Advanced Public Economics, 2016/17, 239.507
1. lecture (October, 3)
1.
Let the transformation curve x  100  2G be given. The utility functions of two
households are u1 (G, x1 )  G 2 ( x1 ) 2 , u 2 (G, x 2 )  G 3 ( x 2 ) 3 . Interpret the parameter 2
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2
in the transformation curve. Find an (arbitrary) allocation (G, x1 , x 2 ) such that
x  x1  x 2 and (G, x) is on the transformation curve. Check whether this allocation is
Pareto-efficient.
2. lecture (October, 10)
2.
Consider the previous exercise. Let individual endowments
x 1 = 60, x 2  40 ( x 1  x 2  100 ) be given. Compute the Lindahl-equilibrium (set px = 1,
that is, marginal costs of the collective good are 2). To this end, consider for household 1
the problem: max u1 (G, x1 ), s. t. pGG  x1  x 1 , and derive the demand curve for G as a
function G = ..., with pG and x 1 on the right-hand side. Then solve for pG . Do the same
for household 2. Then add the two right-hand sides and set the sum equal to c. Sketch the
procedure.
3. lecture (October, 17)
3. a) Let the utility function of some household be uh(G,xh) = ( x h ) G1 , with initial endowment
x h , and price pG of the collective good. Determine the reaction function of this household
depending on the contribution gk of a second household k. That is, consider the optimization
problem
max ( x h ) ( g h  g k )1 ,
s.t. x h  pG g h  x h ,
x h  0, g h  0.
From the first-order condition for this problem and the budget constraint the optimal choice of
gh in the form of gh = ... can be computed, where on the right-hand side gk and pG occur, as well
as x h and . (In the same way, the optimal choice of xh can be computed.)
Find a general solution and specify for  = 0,6, x h  10 , pG = 2
Consider a second household k with identical preferences and initial endowment. Determine the
Nash equilibrium.
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3. b) Show that the Nash equilibrium in a) is not a Pareto-efficient allocation (Compute
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2
-( MRSG , x + MRSG , x ) at the Nash-equilibrium and compare this to - MRTG , x  pG ).
4. lecture (October, 24)
4. Let the utility functions of two households be given by
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u1 (G, x1 ) = x1 + G 2 , u 2 (G, x 2 ) = x 2 + 2G 3
where G describes a discrete collective good and xh, h = 1,2 the amounts of a private good. Let
initial endowments be x 1 , x 2 .
a)
Compute the reservation prices r1, r2 for the collective good from the equation
u h (0,x h ) = u h (1,x h - r h ) , h = 1,2, using the above specifications of the utility functions.
b)
Let c = 2 be the costs of the collective good. Do there exist cost shares g1 and g2 of the two
households such that the provision of the collective good (G = 1) financed by g1 and g2
represents a Pareto improvement compared to G = 0? Explain why this requires to find to g1 and
g2 with g1 + g2 = c and gh < rh for h = 1, 2.
5. lecture (November, 7)
5.
Consider three individuals whose reservation prices (willingness to pay) for a discrete collective
good are r1 = 5 000, r2 = 2 000, r3 = 3 000.
The cost of the collective good is c = 8 500.
The government announces cost shares s1 = 3 000, s2 = 2 500, s3 = 3 000, and it sets Kh = 0, for
all h = 1, 2, 3.
Describe for individual 1 what she/he has to pay and receives as a side payment for both
possible outcomes (that G = 1 or G = 0 is chosen by the government), taking arbitrary w 2 and
w 3 as given. What are the resulting utility positions of individual 1 in both possible outcomes?
Why is choosing w1  w1 optimal for individual 1?
6. lecture (November, 14)
6.
Consider an economy with two households h = 1,2 and two goods. One of the goods, denoted by
xh, is a purely private good, the other good, denoted by zh, causes an external effect on the
respective other household. Let the utility function of the two households be
u1 ( x1 , z1 , z 2 )  x1  4 z1  z 2
u 2 ( x 2 , z 2 , z1 )  x 2  6 z 2  12 z1 .
Let pz be the price of z (in terms of x), px = 1.
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a)
Formulate the two conditions for a Pareto-efficient allocation for these utility functions.
