is strictly decreasing in the selling price. As will be noted later in Section 4, the optimal price as a function of production quantity does not follow the same monotone decreasing behavior under demand
13
uncertainty. Under certain conditions the optimal price can be increasing in the production quantity (and the optimal production quantity can be increasing in price). The value gained from the trading option is the difference between optimal expected profits with and without the option of trading in the open market. The next proposition shows that the value of the trading option is decreasing in the spread between the open market buying and selling functions. Before introducing the proposition, we require additional notation. For any given open market buying and selling functions b(u) and s(u), let bˆ ( u ) = b(u) + Δ/2 and sˆ ( u ) = s(u) – Δ/2. We replace b(u) and s(u) in (11) through (23) with bˆ ( u ) and sˆ ( u ) , and we let E[Π * ( Δ )] denote the optimal expected profit with the option of trading in the open market and E[Π NT * ] denote the optimal expected profit without the option of trading in the open market. The value gained from the trading option is VT*(Δ) = E[Π * ( Δ )] − E[Π NT * ]
(24)
Proposition 8. The value of trading in the open market, VT*(Δ), is decreasing in Δ.
Proposition 8 shows that the contribution of the trading option to the expected profit decreases with increasing spread between the unit purchasing cost and the unit revenue from selling the fruit in the open market. When the spread is sufficiently large, the firm does not engage in any buying or selling activity, and the two profit expressions become equal. 3.3. The Impact of Yield-Dependent Trading Costs
In this section we illustrate the impact of yield-dependent cost and revenue structure on a firm’s decisions and profits. Our cost functions are based on data provided by two of the leading olive oil producers of Turkey. Conversations with managers indicate that the cost to purchase and sell fruit in the open market is convex decreasing in yield and that market demand is approximately linear in price. Figure 1 provides two firms’ estimates for the unit purchasing cost of olives as a function of yield. While one firm uses a convex-decreasing yield-dependent purchasing cost, the other firm uses a linearly decreasing yielddependent cost. We use the data provided by Firm 1 as the primary source of the parameters used in the analysis in this section. In order to illustrate the impact of the various forms of yield-dependent trading costs, we consider the following open market revenue and cost functions in our study: s(u) = αγ – βγuγ – δ/2 b(u) = αγ – βγuγ + δ/2 where
γ ∈ {0, 1, 0.5, 0.25} δ ∈ {2, 3, 4}
14
The case of γ = 0 corresponds to static open market purchase and selling cost, representing the traditional modeling approach utilized in literature (e.g., Li and Zheng 2006, Tang and Yin 2007). The case of γ = 1 represents yield-dependent buying cost and selling revenue functions that are linearly decreasing in the yield. The other two values of γ, 0.50 and 0.25, reflect different degrees of convexity in b(u) and s(u), where convexity increases as γ approaches 0. In our computational experiments, we assume u% is uniform on [0, 1]. bHuL 20
Firm 1
15
10 Firm 2
5
0 0.0
0.2
0.4
0.6
0.8
1.0
u
Figure 2. The purchasing cost functions provided by two of the leading olive oil producers in Turkey.
Recall that δ(u) is the spread, which is the difference between unit purchasing cost and the unit selling revenue (net of selling costs). The open market revenue and cost functions in our illustrations exhibit a constant spread for all values of the yield, and thus, δ(u) = δ for all u. The values of αγ and βγ are listed in Table 1.
γ
αγ
βγ
0.00
7.09
0.00
1.00
12.07
9.96
0.50
17.05
14.94
0.25
27.01
24.90
Table 1. Yield-dependent function parameters.
We set the values of αγ and βγ to be consistent with the firms’ data while maintaining consistent expected unit revenue and unit cost, i.e., E ⎡⎣ s ( u% ) ⎤⎦ + δ / 2 = E ⎡⎣b ( u% ) ⎤⎦ − δ / 2 = 7.09 for all γ.
