Varian Microeconomic Analysis (3rd ed) SOLUTIONS

Ch. 5 COST FUNCTION 9

doesn't make sense|you are producing more output in the plant with the higher costs!

It turns out that this corresponds to a constrained maximum and not to the desired minimum. Check the second-order conditions to verify this.

Since the cost function is concave, rather than convex, the optimal solution will always occur at a boundary. That is, you will produce all output at the cheaper plant so c(y) = 2py.

4.7No, the data violate WACM. It costs 40 to produce 100 units of output, but at the same prices it would only cost 38 to produce 110 units of output.

4.8Set up the minimization problem

min x1 + x2 x1x2 = y:

Substitute to get the unconstrained minimization problem

min x1 + y=x1:

The rst-order condition is

1 − y=x21;

which implies x1 = py. By symmetry, x2 = py. We are given that 2py = 4, so py = 2, from which it follows that y = 4.

Chapter 5. Cost Function

5.1 The rm wants to minimize the cost of producing a given level of output:

c(y) = min y12 + y22

y1;y2

such that y1 + y2 = y:

The solution has y1 = y2 = y=2. Substituting into the objective function yields

c(y) = (y=2)2 + (y=2)2 = y2=2:

5.2 The rst-order conditions are 6y1 = 2y2, or y2 = 3y1. We also require y1 + y2 = y. Solving these two equations in two unknowns yields y1 = y=4 and y2 = 3y=4. The cost function is

	y 2		+	3y		2	3y2
c(y) = 3 h		i				=		:
	4			4			4

10 ANSWERS

5.3 Consider the rst technique. If this is used, then we need to have 2x1 + x2 = y. Since this is linear, the rm will typically specialize and set x2 = y or x1 = y=2 depending on which is cheaper. Hence the cost function for this technique is y minfw1=2; w2g. Similarly, the cost function for the other technique is y minfw3; w4=2g. Since both techniques must be used to produce y units of output,

c(w1; w2; y) = y [minfw1=2; w2g + minfw3; w4=2g] :

5.4 The easiest way to answer this question is to sketch an isoquant. First draw the line 2x1 + x2 = y and then the line x1 + 2x2 = y. The isoquant is the upper northeast boundary of this \cross." The slope is −2 to the left of the diagonal and −1=2 to the right of the diagonal. This means that when w1=w2 < 1=2, we have x1 = 0 and x2 = y. When w1=w2 < 1=2, we have x1 = y and x2 = 0. Finally, when 2 > w1=w2 > 1=2, we have x1 = x2 = y=3. The cost function is then

c(w1; w2; y) = minfw1; w2; (w1 + w2)=3gy:

5.5 The input requirement set is not convex. Since y = maxfx1; x2g, the rm will use whichever factor is cheaper; hence the cost function is c(w1; w2; y) = minfw1; w2gy. The factor demand function for factor 1 has

the form	if w1
y		< w2
x1 = ( either 0 or y	if w1	= w2	:

0if w1 > w2

5.6We have a = 1=2 and c = −1=2 by homogeneity, and b = 3 since

@x1=@w2 = @x2=@w1.

5.7Set up the minimization problem

min x1 + x2 x1x2 = y:

Substitute to get the unconstrained minimization problem min x1 + y=x1:

The rst-order condition is

1 − y=x21;

which implies x1 = py. By symmetry, x2 = py. We are given that 2py = 4, so py = 2, from which it follows that y = 4.

Ch. 5 COST FUNCTION 11

5.8 If p = 2, the rm will produce 1 unit of output. If p = 1, the rstorder condition suggests y = 1=2, but this yields negative pro ts. Therm can get zero pro ts by choosing y = 0. The pro t function is (p) = maxfp2=4 − 1; 0g.

5.9.a d =d = py > 0.

5.9.b dy=d = p=c00(y) > 0.

5.9.c p0( ) = n[y + p=c00]=[D0(p) − n =c00] < 0.

5.10 Let y(p; w) be the supply function. Totally di erentiating, we have

n	@y(p; w)		n	@xi(p; w)		n	@xi(w; y) @y(p; w)
Xi			X			X
		dwi = −		@p dwi = −			@y		@p dwi:
dy =	@wi		i=1			i=1
=1

The rst equality is a de nition; the second uses the symmetry of the substitution matrix; the third uses the chain rule and the fact that the unconditional factor demand, xi(p; w), and the conditional factor demand, xi(w; y), satisfy the identity xi(w; y(p; w)) = xi(p; w). The last expression on the right shows that if there are no inferior factors then the output of the rm must increase.

5.11.a x = (1; 1; 0; 0).

5.11.b minfw1 + w2; w3 + w4gy.

5.11.c Constant returns to scale.

5.11.d x = (1; 0; 1; 0).

5.11.e c(w; y) = [minfw1; w2g + minfw3; w4g]y.

5.11.f Constant.

5.12.a The diagram is the same as the diagram for an inferior good in consumer theory.

5.12.b If the technology is CRS, then conditional factor demands take the form xi(w; 1)y. Hence the derivative of a factor demand function with respect to output is xi(w) 0.

