Microeconomics Solutions 02

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Chapter 2

The Firm

Exercise 2.1 Suppose that a unit of output q can be produced by any of thefollowing combinations of inputs

z1 =

0:20:5

; z2 =

0:30:2

; z3 =

0:50:1

1. Construct the isoquant for q = 1.

2. Assuming constant returns to scale, construct the isoquant for q = 2.

3. If the techniquez4 = [0:25; 0:5] were also available would it be included inthe isoquant for q = 1?

Outline Answer

q = 1

z1

z2

0 0.2

0.5

0.3

0.2

0.5

0.1

z1

z2

z3

z4

Figure 2.1: Isoquant simple case

3


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Microeconomics CHAPTER 2. THE FIRM

0 0.2

0.5

q = 1

z2

z3

0.3

0.2

z4z1

z1

z2

0.5

0.1

Figure 2.2: Isoquant alternative case

0 0.2

0.5

q = 1

z2

z3

0.3

0.2

q = 2

z1

z1

z2

0.5

0.1

Figure 2.3: Isoquants under CRTS

cFrank Cowell 2006 4


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Microeconomics

1. See Figure 2.1 for the simplest case. However, if other basic techniques

are also available then an isoquant such as that in Figure 2.2 is consistentwith the data in the question.

2. See Figure 2.3. Draw the rays through the origin that pass through eachof the corners of the isoquant for q = 1. Each corner of the isoquant forq = 2. lies twice as far out along the ray as the corner for the case q = 1.

3. Clearly z4 should not be included in the isoquant since z4 requires strictlymore of either input to produce one unit of output than does z2 so that itcannot be ecient. This is true whatever the exact shape of the isoquantin see Figures 2.1 and 2.2

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Exercise 2.2 A rm uses two inputs in the production of a single good. The

input requirements per unit of output for a number of alternative techniques aregiven by the following table:

Process 1 2 3 4 5 6 Input 1 9 15 7 1 3 4Input 2 4 2 6 10 9 7

The rm has exactly 140 units of input 1 and 410 units of input 2 at its disposal.

1. Discuss the concepts of technological and economic eciency with refer-ence to this example.

2. Describe the optimal production plan for the rm.

3. Would the rm prefer 10 extra units of input 1 or 20 extra units of input2?

Outline Answer

1. As illustrated in gure 2.4 only processes 1,2,4 and 6 are technically e-cient.

2. Given the resource constraint (see shaded area), the economically ecientinput combination is a mixture of processes 4 and 6.

5

3

0

z

1

z2

.

Economically Efficient

Point

Attainable

Set

4

6

12

Figure 2.4: Economically ecient point

3. Note that in the neighbourhood of this ecient point MRTS=1. So, asillustrated in the enlarged diagram in Figure 2.5, 20 extra units of input2 clearly enable more output to be produced than 10 extra units of input1.

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Microeconomics

Original

Isoquant

20

10

Figure 2.5: Eect of increase in input



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Exercise 2.3 Consider the following structure of the cost function: C(w; 0) =

0, Cq(w

; q) = int(q) where int(x) is the smallest integer greater than or equal tox. Sketch total, average and marginal cost curves.

Outline Answer

From the question the cost function is given by

C(w; q) = kq k + 1; k 1 < q k; k = 1; 2; 3:::

so that average cost is

k +1 k

q; k 1 < q k; k = 1; 2; 3:::

see Figure 2.6.

0q

1 2 3

C(w,q)

Cq(w,q)

C(w,q)/q

Figure 2.6: Stepwise marginal cost

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Microeconomics

Exercise 2.4 Suppose a rms production function has the Cobb-Douglas form

q = z11 z2

2

where z1 and z2 are inputs, q is output and 1, 2 are positive parameters.

1. Draw the isoquants. Do they touch the axes?

2. What is the elasticity of substitution in this case?

3. Using the Lagrangean method nd the cost-minimising values of the inputsand the cost function.

4. Under what circumstances will the production function exhibit (a) decreas-ing (b) constant (c) increasing returns to scale? Explain this using rstthe production function and then the cost function.