15
Similarly, we set the form and parameters of the demand function to be consistent with the firm’s data; d(p) = 270,000 – 9,000 p. The leasing cost per unit is cl = 2.93 and the processing cost per unit is cp = 2.97. As noted above, the impact of the yield-dependent cost structure is apparent in the threshold expressions developed for the buy and sell policies. When the trading costs of the fruit are static, as in the traditional modeling approaches, these thresholds do not change with the realized yield. However, under a yield-dependent purchasing cost and selling revenue, these thresholds depend on the realized yield. Figure 3 exemplifies this behavior by comparing the thresholds under static versus yield-dependent trading costs. TBHuL and TS HuL
TS(u) of yield-dependent cost
110000 TB(u) of yield-dependent cost
100000
TS(u) of static cost
90000 TB(u) of static cost
80000 70000
0.2
0.4
0.6
0.8
1.0
u
Figure 3. The comparison of the two threshold expressions TB(u) and TS(u) under a static cost (γ = 0, δ = 3) and a convex decreasing yield-dependent trading cost (γ = 0.50, δ = 3). 3.3.1. The Impact on Expected Profit
A yield-dependent cost structure is detrimental for the profits of agricultural businesses. In traditional modeling approaches, the buying cost of the fruit and the revenue from selling the fruit are assumed to be static, not changing with the realized yield. Table 2 provides the firm’s optimal decision for the amount of farm space to be leased and the corresponding expected profit under various values of the spread (δ) and different forms of trading cost functions (static, linear and convex). These computational results illustrate that agribusinesses experience a lower expected profit when the trading costs are defined as yielddependent. This can be seen when the expected profit under γ ∈ {1, 0.5, 0.25} is compared with that under γ = 0 using the same spread, i.e., E[Π (Q* δ ,0 < γ ≤ 1)] < E[Π (Q* δ , γ = 0)] . The reasons for this result are twofold. First, when the realized supply is low, the firm prefers to purchase additional fruit from the open market. Under a yield-dependent cost structure, however, the unit purchasing cost is higher under low realizations, and therefore, the firm pays more for the additional fruit. Second, when the realized supply is high, the firm prefers to sell some of its crop in the open market. Un-
16
der a yield-dependent cost structure, the unit revenue is smaller at higher realizations of the yield. Thus, the firm sells the fruit in the open market at a lower return.
δ
γ
Q*
E[Π (Q* )]
2
0.00
∞
∞
2
1.00
127,212
862,831
2
0.50
119,533
853,834
2
0.25
114,555
851,709
3
0.00
302,250
955,312
3
1.00
131,223
851,308
3
0.50
126,017
841,678
3
0.25
122,375
838,768
4
0.00
206,881
927,348
4
1.00
133,529
840,930
4
0.50
129,879
831,096
4
0.25
127,140
827,782
Table 2. The optimal amount of farm space to be leased and the optimal expected profit under various values of spread and different trading cost functions. 3.3.2. The Impact on the Amount of Farm to be Space Leased
