Problems for the fifth seminar

```Problems for the fifth seminar
ECON4230/35 Microeconomic Theory — Fall semester 2010
Solutions to the problems will be presented in week 45.
Please direct any questions to Geir B. Asheim (Room ES1152, Tel: 228 55498,
e-mail: g.b.asheim@econ.uio.no)
Problem 1 (Partial equilibrium analysis) Varian Problems 13.2, 13.4.
Problem 2 (Partial equilibrium analysis) The preferences of a representative consumer are represented by the utility function
u(x, z) = v(x) + z ,
where x is the quantity consumed of a particular good and z is the money
spent on other goods.
(a) What must be imposed on v in order to ensure that u is quasi-concave?
(b) Show that the Marshallian demand function for good x is identical to the
Hicksian demand function for good x. Show that this demand function,
D(p), satisfies
p = v 0 (D(p)) .
(c) Show that
Z
x00
v 0 (x)dx
x0
is the willingness-to-pay for an increase in the consumption of good x
from x0 to x00 .
(d) Suppose that good x is supplied by a competitive industry. Discuss the
welfare properties of a competitive equilibrium in the market for good x.
Problem 3 (Partial equilibrium analysis) Varian Problems 14.2, 14.6,
14.10.
Problem 4 (Partial equilibrium analysis) A producer that sells a product
x is a monopolist in a market with 60 consumers. 20 of these are high-demand
consumers with utility function uH (x, y) = 5x − 21 x2 + z, and 40 of these are
1
low-demand consumers with utility function uL (x, y) = 3x − 12 x2 + z. In these
utility functions z denotes the money spent on other goods. The monopolist
has a constant unit cost equal to 1.
(a) In a welfare optimum, how many units of the product does each highdemand and low-demand consumer consume?
The monopolist sells the product by offering a contract (rH , xH ) to each of
the high-demand consumers, meaning that the output xH can be purchased
for a total payment of rH , and, likewise, a contract (rL , xL ) to each of the
low-demand consumers. It is not possible for consumers to resell the product
to other consumers.
(b) Assume first that the monopolist can differentiate between the highdemand and low-demand consumers. What is the profit-maximizing pair
of contracts, (rH , xH ) and (rL , xL ), and what is the monopolist’s total
profit?
(c) Assume next that the monopolist can not differentiate between the highdemand and low-demand consumers, so that each type of consumers must
(weakly) prefer the contract that is intended for their type. In this case,
what is the profit-maximizing pair of contracts, (rH , xH ) and (rL , xL ),
and what is the monopolist’s total profit? Why does the monopolist
reduce xL in this case, as compared to the case under part (b)?
(d) Redo parts (b) and (c) under the assumption that the 60 consumers
in the market are divided equally between the two types (i.e., 30 are
high-demand consumers and 30 are low-demand consumers).
Problem 5 (Partial equilibrium analysis)
This problem is motivated by an article in the Norwegian newspaper Aftenposten (reproduced below), reporting that organic vegetables are labeled and
sold as conventional vegetables. When asked “Why do you remove the ecolabeling? Why not just reduce the price for organic vegetables?” the producer
responds: “Then we will just compete with ourselves.”
Assume that there are four consumers with quasi-linear utility function

 120(x + x ) − 10(x + x )2 + z if x + x ≤ 6 ,
C
O
C
O
C
O
uC (xC , xO , z) =
 360 + z
if xC + xO &gt; 6 ,
2
and one consumer with quasi-linear utility function



120xC + 180xO − 10(xC + xO )2 + z if xC + xO ≤ 6 ,





if x + x &gt; 6

C
O
360 + 60xO + z
uC (xC , xO , z) =
and xO ≤ 6 ,






 720 + z
if xO &gt; 6 ,
where xC is the consumption of conventionally labeled vegetables, xO is the
consumption of organic vegetables with eco-labeling, and z is money spent
on other goods. Hence, only one of five consumers has preference for organic
vegetables.
The produced quantity of organic vegetables equals 6 and the produced
quantity of conventional vegetables equals 8. Only organic vegetables can be
sold with eco-labeling, while both kinds can be sold with conventional labeling.
In the latter case, all consumer treat them as conventional vegetables.
What is the producer’s revenue if all the organic vegetables are eco-labeled?
How many units of the organic vegetables will be sold with eco-labeling if the
producer seeks to maximize revenue? What is the maximal revenue?