5. Find the conditional demand curve for input 1.

2

z1

Figure 2.7: Isoquants: Cobb-Douglas

Outline Answer

1. The isoquants are illustrated in Figure 2.7. They do not touch the axes.

2. The elasticity of substitution is dened as

ij := @log(zj=zi)@log

j(z)=i(z)

which, in the two input case, becomes

= @log

z1z2

@log

1(z)2(z)

(2.1)

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In case 1 we have (z) = z11 z22 and so, by dierentiation, we nd:

1(z)

2(z)=

12

=z1z2

Taking logarithms we have

log

z1z2

= log

12

log

1(z)

2(z)

or

u = log

12

v

where u := log (z1=z2) and v := log (1=2). Dierentiating u with respectto v we have

@u@v

= 1: (2.2)So, using the denitions of u and v in equation (2.2) we have

= @u@v

= 1:

3. This is a Cobb-Douglas production function. This will yield a unique inte-rior solution; the Lagrangean is:

L(z; ) = w1z1 + w2z2 + [q z11 z22 ] ; (2.3)and the rst-order conditions are:

@L(z; )@z1

= w1 1z111 z22 = 0 ; (2.4)

@L(z; )@z2

= w2 2z11 z212 = 0 ; (2.5)

@L(z; )@

= q z11 z22 = 0 : (2.6)Using these conditions and rearranging we can get an expression for min-imized cost in terms of and q:

w1z1 + w2z2 = 1z11 z

22 + 2z

11 z

22 = [1 + 2] q:

We can then eliminate :w1 1 qz1 = 0w2 2 qz2 = 0

which impliesz1 =

1w1

q

z2 =2w2

q

: (2.7)

Substituting the values of z1 and z2 back in the production function we

have 1w1

q

1 2w2

q

2= q

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Microeconomics

which implies

q =

q

w11

1 w22

2 11+2(2.8)

So, using (2.7) and (2.8), the corresponding cost function is

C(w; q) = w1z1 + w2z

2

= [1 + 2]

q

w11

1 w22

2 11+2:

4. Using the production functions we have, for any t > 0:

(tz) = [tz1]1 [tz2]

2 = t1+2(z):

Therefore we have DRTS/CRTS/IRTS according as 1 + 2 S 1. Ifwe look at average cost as a function of q we nd that AC is increas-ing/constant/decreasing in q according as 1 + 2 S 1.

5. Using (2.7) and (2.8) conditional demand functions are

H1(w; q) =

q

1w22w1

2 11+2

H2(w; q) =

q

2w11w2

1 11+2and are smooth with respect to input prices.

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Exercise 2.5 Suppose a rms production function has the Leontief form

q = min

z11

;z22

where the notation is the same as in Exercise 2.4.

1. Draw the isoquants.

2. For a given level of output identify the cost-minimising input combina-tion(s) on the diagram.

3. Hence write down the cost function in this case. Why would the La-grangean method of Exercise 2.4 be inappropriate here?

4. What is the conditional input demand curve for input 1?

5. Repeat parts 1-4 for each of the two production functions

q = 1z1 + 2z2

q = 1z21 + 2z

22

Explain carefully how the solution to the cost-minimisation problem diersin these two cases.

2

z1

A B

Figure 2.8: Isoquants: Leontief

Outline Answer

1. The Isoquants are illustrated in Figure 2.8 the so-called Leontief case,

2. If all prices are positive, we have a unique cost-minimising solution at A:to see this, draw any straight line with positive nite slope through Aand take this as an isocost line; if we considered any other point B on theisoquant through A then an isocost line through B (same slope as the onethrough A) must lie above the one you have just drawn.

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2

z1

Figure 2.9: Isoquants: linear

2

z1

Figure 2.10: Isoquants: non-convex to origin



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3. The coordinates of the corner A are (1q; 2q) and, given w, this imme-

diately yields the minimised cost.

C(w; q) = w11q+ w22q:

The methods in Exercise 2.4 since the Lagrangean is not dierentiable atthe corner.