A yield-dependent cost structure alters the firm’s initial investment in the leased farm space. Table 2 demonstrates that the optimal amount of farm space to be leased under a yield-dependent cost structure is less than that under static costs, i.e.,
Qδ* ,0 0. ds du
Proof of Proposition 3. The second-stage objective function can be expressed as
{
}
max π ( p, qi Q, u ) = max p min {d ( p ) , qi } − c p qi + s ( u )( Qu − qi ) . p , qi ≥ 0 qi ≤ Qu
p , qi ≥ 0 qi ≤ Qu
36
For any viable p ≥ cp + s(u), it is clear that π(p, qi) is maximized at qi = d(p), and thus the problem reduces to a single variable optimization problem
{
}
max ⎡⎣ p − c p − s ( u ) ⎤⎦ d ( p ) + s ( u ) Qu ,
p≥0 d ( p ) ≤ Qu
which is concave in p. The first-order condition provides the optimal price expression in the event there is sufficient supply; otherwise price is set to clear available supply: ⎪⎧ p (Qu ) when u ≤ us ( Q ) ⎪⎧Qu when u ≤ us ( Q ) . p* = ⎨ ; qi* = ⎨ ⎪⎩TS (u ) when u > us ( Q ) ⎩⎪ p (TS (u )) when u > us ( Q )
where
⎧ul , if Qu > TS (u ) for all u ∈ [ul , uh ] ⎪⎪ us(Q) = ⎨uh , if Qu < TS (u ) for all u ∈ [ul , uh ] ⎪ ⎪⎩TS (u ) / Q otherwise Since qs = Qu − qi , the amount sold in the open market is ⎧⎪0 when u ≤ us ( Q ) . qs* = ⎨ ⎪⎩Qu − TS ( u ) when u > us ( Q )
Proofs of Propositions 4 and 5. The expected profit is
π ( p, qb ) = ⎡⎣ p − c p – b(u ) ⎤⎦ min {d ( p ), qb } . The proofs of Propositions 4 and 5 follow the proofs of Propositions 2 and 3 except that b(u) replaces s(u). We omit the details. Proof of Proposition 6. Because b(u) > s(u) for all u, TS(u) > TB(u) for all u. The rest of the proof fol-
lows from expressions (12) and (15). Proof of Proposition 7. The first derivative of the first-stage objective function is ∂E[Π (Q )] = −cl + ∫ [b(u )u ] g ( u ) du + ∫ ⎡⎣ p '(Qu )Qu 2 + ( p (Qu ) − c p )u ⎤⎦ g ( u ) du + ∫ [ s (u )u ] g ( u ) du B(Q ) N (Q ) S (Q ) ∂Q
Let ub1(Q), ub2(Q), …, ubn(Q) denote the end points of the intervals in B(Q), e.g., Qubi(Q) = TB(ubi(Q)) for all i. From Proposition 4, it is known that at each ubi(Q), the optimal price satisfies: p (Qubi ( Q )) + p '(Qubi ( Q ))Qubi ( Q ) = c p + b(ubi ( Q )) . Similarly, let us1(Q), us2(Q), …, usm(Q) denote the end points of the intervals in S(Q), e.g., Qusi(Q) = TS(usi(Q)) for all i. From Proposition 2, it is known that at each usi(Q), the optimal price satisfies: p ( Qusi ( Q ) ) + p ' ( Qusi ( Q ) ) Qusi ( Q ) = c p + s ( usi ( Q ) ) .
Thus,
37
{
(
)
}
{
(
)
}
⎧ n ⎡ ⎤ ⎪∑ ⎡∂ ⎣ ubi ( Q ) / ∂Q ⎤⎦ ubi ( Q ) b(ubi ( Q )) − ⎣ p ( Qubi ( Q ) ) − c p + p '(Qubi ( Q ))Qubi ( Q ) ⎦ i = 1 ⎪ ⎪ 2 2 ∂ E[Π (Q)]/ ∂Q = ⎨ ∫ [2 p '(Qu )u 2 + p ''(Qu )Qu 3 ] g ( u ) du + N (Q ) ⎪m ⎪ ⎡ ⎤ ⎣ usi ( Q ) / ∂Q ⎤⎦ usi ( Q ) s (usi ( Q )) − ⎣ p ( Qusi ( Q ) ) − c p + p '(Qusi ( Q ))Qusi ( Q ) ⎦ ⎪∑ ⎡∂ ⎩ i =1
[ 2 p '(Qu ) + p ''(Qu )Qu ] u 2 g ( u ) du
=∫
N (Q)
≤ 0. Proof of Proposition 8. Note that E[Π NT * ] is unaffected by changes in Δ. Let QNT* denote the optimal
