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Eco Solution Exam

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ESSEC BUSINESS SCHOOL Département d'économie ECOI 31080/1 Pre-MSc Microéconomie Examen final 19 décembre 2012 Aucun document n’autorisé. Pas de calculette. Il y a trois exercices. Répondez à chacune d'entre eux de manière concise et précise, dans les espaces disponibles. Écrivez proprement. On ne pourra pas accorder de point à une copie illisible. Vous avez 2h. Chaque partie a le même poids dans la note finale. Dernier conseil : réfléchissez avant d'écrire !

Exercice 1
Supposez que le coût du travail est le seul coût pour les firmes dans l’industrie des téléviseurs. Avec S employés qualifiés et U employés non-qualifiés, une firme peut produire S
0.8 0.2

U

téléviseurs par an.

Le salaire d’un employé qualifié est de 80$ et celui d’un employé non-qualifié est de 20$. 1) Calculez le taux marginal de substitution technique entre ces deux inputs en fonction de S et U.

U  PmS ( S , U ) = 0.8   S

0.2

S PmU ( S , U ) = 0.2   U  U  TMST ( S , U ) = 4   S

0.8

Une réponse alternative qui est aussi acceptée serait : TMST ( S , U ) 

1 S   4 U 

Page: 1

Court terme : à court terme, le nombre de travailleurs qualifiés est fixé à S=1 et le nombre de travailleurs non qualifiés est variable. 2) Trouvez le coût total de court terme de production de q unités.

q = S 0.8U 0.2 = U 0.2 ⇒ U = q 5 SCT (q) = 80S + 20U = 80 + 20q5

3) Pour quel niveau de prix une firme produira une quantité non-nulle de téléviseurs ? Pour quel niveau de prix une firme fera un profit positif à court terme ?

SCT (q ) = 80 S + 20U = 80 + 20q 5 SCM (q ) = SCFM + SCVM (q ) = SCm(q ) = 100q 4
SCm est toujours au-dessus du SCVM. Donc, la firme produit une quantité strictement positive pour n’importe quel prix. SCm est au-dessus du SCVM, quand 80q4 est au-dessus de 80/q, ou quand q est supérieure a 1. Cela se produit quand p est supérieur à SCm(1)=SCVM(1)=100.

80 + 20q 4 q

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Long terme : à long terme, les deux facteurs de production S et U sont variables. 4) Supposons qu’une entreprise cherche à produire q téléviseurs. Combien d’unités de travail qualifié et non qualifié va-t-elle décider d’employer de sorte à minimiser ses coûts de long terme ? La combinaison optimale des facteurs de productions serait la solution de ces deux équations : (1)

TMST =

PmS ( S ,U ) U w 80 =4 = S = PmU ( S ,U ) S wU 20

(2) S 0.8U 0.2 = q Si on résout le système, on aura S (q ) = U (q ) = q.

5) Quel est le coût total de long terme de cette firme ? LCT(q) = 80 q + 20 q = 100q.

6) Cette firme présente-t-elle des économies ou déséconomies d’échelle ? LCM (q) = 100. Constant. Cette firme n’a ni des economies ni des diseconomies d’échelle. En d’autres termes, elle est des rendements d’échelle constants.

Page: 3

Exercice 2
On désigne parfois sous le nom de supertankers les plus grands pétroliers. Le coût de location d’un pétrolier est de 125$. Le coût additionnel de transport de q unités de pétrole est de 1 2 20q + q . On suppose qu’une firme active dans cette industrie ne peut gérer au plus qu’un 20 seul pétrolier.

Long terme
(1) Donnez le coût total de long terme, le coût moyen ainsi que le coût marginal d’une firme de ce secteur. LCT(q) = 125 + 20q + q²/20. LCM(q) = 125/q + 20 + q/20. LCm (q) = dLCT/dq = 20 + q/10.

(2) Quelle est la taille critique (minimale) (ou “échelle d’efficience”) d’une firme de ce secteur ? LCM (EE) = LCm (EE) 125/q + 20 + q/20 = 20 + q/10 125/q = q/20 q² = 2500, Donc EE = q = 50

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(3) En supposant que des firmes peuvent entrer et sortir librement du marché, donnez la courbe d’offre de long terme de l’industrie. P = LCM (EE) = LCm (EE) = 25. La courbe d’offre de long terme serait constant et (horizontale) à un niveau de prix = 25.

(4) Supposons que la demande de services de pétroliers est donnée par D0(p) = 3000 – 100p. Trouvez le nombre d’équilibre de pétroliers (n*) dans le marché. D0(25) = 3000 – 2500 = 500. Chaque firme produit EE=50 dans le long-terme et en équilibre. Donc n*=500/50 = 10.

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Court terme : à court terme, chaque entreprise présente sur le marché doit payer le coût de location d’un pétrolier, qu’elle s’en serve pour transporter du pétrole ou non. Par ailleurs, les firmes ne peuvent entrer ou sortir du marché librement.
(5) Calculez la fonction d’offre de court terme d’une firme. SCM(q) = SCFM(q) + SCVM(q) SCFM(q) = 125/q SCVM(q) = 20 + q/20. SCm (q) = 20 + q/10 SCm est toujours au-dessus du SCVM. Donc SCm est identique à la courbe d’offre de la firme dans le court terme.

