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Chapter 6
A Simple Economy
Exercise 6.1 In an economy the activity of digging holes in the ground is carried out by self-employed labourers (single–person …rms). The production of
one standard-sized hole requires a minimum input of one unit of labour. No
self-employed labourer can produce more than one hole.
1. Draw the technology set Q for a single …rm.
2. Draw the technology set Q for two …rms.
3. Which of the basic production axioms are satis…ed by this simple technology?
Outline Answer
q2
q2
1
2
q1
q1
Figure 6.1: Convexi…cation of Production
1. See left-hand panel of Figure 6.1. Good 1 is labour, good 2 is holes.
2. See right-hand panel of Figure 6.1; this has been rescaled to be compatible
with the single-…rm case..
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CHAPTER 6. A SIMPLE ECONOMY
3. Nonconvexities are present so that the divisibility assumption is violated.
Nevertheless the production can be (approximately) convexi…ed by increasing the number of …rms.
c Frank Cowell 2006
82
Microeconomics
Exercise 6.2 Consider the following four examples of technology sets Q:
q : q12 + q22 + q3 + q4 0; q1 ; q2 0; q3 ; q4 0
n
o
:
q : q1 [ q2 ]
[ q3 ]
0; q1 0; q2 ; q3 0
A
:
B
1
log (q2 q3 ) 0; q1 0; q2 ; q3 0
2
n
q4
:
q : q1 + q2 + max q3 ;
0; q1 ; q2 0; q3 ; q4
C
:
D
q : log q1
o
0
1. Check whether basic production axioms are satis…ed in each case.
2. Sketch their isoquants and write down the production functions.
3. In cases B and C express the production function in terms of the notation
used in chapter 2.
4. In cases A and D draw the transformation curve.
Outline Answer:
1. Axioms:
A:
B:
C:
D:
Additivity is not satis…ed
All axioms are satis…ed if
All axioms satis…ed.
All axioms are satis…ed if
=
=
< 1:
> 0:
2. Isoquants and production functions:
A: Isoquants are straight lines.
(q) = q12 + q22 + q3 + q4
B: If
=
=
< 1 isoquants will be similar to hyperbolas.
(q) = q1
[ q2 ]
C: Similar to B.
(q) = log q1
[ q3 ]
1
log (q2 q3 )
2
D: Isoquants are rectangular.
(q) = q1 + q2 + max q3 ;
3. Single-output production functions. Let z2 =
h
i1=
B: q1
[z2 ] + [z3 ]
p
C: q1
z2 z3
q2 and z3 =
4. Transformation curve (the boundary of the set Q).
A: A quarter circle.
D: A straight line.
c Frank Cowell 2006
83
q4
q3 . Then:
Microeconomics
CHAPTER 6. A SIMPLE ECONOMY
Exercise 6.3 Suppose two identical …rms each produce two outputs from a
single input. Each …rm has exactly 1 unit of input. Suppose that for …rm 1 the
amounts q11 , q22 it produces of the two outputs are given by
q11
=
1
q21
=
[1
1
]
where 1 is the proportion of the input that …rm 1 devotes to the production of
good 1 and and depend on the activity of …rm 2 thus
2
=
1+2
=
1 + 2[1
2
]:
where 2 is the proportion of the input that …rm 2 devotes to good 1. Likewise
for …rm 2:
q12
=
0 2
q22
=
0
2
[1
]
1
0
=
1+2
0
=
1 + 2[1
1
]:
1. Draw the production possibility set for …rm 1 if …rm 2 sets
…rm 2’s production possibility set if …rm 1 sets 1 = 12 .
2
=
1
2
and
2. Draw the combined production-possibility set.
Outline Answer:
1
1. The production possibility set for …rm 1 if …rm 2 sets 2 = are given by
2
the top left-hand panel of Figure 6.2 below. Firm 2’s position if …rm 1 sets
1
1
is given in the top right-hand panel. The combined production
=
2
possibility set is as shown in the lower panel.
1
; then by de…nition
=
= 2, and …rm 1’s production
2
1
1
1
boundary is given by all the points (2 ; 2[1
]) where 0
1. A
similar argument applies to …rm 2. Now consider the sum of their outputs.
1
2
This will be given by points such as 1 + 2 + 4 1 2 , 2
+ 4[1
1
2
2
1
2
][1
]. So, for example, if
=
= 0 we get output (0; 6); if
1
1
= 2 = 1 we get (6; 0) but if 1 = 2 = we get (2; 2).
2
2. Let
2
=
c Frank Cowell 2006
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Microeconomics
q2`1
q2`2
Firm 1
Firm 2
q1`1
0
0
q2
Combined
Firms
Figure 6.2: Externalities and nonconvexity
c Frank Cowell 2006
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CHAPTER 6. A SIMPLE ECONOMY
Exercise 6.4 Take the model of Exercise 2.14. Assuming that production is
organised to maximise pro…ts at given prices show that pro…t-maximising net
outputs of goods 1 and 2 are:
q1
=
q2
=
A
p1
2
A
p2
2
where pi is the price of good i expressed in terms of commodity 3, and that
maximised pro…ts are
2
2
[p1 ] + [p2 ]
:
=A
4
Outline Answer:
Using standard notation for net output pro…ts are given by
=
3
X
pi yi
(6.1)
i=1
If pro…ts are maximised then production must take place on the transformation
curve. Therefore, substituting we get:
2
To maximise
FOCs:
2
[y1 ] + [y2 ]
A
= p1 y1 + p2 y2
(6.2)
just maximise (6.2) with respect to y1 and y2 . This gives the
pi
2
yi = 0
A
y1
=
y2
=
(6.3)
i = 1; 2, and so
A
p1
2
A
p2
2
(6.4)
(6.5)
from which, using (6.2), pro…ts are
2
=A
c Frank Cowell 2006
2
[p1 ] + [p2 ]
:
4
86
(6.6)
Microeconomics
Exercise 6.5 Take the model of Exercise 5.3 but suppose that income is exogenously given at y1 for the …rst period only. Income in the second period can
be obtained by investing an amount z in the …rst period. Suppose y2 = (z),
where is a twice-di¤ erentiable function with positive …rst derivative and negative second derivative and (0) = 0, and assume that there is a perfect market
for lending and borrowing.