lease quantity without the option of trading in the open market and let Q*(Δ) denote the optimal lease quantity with the option of trading in the open market. Finally, we define VT(Δ, Q, QNT) = E[Π(Q, Δ)] – E[ΠNT(QNT)], which is the gain from the trading option as a function of Δ and the lease quantities with (Q) and without (QNT) the trading option. Recall that bˆ ( u ) = b(u) + Δ/2 and sˆ ( u ) = s(u) – Δ/2. Accordingly, the value of trading in the open market can be expressed as: VT*(Δ) = VT(Δ, Q*(Δ), QNT*) =∫
(
N Q* ( Δ )
(
)
⎡ p Q* ( Δ ) u ) − c p ⎤ Q* ( Δ ) u − ⎡ p ( QNT *u ) − c p ⎤ QNT *u g ( u ) du + ⎦ ⎣ ⎦ ) ⎣ (
(
)
(
)
∫(
⎡ p (TB ( u ) ) − c p ⎤⎦ TB ( u ) − ⎡ p ( QNT *u ) − c p ⎤ QNT *u g ( u ) du + ⎣ ⎦ ) ⎣
∫(
⎡ p (TS ( u ) ) − c p ⎤⎦ TS ( u ) − ⎡ p ( QNT *u ) − c p ⎤ QNT *u g ( u ) du − ⎣ ⎦ ) ⎣
∫(
⎡( b ( u ) + Δ / 2 ) (TB ( u ) − Q* ( Δ ) u ) ⎤ g ( u ) du + ⎦ )⎣
∫(
)⎣
B Q* ( Δ )
S Q* ( Δ )
B Q* ( Δ )
S Q* ( Δ )
⎡( s ( u ) − Δ / 2 ) ( Q* ( Δ ) u − TS ( u ) ) ⎤ g ( u ) du . ⎦
Note that for any u ∈ B(Q*(Δ)), we have TB(u) > Q*(Δ)u, and that for any u ∈ S(Q*(Δ)), we have TS(u) < Q*(Δ)u. Therefore, for any x ∈ (0, Δ), VT*(Δ – x) – VT*(Δ) = VT(Δ – x, Q*(Δ – x), QNT*) – VT(Δ, Q*(Δ), QNT*) ≥ VT(Δ – x, Q*(Δ), QNT*) – VT(Δ, Q*(Δ), QNT*) =∫
⎡( x / 2 ) (TB ( u ) − Q* ( Δ ) u ) ⎤ g ( u ) du + ⎦
(
)⎣
∫(
)⎣
B Q* ( Δ )
S Q* ( Δ )
⎡( x / 2 ) ( Q* ( Δ ) u − TS ) ⎤ g ( u ) du ⎦
≥ 0. Thus, VT*(Δ) is decreasing in Δ.
38
Proof of Proposition 9. We define
qb(Q, Qf, u) = open market purchase quantity function qs(x, u) = open market sell quantity function where x denotes the total available supply of fruit, e.g., x = Qu + Qf + qb(Q, Qf, u)
(
))
(
E ⎡Π Q, Q f , u% , qb ( Q, Q f , u% ) , qs Qu% + Q f + qb ( Q, Q f , u% ) , u% ⎤ = expected profit given lease quantity Q, ⎣⎢ ⎦⎥ futures quantity Qf, open market buying function qb(Q, Qf, u), open market selling function qs(x, u), and the optimal price and production quantity E ⎡⎣Π * ( Q, Q f , u% ) ⎤⎦ = optimal expected profit given lease quantity Q and futures quantity Qf
Applying the above notation, we have E ⎡⎣Π * ( Q, Q f , u% ) ⎤⎦ =
=
(
max )
qb Q ,Q f ,u ≥ 0, qs ( x ,u ) ≥ 0
max
qb ( Q ,0,u ) ≥ Q f , qs ( x ,u ) ≥ 0
{E ⎡⎢⎣Π (Q, Q , u%, q (Q,Q , u% ) , q (Qu% + Q + q (Q,Q , u% ) , u% ))⎤⎥⎦} f
b
f
s
f
b
f
{E ⎡⎣Π (Q,0, u%, q (Q,0, u% ) , q (Qu% + q (Q,0, u% ) , u% ))⎤⎦} b
s
b
(because buying futures at unit cost E ⎡⎣b ( u% ) ⎤⎦ is equivalent to introducing a lower bound constraint on the number of units to purchase on the open market) ≤
max
qb ( Q ,0,u ) ≥ 0, qs ( x ,u ) ≥ 0
{E ⎣⎡Π (Q,0, u%, q (Q,0, u% ) , q (Qu% + q (Q,0, u% ) , u% ))⎦⎤} ∀Q b
s
b
f
≥0
(because the above problem is a relaxation of the previous problem) = E ⎡⎣Π * ( Q,0, u% ) ⎤⎦ . Proof of Proposition 10. The proof is similar to the proof of Proposition 6 in Kocabıyıkoğlu and Popescu
(2008). In order to simplify notation, we drop ‘|Q = qi = qs = 0, u’ from the profit expressions and we replace cp + b(u) with c and we replace qb with q.
