Econ 301: Topics in Microeconomics: Quiz 1
Sanjaya DeSilva, Spring 2008
Suppose Sanjay is a shoplifter that was caught stealing a six-pack of beer from the
convenience store. The police finds out that he likes only two things, drinking beer
(B) and going running (R). The two activities are described by the following ”consumption” functions. Xi refers to goods used for the activity, and Ti refers to the
time used for the activity.
XB = aB B
(1)
TB = bB B
(2)
TR = bR R
(3)
U = B α R1−α
(4)
The utility function is
Suppose he can get his beer by purchasing at the store, or by stealing from the
store. The quantities purchased and stolen are XBM and XBC respectively.
The total amount of beer (both stolen and bought) is represented by
XB = XBM + (1 − g)XBC
(5)
where g is an index of his ethical standing, i.e. how much discounts a stolen beer
relative to a purchased beer because of guilt.
Purchasing beer costs money, and stealing beer costs time. Specifically, the stealing production function is,
XBC = (1 − e)tνC
(6)
where tC is the time spent towards stealing, and e is a measure of law enforcement
that determines how effective his attempts at stealing are, i.e. if e = 0, all attempts
to steal will succeed, and if e = 1, all attempts to steal will fail.
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The time and budget constraints are
TC + TR + TB + L = T
PB XBM = wL + V
where L is time spent working, T is the time endowment, and V is nonearned
income.
1. State the maximization problem, set up the Lagrangian, construct the FOCs,
and derive functions for the optimal time allocated to stealing, and the demand
for stolen and store bought beer.
The Lagrangian function is
L = B α R1−α + λ(wT + V − PB XBM − wTC − wTR − wTB )
(7)
Ordinarily, the choice variables would be XBM , TC , TR and TB . However, note
that XB , TB and TR enter the production of B and R in fixed proportions.
Therefore, the Lagrangian can be expressed with B, R and TC as the choice
variables as follows,
L = B α R1−α +λ(wT +V −PB aB B −PB (1−g)XBC −wTC −wbR R−wbB B) (8)
which can be simplified to
U = B α R1−α +λ(wT +V −(PB aB +wbB )B −wbR R+PB (1−g)XBC −wTC ) (9)
Now, the choice variables are R, B and TC , and the corresponding FOCs are
∂U
= λwbR
∂R
∂U
= λ(PB aB + wbB )
∂B
∂XBC
λPB (1 − g)
= λw
∂TC
2
(10)
(11)
(12)
From the last FOC, we get the condition for allocation of time to stealing beer,
(1 − g)
∂XBC
w
=
∂TC
PB
(13)
Substitution the functional form,
ν(1 − g)(1 − e)TCν−1 =
w
PB
(14)
The optimal amount of time spent stealing is obtained by solving this equation,
TC∗
=
ν(1 − g)(1 − e)PB
w
1
1−ν
(15)
The demand for stolen beer is obtained by substituting the optimal time to the
stealing production function,
∗
XBC
1
ν 1−ν
ν(1 − g)PB
= (1 − e)
w
(16)
Dividing the first FOC by the second, we get,
∂U
∂R
∂U
∂B
=
wbR
PB aB + wbB
(17)
Substituting the functional form, this reduces to
(1 − α)B
wbR
=
αR
PB aB + wbB
(18)
wT + V = (PB aB + wbB )B + wbR R − PB (1 − g)XBC + wTC
(19)
From the constraint,
Substituting our results so far in the constraint,
ν
1
αaB
(1 − g)(1 − e)PB 1−ν
XB =
wT + V +
ν
− ν 1−ν
(20)
(PB aB + wbB )
wν
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Therefore, the demand for store bought beer is,
∗
∗
XBM
= XB∗ − (1 − g)XBC
"
#
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ν
1
(1 − g)(1 − e)PB 1−ν 1−ν
αaB
ν (1 − PB ) − ν 1−ν
wT + V +
=
(PB aB + wbB )
wν
2. Using your results, explain how each of these demand functions are influenced
by changes in
(a) Wage (w) :Stealing time falls as the wage rises because the opportunity
cost of time increases. Therefore, the demand for stolen beer falls. For
store-bought beer, there are three effects, the income effect that is positive,
the substitution effect between work and leisure that is negative, and the
positive effect through the decrease in stolen beer.
(b) nonearned income (V ): Neither the stealing time nor the demand for stolen
beer is related to nonearned income. The demand for store-bought beer is
positive related due to the income effect.
(c) Law enforcement index (e): Decreases time stealing, and the demand for
stolen beer. This increases the demand for store bought beer, albeit by
less than proportionately.
(d) Guilt index (g): Works just like the law enforcement index.
(e) Preference for beer (α): Does not affect the demand for stolen beer, but
increases the demand for store bought beer.
(f) Price of beer (PB ): Increases the time spent stealing and the demand for
stolen beer, but decreases the demand for store-bought beer.
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