5.12.c The hypothesis can be written as

@c(w; y)2=@y@wi < 0:

But

@c(w; y)2=@y@wi = @c(w; y)2=@wi@y = @xi(w; y)=@y:

12 ANSWERS

5.13.a Factor demand curves slope downward, so the demand for unskilled workers must decrease when their wage increases.

5.13.b We are given that @l=@p < 0. But by duality, @l=@p = −@2 =@p@w =

−@2 =@w@p = −@y=@w: It follows that @y=@w > 0.

5.14 Take a total derivative of the cost function to get:

		Xi			@c			@c
			n
dc =					@wi	dwi +		@y		dy:
		=1
It follows that			dc			n	@c dwi
	@c
						i=1
		=		− P dy			@wi				:
	@y

Now substitute the rst di erences for the dy, dc, dwi terms and you're done.

5.15 By the linearity of the function, we know we will use either x1, or a combination of x2 and x3 to produce y. By the properties of the Leontief function, we know that if we use x2 and x3 to produce y, we must use 3 units of both x2 and x3 to produce one unit of y. Thus, if the cost of using one unit of x1 is less than the cost of using one unit of both x2 and x3, then we will use only x1, and conversely. The conditional factor demands can be written as:

x1 =	3y if w1 < w2 + w3
	0	if w1 > w2 + w3
x2 =	0	if w1	< w2	+ w3
	3y if w1 > w2 + w3
x3 =	0	if w1	< w2 + w3
	3y if w1 > w2 + w3

if w1 = w2 +w3, then any bundle (x1; x2; x3) with x2 = x3 and x1 +x2 = 3y (or x1 + x3 = 3y) minimizes cost.

The cost function is

c(w; y) = 3y min(w1; w2 + w3):

5.16.a Homogeneous:

c(tw; y) = y1=2(tw1tw2)3=4

=t3=2(y1=2(w1w2)3=4)

=t3=2c(w; y) No.

Ch. 5 COST FUNCTION 13

Monotone:

@c

=

3

y1=2w−1=4w3=4

> 0

@c

=

3

y1=2w3=4w−1=4

> 0

Yes.

@w1

4

1

2

@w2

4

1

2

Concave:

Hessian

=

−

3

y1=2w1−5=4w23=4

9

y1=2w1−1=4w2−1=4

16

16

"

9

y1=2w1−1=4w2−1=4 −

3

y1=2w13=4w2−5=4

#

16

16

jH1j < 0

9

81

jH2j =

yw1−1=2w2−1=2 −

yw1−1=2w2−1=2

256

256

72

y

= −

< 0

No

256

p

w1w2

Continuous: Yes

5.16.b Homogeneous:

p

c(tw; y) = y(tw1 + tw1tw2 + tw2)

= ty(w1 + p

+ w2)

w1w2

= tc(y;

)

Yes

w

Monotone:

@c

1

@c

1

w2

w1

= y 1 +

r

> 0

= y 1 +

r

> 0

Yes

@w1

2

w1

@w2

2

w2

Concave:

−41 yw21=2w1−3=2

41 yw2−1=2w1−1=2

#

H =

" 41 yw2−1=2w1−1=2 −41 yw2−3=2w11=2

jH1j < 0

j

H

2j

=

1

yw−1w−1

1

yw−1w−1

= 0

Yes

16

2 1

− 16

2

1

Continuous: Yes

14 ANSWERS

Production Function:

x1(w; y) = y 1 +

1

r

w2

(1)

2

w1

x2(w; y) = y 1 +

1

r

w1

(2)

2

w2

Rearranging these equations:

y

w2

x1 − y =

r

(10)

2

w1

y

w1

x2 − y =

r

(20)

2

w2

2

Multiply (10) and (20): (x1 −y)(x2 −y) = y4 . This is a quadratic equation

which gives y = 32 (x2 + x1) 32 p

.

x12 + x22 + 2 − x1x2

5.16.c Homogeneous:

c(tw; y) = y(tw1e−tw1 + tw2)

= ty(w1e−tw1 + w2)

6= tc(w; y)

No

Monotone:

@c

= y( w

e−w1 + e−w1 ) = ye−w1 (1

−

w

)

@w1

− 1

1

This is positive only if w1 < 1.

@c

= y > 0

No

@w2

Concave:

H =

y(w1 −02)e−

w1

0

0

jH1j = y(w1 − 2)e−w1

This is less than zero only if w1 < 2.

jH2j = 0

No

Continuous: Yes

Ch. 5 COST FUNCTION 15

5.16.d Homogeneous:

p

c(tw; y) = y(tw1 − ptw1tw2 + tw2

=ty(w1 − w1w2 + w2)

=tc(w; y) Yes

Monotone:

1

@c

r

w2

= y(1

−

)

@w1

2

w1

This is greater than 0 only if 1 > 1 q

w2

2

w1

@c

1

r

w1

= y(1

−

)

@w2

2

w2

w2

This is greater than 0 only if 2 > qw1

w2 >

1

w1 (by symmetry)

2p

> p

w1

w2

4

or

1

w1 < 4w2

w1 >

w2

4

Monotone only if 1 w2 < w1 < 4w2. No.