π(p, q) = pE ⎡⎣ min {d ( p, ε% ) , q}⎤⎦ − cq = ( p − c ) q − ( p − s2 )
ε ( p ,q )
∫
F (ε ) d ε
εl
which is concave in q (follows from the standard newsvendor model). Furthermore, π(p, q) is concave in p because (1) pd(p, ε) is concave in p for any ε ∈ [εl, εh] (i.e., from the assumption that p(d,ε)d is concave in demand it follows that pd(p, ε) is concave in price), (2) pq is concave in p, and (3) the minimum of two concave functions (e.g., pd(p, ε) and pq) is concave. Thus, π qq ( p, q ) < 0 and π pp ( p, q ) < 0. Taking derivatives while noting that that ε(p, q) = q – d(p), εq(p, q) = 1, εpq(p, q) = εqq(p, q) = 0,
εp(p, q) = –d′(p), and εpp(p, q) = –d″(p),
π q ( p, q ) = p − c − ( p − s2 ) F ( ε ( p, q ) ) ε q ( p, q )
39
= p − c + ( p − s2 ) F ( q − d ( p ) )
π qq ( p, q ) = − ( p − s2 ) f ( q − d ( q ) ) π p ( p, q ) = q −
ε ( p ,q )
∫
F ( ε ) d ε − ( p − s2 ) F ( ε ( p, q ) ) ε p ( p, q )
εl
q−d ( p)
=q−
∫
F ( ε ) d ε + ( p − s2 ) F ( q − d ( p ) ) d ' ( p )
εl
π pp ( p, q ) = −2 F ( ε ( p, q ) ) ε p ( p, q ) − ( p − s2 ) ⎡ f ( ε ( p, q ) ) ε p ( p, q ) + F ( ε ( p, q ) ) ε pp ( p, q ) ⎤ 2
⎣
⎦
= F ( q − d ( p ) ) ⎡⎣ 2d ' ( p ) + ( p − s2 ) d "( p ) ⎤⎦ − ( p − s2 ) f ( q − d ( p ) ) d ' ( p )
2
⎡ f ( ε ( p, q ) ) ε p ( p , q ) ε q ( p , q ) + ⎤ ⎥ ⎢ F ( ε ( p, q ) ) ε pq ( p, q ) ⎥ ⎣ ⎦
π pq ( p, q ) = 1 − F ( ε ( p, q ) ) ε q ( p, q ) − ( p − s2 ) ⎢
= 1 − F ( q − d ( p ) ) + ( p − s2 ) f ( q − d ( p ) ) d ' ( p ) ⎡ ⎛ p − s2 ⎞ ⎤ = ⎡⎣1 − F ( q − d ( p ) ) ⎤⎦ ⎢1 − ⎜ ⎟ ∈ ( p, q ) ⎥ ⎣ ⎝ p ⎠ ⎦ Now consider the objective function on the optimal price path: π * ( q ) = π ( p* (q ), q ) . Note that
= −
π pq ( p* ( q ) , q ) π pp ( p* ( q ) , q )
dp* ( q ) dq
(by the Implicit Function Theorem). We have
⎛ dp* ( q ) ⎞ ⎟⎟ ⎝ dq ⎠ p = p* ( q )
π qq ( q ) = π qq ( p, q ) + π pp ( p, q ) ⎜⎜ *
= π qq ( p, q )
=
1
( π ( p, q ) ) − π pp ( p, q )
π pp ( p ( q ) , q ) *
2
pq
p = p* ( q )
Δ ( p* ( q ) , q )
(
)
where Δ ( p* ( q ) , q ) = π pp ( p* ( q ) , q ) π qq ( p* ( q ) , q ) − π pq ( p* ( q ) , q ) (i.e., the determinant of the Hes2
sian evaluated at (p*(q), q)). Recall that π qq ( p, q ) < 0 and π pp ( p, q ) < 0. Thus, if Δ ( p* ( q ) , q ) ≥ 0, then
π *qq ( q ) ≤ 0 and π*(q) is concave in q. Substituting π qq ( p* ( q ) , q ) , π pp ( p* ( q ) , q ) , and π pq ( p* ( q ) , q ) into Δ ( p* ( q ) , q ) , we get
40
(
) ( ( )) ( ( ⎡( p ( q ) − s ) f ( q − d ( p ( q ) ) ) d ' ( p ( q ) ) ⎤ ⎣ ⎦
)
))
(
(
) (
) (
)
) (
) (
)
