An Economic Analysis of Joint
Products under Demand Uncertainty
Mahito Okura
Doshisha Women’s College of Liberal Arts
December 8, 2018
The Workshop on Insurance and Finance(jointly with Meiji Institute for Advanced Study of Mathematical Sciences (MIMS))

11. Introduction: demand information• Demand information is an important factor in determining
production quantities of firms.
• In some cases, some or all firms do not have demand
information for their market.
• In such cases, firms only use information about “expected”
demand and there may be a gap between expected andactual demand levels.
• Asymmetric information problem might be realized.

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1. Introduction: previous studies• Microeconomics of uncertainty: Basar and Ho (1974),Ponssard (1979),Vives (1984), Sakai and Yoshizumi (1991),
Sakai and Sasaki (1992), Sakai (1993), Asplund (2002),
Chokler et al. (2006), Arya et al. (2013), Afeche et al. (2014),
Okura (2014)• Flexible technology under demand uncertainty: Goyal andNetessaine (2011), Boyabatli (2015), Boyabatli et al. (2015)3

1. Introduction: joint products• Joint products: Products indivisibly produced from a single input.
• Example: petroleum products are produced simultaneously, andthe production quantity ratio of these products is exogenously
decided.
• Gasoline: 27.3%• Naphtha: 10.2%• Jet fuel: 7.7%
• Kerosene: 8.8%
• Light oil: 21.6%• Heavy oil: 18.4%(in fiscal 2013. Source: website of the Petroleum Association of Japan)
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1. Introduction: research purposes• This study analyzes the market of joint products under demand
uncertainty.
• It is related to economic models of uncertainty and flexible
technology, but...

• the market of joint products has not been discussed in economics ofuncertainty.
• the main focus of the studies in flexible technology is on thecharacteristics of or differences among technologies, while demand
uncertainty is considered as a market condition.
• In contrast, this study discusses the effect of demand uncertainty andasymmetric information, and investigates how results differ under different
information structures, while treating technologies for producing multiple
products are an exogenous variable.
52. Model setting: notation• Two firms (firm A and B)• Two joint products (product 1 and 2)• Unit cost of the input :• Production quantity of product 1:• Production quantity of product 2:

where

•
: product 1’s ratio (exogenous variable)•: product 2’ ratio• “one-input, one-output” situation:and•: Amount of input of firm6

2. Model setting: demand function andinformation structure• Demand function:

where• The firm might not have demand information for the market ,represented by in the above demand function.
: Firm ’s demand information situation in market•••௜௝௜
௝: firm does not have demand information in market: firm has full demand information in market• Information structure:7

2. Model setting: demand information andprofit function• Demand information in the two markets ଵ ଶ has a joint
probability distribution and is normally distributed as ଵ ଶ ,ଶ ଶ.
ଵ ଶ , and•••௝ଶ௝௝ଶ
௝௝: Correlation coefficient of the two demand information௜௜௜• The profit function of each firm: ௜ଵ ଵଶ ଶ
• Each firm competes on the basis of production quantity anddetermines the amount of inputs for their products to maximize
their own expected profit.
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3. Information structures• The number of information structures: 16• The following 7 situations can arise:• Situation 1: no firm has any demand information (.
• Situation 2: one firm has demand information in either market, but
the other firm does not (,,,.
• Situation 3: each firm has demand information in either market, but
demand information of each firm is different (.

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3. Information structures• Situation 4: each firm has demand information in either market anddemand information of each firm is the same (.
• Situation 5: one firm has demand information in both markets (thatis, perfect information), but the other firm has no demandinformation (.
• Situation 6: one firm has demand information for both markets, but
the other firm has demand information in one market (,,,.
• Situation 7: both firms have demand information in both markets,that is, no demand uncertainty exists (.
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4. Deriving equilibrium expected profit• The equilibrium expected profit is derived for each situation.
• The results of the computations in all seven situations aresummarized in Table 1 (see separated document).
•: the equilibrium expected profit of firm when஺ ஺ ஻ ஻the information structure isଵ ଶ ଵ ଶ .
௜஺ଵ஺ଶ஻ଵ

஻ଶ

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5. Investigation of the equilibrium expected
profit in each information structure: Result 1
•
•
•

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5. Investigation of the equilibrium expected
profit in each information structure: Result 1
Result 1:• The equilibrium expected profit of firm A when it has nodemand information or demand information for one market
are the same, regardless of the information structure of firm
B.

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5. Investigation of the equilibrium expected
profit in each information structure: Result 2••••• If(wherethen145. Investigation of the equilibrium expected
profit in each information structure: Result 2
Result 2:• The equilibrium expected profit of firm A when it has demand
information for both markets depend on the information structureof firm B. If firm B has no (either, both) demand information, theequilibrium expected profit of firm A is high (middle, low).
• If the variance of demand information in market 1 is higher(lower) than that in market 2, the equilibrium expected profit offirm A, when it has demand information for both markets but firm
B has demand information for only market 1, is smaller (larger)
than when firm B has demand information for only market 2.
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5. Investigation of the equilibrium expected
profit in each information structure: Result 3• If(where••

஺஺

஺

஺

then16

5. Investigation of the equilibrium expected
profit in each information structure: Result 3
Result 3:
If the variance of demand information in market 1 is higher(lower) than that in market 2, the equilibrium expected profit
of firm A, which has demand information in market 1, is larger(smaller) than if it has demand information in market 2, whenfirm B has either no demand information or demand
information for one market.