(Note: v1 ( z1 )  4 z1 , 1 ( z 2 )   z 2 . The same for household 2.) Compute the optimal z1*
and z2* from these conditions.
b)
Compute the private demand z1 , z 2 (as a function of pz ). Hint: compute z1 from the
equation dv1 / dz1  pz , and the same for z 2 ).
c)
Compute z1*, z 2 *, z1 , z 2 for the special value pz  2. Interpret, why z h  z h * .
Note: Only in case of a quasilinear utility function it is possible to compute z h , z h * without
reference to the initial endowments x h .
7. The problem of the commons: Lake Ec can be freely accessed by fishermen. The cost of sending a
boat out on the lake is r > 0. When b boats are sent out onto the lake, f(b) fish are caught in total [so
each boat catches f(b)/b fish], where f'(b) > 0 and f"(b) < 0 at all b  0. The price of fish is p > 0, which
is unaffected by the level of the catch from Lake Ec.
a)
Characterize the equilibrium number of boats that are sent out on the lake.
b)
Characterize the optimal number of boats that should be sent out on the lake. Compare this with
your answer to (a).
c)
What per-boat fishing tax would restore efficiency?
d)
Suppose that the lake is instead owned by a single individual who can choose how many boats to
send out. What level would this owner choose?
Hint: Consider the precise meaning of f(b): the total amount of fish which is caught if b boats are sent
out. Obviously then, the derivative of f can be interpreted as the increase of the total catch, if a further
boat is sent out.
In contrast: The amount of fish, which is caught by this particular further boat itself, can be described
by the average f(b)/b.
Assume that decisions to send out a further boat are made subsequently by independent fishermen.
Given that b boats are already on the lake, a further boat will catch the average amount of fish. If the
value of the catch (note: price p per unit of fish) exceeds the cost, the boat will be sent out. The
equilibrium number of boats will therefore be determined by the condition that the value of (average)
catch of the last boat equals cost r.
On the other hand, from an efficiency point of view, it is clearly optimal to send out a further boat, if
the value of the additional amount of fish, which is caught if one more boat is sent out, exceeds (or at
least covers) the cost of the further boat.
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With f ' > 0, f " < 0 (diminishing marginal returns): which relation must hold between b and b̂ , if b
is obtained from the condition f ' ( b ) = r/p, and b̂ is obtained from the condition f( b̂ )/ b̂ = r/p (let r
and p be fixed positive numbers)? Consider the diagram, where f(b) and the straight line with slope r/p
are drawn. Illustrate how b and b̂ are determined.
f(b)
b
tan   r / p
7. lecture (November, 21)
8.
Let the production f(K,z) of a firm producing a single good be
f(K,z) = K0,8z0,4,
(1)
where K is an input (e. g. capital) and z is a level of emission.
The firm wants to produce 100 units of the good.
a)
The government restricts the firm to emit not more than z  40 . Which amount of the factor
K has to be employed by the firm?
b)
Let the price per unit of K be pK = 2 and the government imposes a tax tz = 3 per unit of z.
Which combination of K and z is chosen by the firm in order to minimize costs? (Hint: use
(1) and the condition that the marginal rate of technical substitution between z and K is equal
to the price ratio, that is (f / z ) / (f / K )  t z / pK .)
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8. lecture (November, 28)
9. An approximate formula (for "small" tax rates) to calculate the deadweight loss (excess burden) of a
tax in a partial market is:
DWL  12  2 pxη ,
where  denotes the tax rate, p the (fixed) producer price without tax, q the consumer price, x the
quantity demanded in case of no tax, and  the elasticity of demand (in absolute value).
q
C
p(1+ )
q
p
E
B
x
x
x
To arrive at this formula one starts with the formular for the area of the triangle BEC, (q   x) / 2 ,
eliminates  x by using the definition of the demand elasticity  
x / x
, and sets q = p, q  p .
q / q
Show this derivation exactly. Calculate DWL for the demand function x = 100 - 2q for p (=q) = 20 and
 = 0,1.
Note: According to this formula, the DWL is quadratic in the tax rate. A more precise definition of the
DWL according to the Hicksian measure would require to work with the compensated demand
elasticity. See Stiglitz (2000), Economics of the Public Sector, pp. 527-528 (note: different symbols).
9. lecture (December, 12)
10. Consider the utility function u(c, F )  c0,4 F 0,6 , where c is a (universal) consumer good
and F denotes leisure. Furthermore, the upper limit of the available time is given with F  16 ,
the wage rate is w = 5 and the price of the consumer good is p = 1.