Δ p* ( q ) , q = − p* ( q ) − s2 f q − d p* ( q ) F q − d p* ( q ) ⎡⎣ 2d ' p* ( q ) + p* ( q ) − s2 d " p* ( q ) ⎤⎦ + *
*
*
2
2
⎛ ⎡ ⎛ p* ( q ) − s2 * ⎡ ⎤ ⎜ 1 − F q − d p ( q ) ⎢1 − ⎜ ⎦ ⎢ ⎜ p* ( q ) ⎜⎣ ⎣ ⎝ ⎝
(
(
(
))
) (
(
)) (
−
⎞ * ⎟⎟ ∈ p ( q ) , q ⎠
(
))
(
) (
⎤⎞ ⎥⎟ ⎟ ⎦⎥ ⎠
2
= − p* ( q ) − s2 f q − d p* ( q ) F q − d p* ( q ) ⎡ 2d ' p* ( q ) + p* ( q ) − s2 d " p* ( q ) ⎤ − ⎣ ⎦ 2 ⎡ ⎛ * ⎡1 − F q − d p* ( q ) ⎤ ⎢1 − 2 ⎜ p ( q ) − s2 * ⎜ ⎣ ⎦ ⎢ ⎝ p (q) ⎣
⎤ ⎞ * ⎟⎟ ∈ ( q ) ⎥ ⎠ ⎦⎥ The first term is positive because 2d ' ( p ) + ( p − s2 ) d '' ( p ) < 0 (follows from 2d ' ( p ) + pd '' ( p ) < 0 and
(
))
(
d′(p) < 0). Observe that p*(q) – cp – b(u) ≥ 0 (i.e., the margin of finished product is nonnegative), which implies p*(q) ≥ cp + b(u) ≥ cp + b(1). Therefore, if ∈* ( q ) ≥ * 1 ⎛ p (q) 2 ⎝ p * ( q ) − s2
≥ ⎜⎜
1 ⎛ c p + b (1) ⎜ 2 ⎝⎜ c p + b (1) − s2
⎞ * ⎟⎟ , then ∈ ( q ) ⎠
⎞ * * ⎟⎟ , Δ ( p ( q ) , q ) ≥ 0, and π qq ( q ) ≤ 0. ⎠
Since π(p, q) is jointly concave in p and q, the optimal quantity solves
π q ( p, q ) = p − c + ( p − s2 ) F ( q − d ( p ) ) = 0 ⎛ p−c ⎞ ⇒ q* = d(p) + F −1 ⎜ ⎟ ⎝ p − s2 ⎠ and the optimal price solves
π p ( p, q ) = q −
q−d ( p)
∫
F ( ε ) d ε + ( p − s2 ) F ( q − d ( p ) ) d ' ( p ) = 0,
εl
which at the optimal price and quantity yields (27), i.e., ⎛ p* − c F −1 ⎜ * ⎜ p −s 2 ⎝
∫ ε
q* =
⎞ ⎟ ⎟ ⎠
) (
(
( )) ( )
F ( ε ) d ε − p* − s2 F q* − d p* d ' p*
l
(
) ( )
= − p* − c d ' p * +
⎛ p* − c ⎞ F −1 ⎜ * ⎟ ⎜ p −s ⎟ 2 ⎠ ⎝
∫
F (ε ) d ε .
εl
Proof of Proposition 11. a) The proof is similar to the proofs of Propositions 2 and 3 of Kocabıyıkoğlu
and Popescu (2008). In order to simplify notation, we replace cp + b(u) with c and we replace qb with q. From the Implicit Function Theorem,
41
dp* ( q )
= −
dq dq* ( p )
π pq ( p* ( q ) , q )
π pp ( p* ( q ) , q )
= −
dp
π pq ( p, q* ( p ) ) π qq ( p, q* ( q ) )
.
Furthermore, π qq ( p, q ) < 0 and π pp ( p, q ) < 0, and thus the sign of
dp* ( q ) dq
and
dq* ( p ) dp
is determined
by the sign of π pq ( p, q ) . As shown in the proof of Proposition 10, ⎡ ⎛ p * ( q ) − s2 ⎞ ⎤ * π pq ( p* ( q ) , q ) = ⎡1 − F q − d ( p* ( q ) ) ⎤ ⎢1 − ⎜⎜ ⎟⎟ ∈ ( p ( q ) , q ) ⎥ , and * ⎣
(
)⎦ ⎢
(
))
(
p (q)
⎝
⎣
⎡
Therefore, p is decreasing in q iff ∈ ( q ) ≥
( ) p(q ) − s p q* *
*
( ) ( )
⎛ p q* − s2 ⎞ ⎟ ∈ p, q* ( p ) * ⎟ ⎝ p q ⎠
π pq ( p, q* ( p ) ) = ⎢⎡1 − F q* ( p ) − d p ( q* ) ⎥⎤ ⎢1 − ⎜ ⎣ ⎦⎢ ⎜ *
⎥⎦
⎠
⎣
p* ( q )
p ( q ) − s2 *
(
)
⎤ ⎥. ⎥ ⎦
, and q* is decreasing in p iff ∈* ( p ) ≥
.