4. Conditional demand is constant if all prices are positive

H1(w; q) = 1q

H2(w; q) = 2q:

5. Given the linear case

q = 1z1 + 2z2

Isoquants are as in Figure 2.9. It is obvious that the solution will be either at the corner (q=1; 0)

if w1=w2 < 1=2 or at the corner (0;q=2) if w1=w2 > 1=2, orotherwise anywhere on the isoquant

This immediately shows us that minimised cost must be.

C(w; q) = qmin

w11

;w22

So conditional demand can be multivalued:

H1(w; q) =

8>>>>>>>>>:

q1

if w1w2 12

H2(w; q) =

8>>>>>>>>>:

0 if w1w2 12

Case 3 is a test to see if you are awake: the isoquants are not convexto the origin: an experiment with a straight-edge to simulate anisocost line will show that it is almost like case 2 the solution willbe either at the corner (

pq=1; 0) if w1=w2 p

1=2 (but nowhere else). So thecost function is :

C(w; q) = min

w1

rq

1; w2

pq=2

:



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Microeconomics

The conditional demand function is similar to, but slightly dierent

from, the previous case:

H1(w; q) =

8>>>>>>>>>>>>>:

q1

if w1w2 q

12

H2(w; q) =

8>>>>>>>>>>>>>:

0 if w1w2 q

12

Note the discontinuity exactly at w1=w2 =p

1=2



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Exercise 2.6 Assume the production function

(z) =h

1z1 + 2z

2

i 1

where zi is the quantity of input i and i 0 , 1 < 1 are parameters.This is an example of the CES (Constant Elasticity of Substitution) productionfunction.

1. Show that the elasticity of substitution is 11 .

2. Explain what happens to the form of the production function and the elas-ticity of substitution in each of the following three cases: ! 1, ! 0,! 1.

3. Relate your answer to the answers to Exercises2.4 and 2.5.

Outline Answer

1. Writing the production function as

(z) :=h

1z1 + 2z

2

i 1

it is clear that the marginal product of input i is.

i(z

) :=h

1z

1 + 2z

2i 11

iz

1

i (2.9)

Therefore the MRTS is

1(z)

2(z)=

12

z1z2

1(2.10)

which implies

log

z1z2

=

1

1 log12

11 log

1(z)

2(z)

:

Therefore

= @log z1

z2

@log1(z)2(z)

= 11

2. Clearly ! 1 yields = 0 ((z) = min f1z1; 2z2g), ! 0 yields = 1 ((z) = z11 z

22 ), ! 1 yields = 1 ((z) = 1z1 + 2z2).

3. The case ! 1 corresponds to that in part 1 of Exercise 2.5; ! 0.corresponds to that in Exercise 2.4; ! 1. corresponds to that in part 5of Exercise 2.5 .



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Microeconomics

Exercise 2.7 For the CES function in Exercise 2.6 nd H1(w; q), the condi-

tional demand for good 1, for the case where 6= 0; 1. Verify that it is decreasingin w1 and homogeneous of degree 0 in (w1,w2).Outline Answer

From the minimization of the following Lagrangean

L(z; ;w; q) :=mXi=1

wizi + [q (z)]

we obtain

1 [z1 ]1

q1 = w1 (2.11)

2 [z2 ]

1

q1

= w2 (2.12)

On rearranging:

w11

1

q1= [z1 ]

1

w22

1

q1= [z2 ]

1

Using the production function we get

1

w11

1

q1

1

+ 2

w22

1

q1

1

= q

Rearranging we nd

q1 =

11

1 [w1]

1 + 11

2 [w2]

1

1

q1

Substituting this into (2.11) we get:

w1 = 1 [z1 ]1

11

1 [w1]

1 + 11

2 [w2]

1

1

q1

Rearranging this we have:

z1 =

"1 + 2

12

w2w1

1

#1

q

Clearly z1 is decreasing in w1 if < 1. Furthermore, rescaling w1 and w2by some positive constant will leave z1 unchanged.