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5. Investigation of the equilibrium expected
profit in each information structure: Result 4
• If,
•and
•

is realized: values for which the result of the following equation is zero஺஺ହఊమ ఙభమ ାଽ ଵିఊ మ ఙమమ ఘమ ାଵ଴ ଵିఊ ఘఙభ ఙమ ିସ ଵିఊ మ ఙమమଷ଺ ఊమ ௕భ ା ଵିఊ మ ௕మ18

5. Investigation of the equilibrium expected
profit in each information structure: Result 4
Result 4:
Suppose that firm A has demand information in market 1(market 2) and firm B has no demand information for thismarket. Where the absolute value of the correlation coefficientis relatively large (small), the equilibrium expected profit offirm A decreases (increases) when both firms have demand
information in market 2 (market 1).

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5. Investigation of the equilibrium expected
profit in each information structure: Result 5• In the case ofor:••஺஺஺஺஺஺஺஺஺஺஺஺•஺஺஺஺• Then,20

5. Investigation of the equilibrium expected
profit in each information structure: Result 5
Result 5:
Where demand information in two markets are perfectly
correlated, equilibrium expected profit will be equal in each of
the following three cases: (1) firm A has demand informationin either or both markets, firm B has no demand information;(2) both firms have demand information in either or bothmarkets, (3) firm A has no demand information, firm B hasdemand information in either or both markets. Results forthese information structures are therefore the same in a singlemarket.
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5. Investigation of the equilibrium expected
profit in each information structure: Result 6•భమ•భమ•భమ•భమభభభభమమమమమమమమమమమమమమమ
మ• Then,22

5. Investigation of the equilibrium expected
profit in each information structure: Result 6
Result 6:
Where firm A has no demand information, its equilibrium
expected profit always increases when it obtains demand
information for either market. The increase in the equilibrium
expected profit of firm A when firm B has no demand
information is larger than when firm B has demand informationfor either or both markets. Furthermore, the increase in theequilibrium expected profit of firm A when firm B has demand
information for one market is the same as when firm B has
demand information for both markets.
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5. Investigation of the equilibrium expected
profit in each information structure: Result 7•

଴଴

•

଴ଵ

•

ଵ଴

•

஺
஺
஺
஺

ଵଵ

• Then,• If• If஺ଵିఘమ ఊమ ఙభమ஺ସ ఊమ ௕భ ା ଵିఊ మ ௕మଵିఘమ ఊమ ఙభమ஺ସ ఊమ ௕భ ା ଵିఊ మ ௕మଵିఘమ ସఊమ ఙభమ ାହ ଵିఊ మ ఙమమ஺ଷ଺ ఊమ ௕భ ା ଵିఊ మ ௕మଵିఘమ ఊమ ఙభమଽ ఊమ ௕భ ା ଵିఊ మ ௕మ

ଶ ଶଵ
ଶ ଶଵ

ଶ ଶ
ଶ , thenଶ ଶ , thenଶ଴଴଴ଵଵ଴ଵଵଵ଴଴଴଴ଵଵଵ

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5. Investigation of the equilibrium expected
profit in each information structure: Result 7
Result 7:
Where firm A has demand information in market 2, then obtains
demand information in market 1, the equilibrium expected profit offirm A will always increase. Where the variance of demand
information in market 1 is larger (smaller) than market 2 and firm B
has no demand information or demand information in market 2 only,the increase in the equilibrium expected profit of firm A on
obtaining demand information in market 1 is largest (secondlargest); when firm B has demand information in market 1 only, theincrease is second largest (largest). When firm B has demand
information in both markets, the increase in the equilibrium
expected profit of firm A is lowest.
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5. Investigation of the equilibrium expected
profit in each information structure: Result 8• Investigate the possibility of exchanging demand information
between two firms:• Firm A has demand information in market 1, but not in market 2.
• Firm B has demand information in market 2, but not in market 1.
• Here, the information structure is.
• Each firm has two choices: “disclose own demand information” or“do not disclose own demand information.”26

5. Investigation of the equilibrium expected
profit in each information structure: Result 8Firm A

Firm B

Disclose

Not disclose

Disclose
Not disclose
Table 2: Firm A and B’s choices and corresponding information structures27

5. Investigation of the equilibrium expected
profit in each information structure: Result 8•••• Nash equilibrium: both firms choose “Not disclose”(dominant strategy) andstill holds.
• Outcome in Nash equilibrium is Pareto-inferior outcome.
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5. Investigation of the equilibrium expected
profit in each information structure: Result 8
Result 8:
Where firm A has demand information in market 1 (market 2)
and firm B has demand information in market 2 (market 1), if
both firms decide their disclosure policy independently, wewould expect no exchange of demand information because aprisoners’ dilemma situation is realized. Then, Pareto-inferior
outcome cannot be improved.

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6. Conclusion• From the results, for example, we know

• Policymakers must know the variances and correlation coefficientsof demand information in two markets to realize an optimal outcome.
• Some kind of coordination or incentive schemes are needed forrealizing desirable outcomes through exchange of demand
information. One solution is that, instead of each firm, industrial
association investigates the demand information and discloses it to
all member firms.

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