2
Proof of Proposition 12. The proof follows the proof of Proposition 10 except that b(u) replaces s(u). We
omit the details. The following result (stated without proof) follows from Proposition 10 and 1 ⎛ c p + b (1) ⎜ 2 ⎜⎝ c p + b (1) − s2
1 ⎛ c p + s (1) ⎜ 2 ⎜⎝ c p + s (1) − s2
⎞ ⎟⎟ ⎠
1 ⎛ c p + s (1) Lemma A1. If ∈* ( q ) ≥ ⎜ 2 ⎜⎝ c p + s (1) − s2
⎞ * ⎟ for any q, then π p (q ), q is concave in q. ⎟ ⎠
(
Proof of Proposition 13. The first-stage objective function is
42
)
⎞ ⎟⎟ ≥ ⎠
⎧ ⎡ ⎡ p* (TB(u ) ) − c p − b(u ) ⎤ TB (u ) + b(u )Qu − ⎤ ⎦ ⎪ ⎢⎣ ⎥ * ε ( p (TB ( u ) ),TB ( u ) ) ⎪ ⎢ ⎥ g ( u ) du + ⎪ ∫B( Q ) ⎢( p* (TB (u ) ) − s ) ⎥ ε ε F d ( ) 2 ∫ ⎪ ⎢ ⎥ ε l ⎣ ⎦ ⎪ * ⎪ ε p Qu Qu , ( ( ) ) ⎡ ⎤ ⎪ ⎥ g ( u ) du + E[Π(Q)] = –clQ + ⎨ ∫ ⎢ ( p* ( Qu ) − c p )Qu − ( p* ( Qu ) − s2 ) F d ε ε ( ) ∫ N (Q ) ⎢ ⎥ εl ⎪ ⎣ ⎦ ⎪ ⎪ ⎡ ⎡ p* (TS (u ) ) − c p − s (u ) ⎤ TS (u ) + s (u )Qu − ⎤ ⎦ ⎪ ⎢⎣ ⎥ * ε p TS u ) ),TS ( u ) ) ( ( ( ⎪∫ ⎢ ⎥ g ( u ) du ⎪ S (Q ) ⎢( p* (TS (u ) ) − s ) ⎥ ε ε F d ( ) 2 ∫ ⎪ ⎢ ⎥ εl ⎣ ⎦ ⎩ Alternatively, the above first-stage objective function can be written as:
⎧ ⎡ ⎡ p* (TB (u ) ) − c p − b(u ) ⎤ TB (u ) + b(u )Qu − ⎤ ⎦ ⎪ ⎢⎣ ⎥ * ⎪ ε ( p (TB ( u ) ),TB ( u ) ) ⎢ ⎥ g ( u ) du + ∫ ⎪ B(Q ) ⎢ * ⎥ F (ε ) d ε ( p (TB(u) ) − s2 ) ∫ε ⎪ ⎢ ⎥ l ⎣ ⎦ ⎪ ⎪ * E[Π(Q)] = –clQ + ⎨ ∫ ⎣⎡π ( p ( Qu ) , Qu ) ⎦⎤ g ( u ) du + N (Q ) ⎪ ⎪ ⎡ ⎡ p* (TS (u ) ) − c p − s (u ) ⎤ TS (u ) + s (u )Qu − ⎤ ⎦ ⎪ ⎢⎣ ⎥ * p TS u ) ),TS ( u ) ) ( ε ( ⎪ ( ⎢ ⎥ g ( u ) du ⎪ ∫S (Q ) ⎢( p* (TS (u ) ) − s ) ⎥ ε ε F d ( ) 2 ∫ε ⎪ ⎢ ⎥ l ⎣ ⎦ ⎩ The first derivative of the first-stage objective function is ∂E[Π (Q )] = −cl + ∫ b(u )ug ( u ) du + ∫ ⎡⎣π Q ( p* ( Qu ) , Qu ) ⎤⎦ g ( u ) du + ∫ s (u )ug ( u ) du . B(Q ) N (Q) S (Q ) ∂Q