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Exercise 2.8 For any homothetic production function show that the cost func-

tion must be expressible in the form

C(w; q) = a (w) b (q) :

0

z2

z1

Figure 2.11: Homotheticity: expansion path

Outline Answer

From the denition of homotheticity, the isoquants must look like Figure2.11; interpreting the tangents as isocost lines it is clear from the gure thatthe expansion paths are rays through the origin. So, if Hi(w; q) is the demandfor input i conditional on output q, the optimal input ratio

Hi(w; q)

Hj(w; q)

must be independent of q and so we must have

H

i

(w

; q)Hi (w; q0) = H

j

(w

; q)Hj (w; q0)

for any q; q0. For this to true it is clear that the ratio Hi(w; q)=Hi (w; q0) mustbe independent ofw. Setting q0 = 1 we therefore have

H1(w; q)

H1(w; 1)=

H2(w; q)

H2(w; 1)= ::: =

Hm(w; q)

Hm(w; 1)= b(q)

and so

Hi(w; q) = b(q)Hi(w; 1):

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Therefore the minimized cost is given by

C(w; q) =mXi=1

wiHi(w; q)

=mXi=1

wib(q)Hi(w; 1)

= b(q)

mXi=1

wiHi(w; 1)

= a(w)b(q)

where a(w) =

Pmi=1 wiH

i(w; 1):



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Exercise 2.9 Consider the production function

q =

1z11 + 2z

12 + 3z

13

11. Find the long-run cost function and sketch the long-run and short-run

marginal and average cost curves and comment on their form.

2. Suppose input 3 is xed in the short run. Repeat the analysis for theshort-run case.

3. What is the elasticity of supply in the short and the long run?

Outline Answer

1. The production function is clearly homogeneous of degree 1 in all inputs i.e. in the long run we have constant returns to scale. But CRTS impliesconstant average cost. So

LRMC = LRAC = constant

Their graphs will be an identical straight line.

2

z1

Figure 2.12: Isoquants do not touch the axes

2. In the short run z3 = z3 so we can write the problem as the followingLagrangean

L(z; ) = w1z1 + w2z2 + h

q 1z11 + 2z12 + 3z13 1i ; (2.13)or, using a transformation of the constraint to make the manipulationeasier, we can use the Lagrangean

L(z; ) = w1z1 + w2z2 +

1z11 + 2z

12 k

(2.14)

where is the Lagrange multiplier for the transformed constraint and

k := q1 3z13 : (2.15)



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Note that the isoquant is

z2 = 2

k 1z11:

From the Figure 2.12 it is clear that the isoquants do not touch the axesand so we will have an interior solution. The rst-order conditions are

wi iz2i = 0; i = 1; 2 (2.16)

which imply

zi =

riwi

; i = 1; 2 (2.17)

To nd the conditional demand function we need to solve for . Using the

production function and equations (2.15), (2.17) we get

k =2X

j=1

j

jwj

1=2(2.18)

from which we nd p =

b

k(2.19)

whereb :=

p1w1 +

p2w2:

Substituting from (2.19) into (2.17) we get minimised cost as

~C(w; q; z3) =2Xi=1

wizi + w3z3 (2.20)

=b2

k+ w3z3 (2.21)

=qb2

1 3z13 q+ w3z3: (2.22)

Marginal cost isb2

1 3z13 q2 (2.23)

and average cost is

b2

1 3z13 q+ w

3z3q

: (2.24)

Let q be the value of q for which MC=AC in (2.23) and (2.24) at theminimum of AC in Figure 2.13 and let P be the corresponding minimumvalue of AC. Then, using p =MC in (2.23) for p p the short-run supply

curve is given by q = S(w; p) =

8>>>>>>>>>:

0 if p p

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3. Dierentiating the last line in the previous formula we get

d ln qd lnp

=pq

dqdp

=12

1pp=b 1 > 0

Note that the elasticity decreases with b. In the long run the supply curvecoincides with the MC,AC curves and so has innite elasticity.

marginal

cost

average

cost

q

Figure 2.13: Short-run marginal and average cost



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Microeconomics

Exercise 2.10 A competitive rms output q is determined by

q = z11 z22 :::z

mm

where zi is its usage of input i andi > 0 is a parameter i = 1; 2;:::;m. Assumethat in the short run only k of the m inputs are variable.

1. Find the long-run average and marginal cost functions for this rm. Underwhat conditions will marginal cost rise with output?