——————
Begge er &oslash;kosalat — bare én merkes
&Oslash;kologiske gr&oslash;nnsaker blir merket og solgt som konvensjonelle fordi det er produsert
for mye. Det betyr tapte inntekter for b&oslash;ndene.
BERIT E. BAUMBERGER, Sande, Vestfold
F&oslash;rst publisert: 09.08.07 — Oppdatert: 09.08.07 kl. 00:01
http://www.aftenposten.no/nyheter/iriks/article1927574.ece
– I år hadde mange stor tro på det norske &oslash;kologiske gr&oslash;nnsaksmarkedet, og
b&oslash;nder satset stort. Men nå sliter vi med å få solgt en del av produktene, sier
ansvarlig for &oslash;kologiske produkter i Gartnerhallen, Anders H&oslash;rthe.
Gartnerhallen leverer gr&oslash;nnsaker til blant annet NorgesGruppen og BAMA. De
opplyser at de står for rundt 50 prosent av de norske &oslash;kogr&oslash;nnsakene.
– L&oslash;sningen for en del av b&oslash;ndene er å selge &oslash;kovarene som billigere, konvensjonelle varer. Det er trist at det må v&aelig;re slik, sier H&oslash;rthe.
Han sier det koster mer å produsere &oslash;kologisk, og at prisen må gjenspeile det.
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Merker om.
&Oslash;kologisk Drift AS er den klart st&oslash;rste leverand&oslash;ren av &oslash;kologiske gr&oslash;nnsaker
til Gartnerhallen. I år sliter de med å få omsatt blant annet isbergsalat, squash,
blomkål og brokkoli.
– Halvparten av all isbergsalaten og nesten all squashen blir solgt konvensjonelt,
selv om den er produsert &oslash;kologisk, forteller eier og bonde Ole Martin Siem, som
driver kologisk Drift AS sammen med Arild Hermansen.
– Vi går i null eller taper penger på &oslash;kogr&oslash;nnsakene som vi må selge konvensjonelt, men det er bedre enn at gr&oslash;nnsakene råtner bort på lager, sier Hermansen.
– Hvorfor fjerner dere &oslash;ko-merket, hvorfor ikke bare sette ned prisen på de &oslash;kologiske varene?
– Da konkurrerer vi jo mot oss selv. Vi skulle gjerne solgt &oslash;koproduktene billigere
hvis det var en del av en ordentlig kampanje ut mot kunden. Men det er vanskelig
å få til på så kort tid, sier Siem.
For ambisi&oslash;st.
H&oslash;stingen av salat er i full gang på jordene i Sande i Vestfold, og på lageret ved
Oreberg gård står det palle på palle med &oslash;kologisk squash.
– Har dere rett og slett satset for ambisi&oslash;st i forhold til markedet?
– Det er vanskelig å si, vi vet at ettersp&oslash;rselen blant forbrukere er der. Vi mener
de som selger produktene, rett og slett ikke er flinke nok til å tilby kundene varene,
sier H&oslash;rthe.
Han understreker at problemet dreier seg om de nye &oslash;koproduktene, som salat,
brokkoli og squash. For andre &oslash;kogr&oslash;nnsaker går nemlig tallene i v&aelig;ret.
– Vi har aldri solgt så mye &oslash;kogulr&oslash;tter og -poteter. Det er selvsagt sv&aelig;rt
positivt, sier H&oslash;rthe.
Unngår å merke om.
Coop er den andre store akt&oslash;ren på det &oslash;kologiske markedet i Norge, og den
mest erfarne. De har direkte kontakt med sine produsenter, og sier de nesten alltid
unngår å merke &oslash;kologiske varer som konvensjonelle.
– Hvis det er produsert for mye &oslash;kologisk, kj&oslash;rer vi heller en priskampanje enn
å merke gr&oslash;nnsakene som konvensjonelle, sier &Oslash;yvind Bergstr&oslash;m, leder av frukt- og
gr&oslash;ntavdelingen i Coop.
Coop har drevet lenge med &oslash;kologiske gr&oslash;nnsaker og mener de har gode rutiner
for å unngå at &oslash;kologiske varer blir merket konvensjonelt.
– Vi &oslash;ker omsetningen sv&aelig;rt mye på nesten alle &oslash;kovarer, men i sommer har det
v&aelig;rt litt vanskeligere å få solgt salat, sier Bergstr&oslash;m
4
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