⎧ n ⎧b ( u ( Q ) ) − ⎫ ⎪ ⎪ ⎡∂u Q / ∂Q ⎤ u Q ⎪ bi ⎬ ⎣ bi ( ) ⎦ bi ( ) ⎨ ⎡ ⎪∑ ⎪⎩ ⎣ p ( Qubi ( Q ) ) − c p + p '(Qubi ( Q ))Qubi ( Q ) ⎤⎦ ⎪⎭ ⎪ i =1 ⎪ ∂ 2 E[Π (Q)]/ ∂Q 2 = ⎨ ∫ ⎡⎣π QQ ( p* ( Qu ) , Qu ) ⎤⎦ g ( u ) du + N (Q) ⎪ ⎪m ⎧ s ( usi ( Q ) ) − ⎫ ⎪ ⎪ ⎡∂u ( Q ) / ∂Q ⎤ u ( Q ) ⎪⎨ ⎬ ⎣ si ⎦ si ⎪∑ ⎡ ⎤ p ( Qusi ( Q ) ) − c p + p '(Qusi ( Q ))Qusi ( Q ) ⎪ i =1 ⎪ ⎣ ⎦ ⎩ ⎭ ⎩
(
)
(
)
Let ub1(Q), ub2(Q), …, ubn(Q) denote the end points of the intervals in B(Q), e.g., Qubi(Q) = TB(ubi(Q)) for all i. From Proposition 10, it is known that at each ubi(Q), the optimal price satisfies:
43
⎧ ⎫ ⎪ p(TB (u ( Q ))) − c b(u ( Q )) ⎪ bi p bi ⎪ ⎪ ε ( p (TB ( ubi ( Q ) )),TB ( ubi ( Q ) )) ⎪ ⎪ ⎪ ⎪ − ε ε + p '( TB ( u Q ))[ TB ( u Q ) F ( ) d ⎨ ⎬=0. bi ( ) bi ( ) ∫ε ⎪ ⎪ l ⎪ ⎪ ⎪ −( p (TB (u ( Q ))) − s ) ∂ε ( p(TB (ubi ( Q ))), TB (ubi ( Q ))) F (ε ( p(TB (u ( Q ))), TB (u ( Q )))) ⎪ bi bi bi 2 ⎪⎭ ∂Q ⎩⎪ Similarly, let us1(Q), us2(Q), …, usm(Q) denote the end points of the intervals in S(Q), e.g., Qusi(Q) = TS(usi(Q)) for all i. From Proposition 12, it is known that at each usi(Q), the optimal price satisfies: ⎧ ⎫ ⎪ p(TS (u ( Q ))) − c − s (u ( Q )) ⎪ si p si ⎪ ⎪ ε ( p (TS ( usi ( Q ) )),TS ( usi ( Q ) )) ⎪ ⎪ ⎪ ⎪ − ε ε + p '( TS ( u Q ))[ TS ( u Q ) F ( ) d ⎨ ⎬=0. si ( ) si ( ) ∫ε ⎪ ⎪ l ⎪ ⎪ ⎪ −( p (TS (u ( Q ))) − s ) ∂ε ( p(TS (usi ( Q ))), TS (usi ( Q ))) F (ε ( p(TS (u ( Q ))), TS (u ( Q )))) ⎪ si si si 2 ⎪⎭ ∂Q ⎩⎪ Therefore, ∂ 2 E[Π (Q)]/ ∂Q 2 = ∫
N (Q )
⎡π QQ ( p* ( Qu ) , Qu ) ⎤ g ( u ) du . ⎣ ⎦
Let q = Qu. Observe that π Q ( p* (Qu ), Qu ) = u × π q ( p* (q ), q u ) , and
π QQ ( p* (Qu ), Qu ) = u 2 × π qq ( p* ( q), q u ) . Therefore, the sign of π qq ( p* (q), q u ) determines the sign of 1 ⎛ c p + s (1) d2E[Π(Q)]/dQ2. From Lemma A3, if ∈* ( q ) ≥ ⎜ 2 ⎜⎝ c p + s (1) − s2
∂ 2 E[Π (Q)]/ ∂Q 2 = ∫
N (Q )
⎞ * ⎟ , then π qq ( p (q ), q) < 0, and thus, ⎟ ⎠
⎡u 2π qq ( p* (q ), q u ) ⎤ g ( u ) du < 0 . ⎣ ⎦
44