2. Find the short-run marginal cost function.

3. Find the rms short-run elasticity of supply. What would happen to thiselasticity if k were reduced?

Outline AnswerWrite the production function in the equivalent form:

log q =

mXi=1

i log zi (2.25)

The isoquant for the case m = 2 would take the form

z2 =

qz11

12 (2.26)

which does not touch the axis for nite (z1; z2).

1. The cost-minimisation problem can be represented as minimising the La-grangean

mXi=1

wizi +

"log q

mXi=1

i log zi

#(2.27)

where wi is the given price of input i, and is the Lagrange multiplierfor the modied production constraint. Given that the isoquant does nottouch the axis we must have an interior solution: rst-order conditions are

wi iz1i = 0; i = 1; 2;::;m (2.28)

which imply

zi =iwi

; i = 1; 2;::;m (2.29)

Now solve for . Using (2.25) and (2.29) we get

zii =

iwi

i; i = 1; 2;::;m (2.30)

q =

mYi=1

zii =

A

mYi=1

wii (2.31)



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where := Pmj=1 j and A := [Q

mi=1

ii ]

1=are constants, from which

we nd

= A

qQm

i=1 wii

1== A [qw11 w

22 :::w

mm ]

1= : (2.32)

Substituting from (2.32) into (2.29) we get the conditional demand func-tion:

Hi (w; q) = zi =iwi

A [qw11 w22 :::w

mm ]

1= (2.33)

and minimised cost is

C(w; q) =

mXi=1

wizi = A [qw11 w

22 :::w

mm ]

1=

(2.34)

= B q1= (2.35)

where B := A [w11 w22 :::w

mm ]

1=. It is clear from (2.35) that cost isincreasing in q and increasing in wi if i > 0 (it is always nondecreasingin wi). Dierentiating (2.35) with respect to q marginal cost is

Cq (w; q) = Bq1 (2.36)

Clearly marginal cost falls/stays constant/rises with q as T 1.

2. In the short run inputs 1;:::;k (k m) remain variable and the remaininginputs are xed. In the short-run the production function can be writtenas

log q =kX

i=1

i log zi + log k (2.37)

where

k := exp

mX

i=k+1

i log zi

!(2.38)

and zi is the arbitrary value at which input i is xed; note that B isxed in the short run. The general form of the Lagrangean (2.27) remainsunchanged, but with q replaced by q=k and m replaced by k. So the

rst-order conditions and their corollaries (2.28)-(2.32) are essentially asbefore, but and A are replaced by

k :=kX

j=1

j (2.39)

and Ak :=hQk

i=1 ii

i1=k. Hence short-run conditional demand is

~Hi (w; q; zk+1;:::; zm) =iwi

Ak

q

kw11 w

22 :::w

kk

1=k(2.40)

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and minimised cost in the short run is

~C(w; q; zk+1;:::; zm) =kX

i=1

wizi + ck

= kAk

q

kw11 w

22 :::w

kk

1=k+ ck (2.41)

= kBkq1=k + ck (2.42)

where

ck :=mX

i=k+1

wizi (2.43)

is the xed-cost component in the short run and Bk := Ak [w11 w

22 :::w

kk =k]

1=k .

Dierentiating (2.42) we nd that short-run marginal cost is

~Cq (w; q; zk+1;:::; zm) = Bkq1kk

3. Using the Marginal cost=price condition we nd

Bkq1kk = p (2.44)

where p is the price of output so that, rearranging (2.44) the supply func-tion is

q = S(w; p; zk+1;:::; zm) = p

Bk k

1k

(2.45)

wherever MCAC. The elasticity of (2.45) is given by@log S(w; p; zk+1;:::; zm)

@logp=

k1 k

> 0 (2.46)

It is clear from (2.39) that k k1 k2::: and so the positive supplyelasticity in (2.46) must fall as k falls.



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Exercise 2.11 A rm produces goods 1 and 2 using goods 3,...,5 as inputs. The

production of one unit of good i (i = 1; 2) requires at least aij units of good j, (j = 3; 4; 5).