4

Concave:

41 yw1−3=2w21=2

41 yw1−1=2w2−1=2

#

H = " −41 yw1−1=2w2−1=2

−41 yw11=2w2−1=2

1

yw1−3=2w21=2 > 0

jH1j =

4

jH2j = 0

No (it is convex)

Continuous: Yes

5.16.e

Homogeneous:

c(tw; y) = (y + 1 ptw1tw2) y

= tc(y; w) Yes

if wi < wj if wi > wj if wi = wj .

16 ANSWERS

Monotone in w:

@c

1

1

@c

1

1

w2

w1

=

(y +

)r

> 0

=

(y +

)r

> 0 Yes

@w1

2

y

w1

@w2

2

2

w2

Concave:

H = "

−41 (y + y1 w1−3=2w21=2

41 (y + y1 )w1−1=2w2−1=2

#

But not in y!

41 (y; y1 )w1−1=2w2−1=2

−41 (y + y1 )w11=2w2−3=2

jH1j < 0

Yes

jH2j = 0

Continuous: Not for y = 0.

p

5.17.a y = ax1 + bx2

5.17.b Note that this function is exactly like a linear function, except that the linear combination of x1 and x2 will produce y2, rather than just y. So, we know that if x1 is relatively cheaper, we will use all x1 and no x2, and conversely.

5.17.c The cost function is c(w; y) = y2 min(wa1 ; wb2 ):

Chapter 6. Duality

6.1 The production function is f(x1; x2) = x1 + x2. The conditional factor demands have the form

8 < y

xi = : 0any amount between 0 and y

6.2 The conditional factor demands can be found by di erentiating. They are x1(w1; w2; y) = x2(w1; w2; y) = y. The production function is

f(x1; x2) = minfx1; x2g:

6.3 The cost function must be increasing in both prices, so a and b are both nonnegative. The cost function must be concave in both prices, so a and b are both less than 1. Finally, the cost function must be homogeneous of degree 1, so a = 1 − b.

Ch. 8 CHOICE 17

Chapter 7. Utility Maximization

7.1The preferences exhibit local nonsatiation, except at (0; 0). The consumer will choose this consumption point when faced with positive prices.

7.2The demand function is

			m=p1	if p1	< p2
x1	=	8 any x1 and x2 such that p1x2 + p2x2 = m		if p1	= p2
		:	0	if p1	> p2
		<

The indirect utility function is v(p1; p2; m) = maxfm=p1; m=p2g, and the expenditure function is e(p1; p2; u) = u minfp1; p2g:

7.3 The expenditure function is e(p1; p2; u) = u minfp1; p2g. The utility function is u(x1; x2) = x1 + x2 (or any monotonic transformation), and the demand function is

			m=p1	if p1	< p2
x1	=	8 any x1 and x2 such that p1x1 + p2x2 = m		if p1	= p2
		:	0	if p1	> p2
		<

7.4.a Demand functions are x1 = m=(p1 + p2), x2 = m=(p1 + p2).

7.4.b e(p1; p2; u) = (p1 + p + 2)u

7.4.c u(x1; x2) = minfx1; x2g

7.5.a Quasilinear preferences.

7.5.b Less than u(1).

7.5.c v(p1; p2; m) = maxfu(1) − p1 + m; mg

7.6.a Homothetic.

7.6.b e(p; u) = u=A(p)

7.6.c (p; q; m) = mA(q)=A(p)

7.6.d It will be the same, since this is just a monotonic transformation.

Chapter 8. Choice

8.1 We know that

xj(p; m) hj (p; v(p; m)) @e(p; v(p; m))=@pj:

(0:1)

18 ANSWERS

(Note that the partial derivative is taken with respect to the rst occurrence of pj .) Di erentiating equation (0.1) with respect to m gives us

@xj	=	@2e(p; v(p; m)) @v(p; m)
			:
		@pj@u
@m			@m

Since the marginal utility of income, @v=@m, must be positive, the result follows.

8.2 The Cobb-Douglas demand system with two goods has the form

x1 = a1m p1

x2 = a2m p2

where a1 + a2 = 1. The substitution matrix is
−	a1mp−2	a12mp−2			a1a2mp−1p−1				:
	12	− 1	p2−	11	−	2	1 2 2	2
	−a1a2mp1−				−a2mp2−		− a2mp2−

This is clearly symmetric and negative de nite.

8.3 The equation is d =dt = at+b + c. The indirect money metric utility function is

c a c

c

a

aq

(q; p; m) = eb(q−p) hm +

+

+

pi

−

−

−

:

b

b2

b

b

b2

b

8.4 The demand function can be written as x = ec+ap+bm. The integrability equation is

d = eat+b +c:

dt

Write this as

e−b ddt = eceat:

Integrating both sides of this equation between p and q, we have

	e−b	q	eceat q
−		ip =				ip:
	b			a
Evaluating the integrals, we have
eb (q;p;m) = e−bm −				bec	[eap − eaq] :
				a