(6) Supposons que la demande de services de pétrolier se déplace de façon inattendue et vaut désormais D1(p) = 3200 – 100p. Trouvez le prix d’équilibre de court terme avec n* entreprises présentes sur le marché. Combien chaque firme va-t-elle produire ? Rappelons que la fonction de l’offre suit la courbe du coût marginal: SCm (q) = 20 + q/10, or p=20 + q/10 Inversons cette fonction pour q = 10p – 200 Dans le court terme, on a n* = 10 firms. L’offre du marché dans le court terme est 100p – 2000. D1(p) = 3200 – 100p = 100p – 2000. 5200 = 200p. Prix d’équilibre = 26. Chaque firme produit 10p – 200 = 60 unités à ce prix. La quantité totale produite par les 10 firmes est 600.

Page: 6

Dynamiques d’équilibres
(7) Supposons que les firmes anticipent que le demande va rester au niveau de D1(p) =3200 – 100p. Donnez le nouveau nombre de firmes présentes sur le marché à long terme, n**. Le prix d’équilibre à long terme est 25 comme décrit si-dessus. D1(25) =3200 – 2500 = 700. Chaque firme produit son échelle d’efficience EE = 50. n**=700 / 50 =14.

(8) De façon (à nouveau) inattendue, la demande revient à son niveau précédent, D0(p) = 3000 – 100p. Trouvez le prix d’équilibre de court terme avec n** firmes après ce changement non anticipé. Vous pouvez donner votre réponse sous forme de fraction (i.e. x/y). Rappelons la fonction d’offre du court terme: q = 10p – 200 Dans le court terme on a n** = 14 firmes. L’offre du marché du court terme est 140p – 2800. D0(p) = 3000 – 100p = 140p – 2800. 5800 = 240p. Prix d’equilibre = 580/24 = 145/6.

(9) Supposons que la demande reste au niveau D0(p) = 3000 – 100p. Que devient le prix d’équilibre à long terme ? Avec 14 firmes dans le marché, le profit économique des firmes à long-terme est négatif. Donc, des firmes vont sortir du marché jusqu’à ce que le prix ajuste et reprend son niveau de long terme qui est égal au minimum du coût moyen = 25. D’après les réponses des parties précédentes de l’exercice, on a le nombre de firmes à l’equilibre (quand la demande est D₀(p)) est 10. Avec un niveau de firmes = 14, 4 firmes doivent sortir du marché et le prix d’equilibre augmente jusqu’à son niveau initial de 25.

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Exercice 3
On considère un problème de type “tacots” (“lemons”) vu en classe, avec de nombreux vendeurs ainsi que de nombreux acheteurs identiques, qu’on suppose neutres face au risque. La valeur d’une voiture pour un vendeur est v et (3/2)v pour un acheteur. On suppose que la valeur d’une voiture est tirée (de façon indépendante) à partir d’une distribution uniforme sur l’intervalle [1/2,1]. 1) Pour qu’une allocation soit Pareto-efficiente (ou qu’elle maximise le surplus), que devraiton faire de ces voitures ? La maximisation de la somme du surplus du vendeur et de l’acheteur implique d’allouer la voiture à l’acheteur puisque la valeur de la voiture est plus élevée pour l’acheteur (3v/2) que pour le vendeur (v). 2) Supposons que le vendeur et l’acheteur connaissent tous deux la valeur de v. Pour quels niveaux de prix le marché s’équilibrerait-il ? Le marché s’équilibre pour tout prix compris entre v et 3v/2, et la voiture est échangée à ce prix. 3) Supposons qu’aucun des vendeurs ou acheteur ne connaisse la valeur exacte de v. Les deux parties connaissent uniquement la distribution de v. Pour quels niveaux de prix le marché s’équilibrerait-il cette fois-ci ? Dans ce cas, la valeur espérée de v est ¾. C’est la valeur espérée de la voiture pour le vendeur. Pour l’acheteur, la valeur espérée est 3/2 fois E[v], ce qui est égal à 9/8. Aussi, le marché s’équilibre pour tout prix compris entre ¾ et 9/8 et la voiture est échangée à ce prix. 4) Supposons à présent que le vendeur connait le type de voiture qu’il possède et que l’acheteur ne connaît que la distribution de cette valeur. Y-a-t-il un équilibre tel que toutes les voitures sont vendues ? Si oui, montrez cet équilibre et sinon expliquez pourquoi il n’existe pas. Pour tous les vendeurs prêts à vendre, p≥1. La qualité (v) moyenne est 3/4. La valeur espérée pour un acheteur de l’achat d’une voiture est (3/2)(3/4)=(9/8). Pour les acheteurs prêts à acheter, p≤ (9/8). Ainsi on peut construire un équilibre où toutes les voitures sont vendues en choisissant un prix compris entre 1 et 9/8. 5) Comme dans la question précédente, on suppose que le vendeur connait le type de voiture qu’il possède et que l’acheteur ne connaît que la distribution de cette valeur. Y-a-t-il un équilibre dans lequel le prix vaut ¾ ? Si oui, montrez cet équilibre (donnez les types de voitures vendues à ce prix) et sinon expliquez pourquoi cet équilibre n’existe pas. A ce prix, les vendeurs avec un type de voiture sur l’intervalle [(1/2),(3/4)] vendraient. La qualité espérée serait de (5/8). La valeur espérée pour l’acheteur de la voiture serait de (3/2)(5/8)=(15/16). Puisque cette dernière valeur est supérieure au prix = 3/4, il existe un équilibre.

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