1. Assuming constant returns to scale, how much of resourcej will be neededto produce q1 units of commodity 1?

2. For given values of q3; q4; q5 sketch the set of technologically feasible out-puts of goods 1 and 2.

Outline Answer

1. To produce q1 units of commodity 1 a1jq1 units of resource j will be

needed.q1a1i + q2a2i Ri:

2. The feasibility constraint for resource j is therefore going to be

q1a1j + q2a2j Rj :

Taking into account all three resources, the feasible set is given as in Figure2.14

q1

q2

Feasible

Set

points satisfying

q1a13 + q2a23 R3

points satisfying

q1a13 + q2a23 R3

points satisfying

q1a

14+ q

2a

24 R4

points satisfying

q1a14 + q2a24 R4

points satisfying

q1a15 + q2a25 R5

points satisfying

q1a

15+ q

2a

25R

5

Figure 2.14: Feasible set

Exercise 2.12 [see Exercise 2.4]



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Microeconomics

Exercise 2.13 An agricultural producer raises sheep to produce wool (good 1)

and meat (good 2). There is a choice of four breeds (A, B, C, D) that can beused to stock the farm; each breed can be considered as a separate input to theproduction process. The yield of wool and of meat per 1000 sheep (in arbitraryunits) for each breed is given in Table 2.1.

A B C D

wool 20 65 85 90meat 70 50 20 10

Table 2.1: Yield per 1000 sheep for breeds A,...,D

1. On a diagram show the production possibilities if the producer stocks ex-actly 1000 sheep using just one breed from the set {A,B,C,D} .

2. Using this diagram show the production possibilities if the producers 1000sheep are a mixture of breeds A and B. Do the same for a mixture ofbreeds B and C; and again for a mixture of breeds C and D. Hence drawthe (wool, meat) transformation curve for 1000 sheep. What would be thetransformation curve for 2000 sheep?

3. What is the MRT of meat into wool if a combination of breeds A and B areused? What is the MRT if a combination of breeds B and C are used?Andif breeds C and D are used?

4. Why will the producer not nd it necessary to use more than two breeds?

5. A new breed E becomes available that has a (wool, meat) yield per 1000sheep of (50,50). Explain why the producer would never be interested instocking breed E if breeds A,...,D are still available and why the transfor-mation curve remains unaected.

6. Another new breed F becomes available that has a (wool, meat) yield per1000 sheep of (50,50). Explain how this will alter the transformationcurve.

Outline Answer

1. See Figure 2.15.

2. See Figure 2.15.

3. The MRT if A and B are used is70 5020 65 =

4

9. If B and C are used it

is going to be20 5085 65 =

3

2:

4. In general for m inputs and n outputs if m > n then m n inputs areredundant.

5. As we can observe in Figure 2.15, by using breed E the producer cannotmove the frontier (the transformation curve) outwards.

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0

20

40

60

80

0 20 40 60 80 100

wool

meat

0

20

40

60

80

0 20 40 60 80 100

wool

meat

Figure 2.15: The wool and meat tradeo

6. As we can observe in Figure 2.16 now the technological frontier has movedoutwards: one of the former techniques is no longer on the frontier.

0

20

40

60

80

0 20 40 60 80 100

wool

meat

0

20

40

60

80

0 20 40 60 80 100

wool

meat

Figure 2.16: Eect of a new breed

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Microeconomics

Exercise 2.14 A rm produces goods 1 and 2 uses labour (good 3) as input

subject to the production constraint

[q1]2 + [q2]

2 + Aq3 0

where qi is net output of good i and A is a positive constant. Draw the trans-formation curve for goods 1 and 2. What would happen to this transformationcurve if the constant A had a larger value?

Outline Answer

1. From the production function it is clear that, for any given value q3,the transformation curve is the boundary of the the set of points (q1; q2)

satisfying

[q1]2 + [q2]

2 Aq3q1; q2 0

where the right-hand side of the rst expression is positive because q3 isnegative. This is therefore going to be a quarter circle as in Figure 2.17.

2. See Figure 2.17.

q2

q1

Small A

Large A

Figure 2.17: Transformation curves


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Microeconomics Solutions 02