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a)
Draw an (arbitrary) indifference curve for the utility function (calculate some points on
the indifference curve).
b) Formulate and draw the budget line without tax and in case of a proportional income tax
 l  0.2 .
c)
What is the tax rate  on the consumer good that would result in the same budget
constraint as the income tax  l  0.2 ?
d) Additional task: Determine the optimal household decision with and without tax.
11. Let two goods and labor be given, as well as three taxes: 1, 2 on the two goods and l on
labor income. The prices of the goods are p1, p2 and the wage rate is w.
a)
Formulate the budget condition of the household.
Transform the budget constraint to describe the following situations
b) There is no tax on labor income.
c)
There is no tax on good 2.
Specify these three variants of the budget condition for 1 = 0.1, 2 = 0.2, l = 0.3. Calculate
the equivalent tax rates  1' , 2' for the formulation b), analogous to the formulation c).
10. lecture (January, 10)
12. Consider the utility function of exercise 10, with symbol x instead of c:
u( x, F )  x0,4 F 0,6 . Let a linear income tax    wL be given, with L  F  F being
labor time and w the wage rate (wL is pre-tax income);  is a uniform lump-sum tax,  (<
1) is the marginal tax rate.
Compute x and y from the individual maximization problem. Hint: the condition
MRSFx  w(1   ) is the same as before (see exercise 10), while the budget constraint
now includes . From these two conditions determine x and F (thus also L  F  F ) in
terms of ,  (clearly, also F and w occur). Explain whether wL increases with w.
13. Continue with the results of example 12. Set F  1 in the following
a)
Show that labor supply L (and, thus, pre-tax income wL) increases with an increase of the
lump-sum element  . (Compute L /  and consider its sign.)
Note: An increase of  has only an income effect on household behavior. Leisure is a
normal good in our case of Cobb-Douglas preferences, therefore leisure goes down (labor
goes up) with a reduction of the budget, as a consequence of an increase of  .
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b) Show that labor supply L (and, thus, pre-tax income wL) decreases with an increase of the
marginal tax rate  . Assume  < 0. (Compute L /  and consider its sign.)
11. lecture (January, 16)
continue example 13
c)
Consider the revenues from the proportional part of the tax,  wL . Let w = 8 and    13
and express  wL in terms of  only. Compute that value of  , for which revenues
 wL are at their maximum. (Set ( wL) /  equal to zero and determine  from this
equation; choose the smaller value of the two solutions (  <1)).
Note that at larger values of  , tax revenues are declining in  (inefficient side of the
Laffer curve.)
14. Consider an overlapping-generations model of an economy, with individuals living for
two periods, being active and working in their first and retired in their second period of
life. Labor time in the first period is 1.
Let the number Nt of active individuals grow by 5% each period t; their (uniform) wage
rate wt grows by 40% each period t, t = 0,1, ... .1 Initially, the number of individuals is
N0 = 100, the wage rate is w0 = 10.
a) If the contribution rate in period t = 1 is 1 = 0,25, which average pension x1 can be
paid to a pensioneer in t = 1 (note that the number of pensioneers is N0). Compute the
ratio x1/w1.
b) Assume that 0 = 1 = 0,25. Compute the internal rate of return i1 to the contribution
ˆ  N) , where ŵ
paid in period 0 (use 1  i1  x1 / (0 w 0 ) ). Check that 1  i1  (1  w)(1
and N are the respective growth rates introduced above.
c) Consider period t = 10. Compute the wage rate w10 and the number of working
individuals N10. Compute x10 (number of pensioneers is N9) with 10 = 0 = 0,25 and
check that the ratio x10/w10 is equal to x1/w1. Moreover, let 9 = 0 = 0,25 and check
that i10 = i1.
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Note that a period here could be 30 years, for example.
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12. lecture (January,23)
d) Now assume that in the initial period t = 0, instead of contributing to a pension system
the individuals purchase government bonds, for which each individual again spends
25% of the wage income w0. Let the interest rate be 60% and the government bonds
are purchased back by the government one period later at a price including interest.
Which amount z1 has to be paid to a (retired) individual in period t = 1 by the
government? What is the to aggregate volume z1N0 of these payments? To finance
z1N0 in period t = 1, the government again sells bonds to the active individuals in this
period. Which share of her/his income w1 must be used for the purchase of
government bonds by each active individual?
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