1. Write down the budget constraint.
2. Explain the rôle of the Decentralisation Theorem in this model
3. Find the household’s optimum and compare it with that of Exercise 5.3.
4. Suppose (z) were to be replaced by
a¤ ect the solution?
(z) where
> 1; how would this
Outline Answer:
1. The budget constraint is
x1 +
x2
1+r
A(z)
(6.7)
where the present value of lifetime income is
A(z) := y1
z+
(z)
1+r
(6.8)
x2
iy
*
ix′A= y ′
ix
*
1+r
x1
Figure 6.3: Lifetime income: self investment
2. Use the decentralisation result. Given that borrowing/lending is possible
at the rate of interest r the right thing to do is to maximise the value of
lifetime income (“pro…ts”) at point y in Figure 6.3 and then maximise
utility using this maximised lifetime income (point x ).
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Microeconomics
CHAPTER 6. A SIMPLE ECONOMY
y2
φ (z)
z
Figure 6.4: The production of human capital
3. Otherwise, without a bank loan one would be constrained to point x0 =
y0 := (y1 ; y2 ). Maximising lifetime income implies
@A(z)
=
@z
1+
z (z)
=0
1+r
(6.9)
for an interior solution. This …rst-order condition implies that the optimal
value z must satisfy
(6.10)
z (z ) = 1 + r;
the marginal rate of transformation of current income into future income
equals the price ratio.
1. It is obvious that if were to be replaced by
( > 1) the opportunity
set is “stretched” in the direction of the vertical axis in Figure 6.4. We
expect that maximised lifetime income will increase. So consumption will
increase in both periods. What happens to optimal investment? From the
…rst-order condition (6.9) we …nd
z (z
)=1+r
(6.11)
from which we could get the optimal investment z . If we increase then
the z obtained from (6.11) will change. To see by how much z changes
di¤erentiate (6.11) thus:
z (z
from which we have
c Frank Cowell 2006
) d +
dz
=
d
zz (z
z (z
zz (z
88
) dz = 0
)
)
>0
(6.12)
Microeconomics
Exercise 6.6 Apply the model of Exercise 6.5 to an individual’s decision to
invest in education.
1. Assume the parameter represents talent. Will more talented people demand purchase more education?
2. How is the demand for schooling related to exogenous …rst-period income
y1 ?
Outline Answer:
The optimal purchase of education z increases with ability , but is independent of initial money income y1 .
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CHAPTER 6. A SIMPLE ECONOMY
Exercise 6.7 Take the savings model of Exercise 5.4. Suppose now that by
investing in education in the …rst period the consumer can augment his future
income. Sacri…cing an amount z in period 1 would yield additional income in
period 2 of
1 e z
where
> 0 is a productivity parameter.
1. Explain how investment in education is a¤ ected by the interest rate. What
would happen if the interest rate were higher than
1?
2. How is the demand for borrowing a¤ ected by (i) an increase in the interest
rate r and (ii) an increase in the person’s productivity parameter ?
Outline Answer
1. By the separation theorem the right thing for the consumer to do is to
maximise net income and then choose optimal consumption. By de…nition
net income is
y2 + [1 e z ]
M = y1 z +
1+r
The FOC for a maximum is
e z
@M
=
@z
1+r
1=0
which has as a solution
z = z (r; ) := log
(6.13)
1+r
Clearly this only makes sense if
1+r
(6.14)
otherwise investment is zero. If (6.14) holds then z is decreasing in its
…rst argument and increasing in its second argument.
2. Assuming (6.14) holds, maximised income is:
M
= M (r; ) := y1
= y1 + log
log
1+r
y2 +
1+
1+r
1+r
+
y2 +
[1 e
1+r
z
]
(6.15)
Hence in this model optimal period-1 consumption is
x1 = k +
y1
k + log
1+r
1+
y2
k+
1+r
Borrowing is given by
B (r; ) := x1 + z (r; )
c Frank Cowell 2006
90
y1
:
(6.16)
Microeconomics
and so, on evaluating this from (6.13) and (6.16), we have
B (r; ) =
y2
k+
1+r
1 + [1
] log
1+r
[y1
k]
Dei¤erentiating we have
@B (r; )
@r
=
@B (r; )
@
=
y2
k+
2
[1 + r]
1
+
1+r
1
1+r
>0
If borrowing decreased with r in the no-education case – see equation
(5.17) in Exercise 5.4 – it must certainly decrease in this case too. Borrowing increases with .
c Frank Cowell 2006
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