Advanced Microeconomics, General Equilibrium Theory � Homework #21

#1. Consider the production function q = f(z1, z2) = z21 + z2. What can you say about the returns to scale in this case?

• Increasing Returns to Scale: f(αz) > αf(z) for any α > 1f(αz) = α2z21 + αz2 > αf(z) = αz21 + αz2

#2. Consider the production function q = f(z1, z2) = z1 + z2.[(a)]

1. draw some isoquants of this production function.

2. derive the conditional factor demands (watch out for corner solutions).

3. derive the cost function C(q, w1, w2).

4. draw the cost function for w1 = w2 = 1 (on the horizontal axis put q and C(q, 1, 1) on the vertical axis).

[a:]

1. Some isoquants from q = 1 to q = 10:

2. To �nd the conditional factor determine the conditional factor demand, we solve the following

minw · z such that f(z) = q

Considering corner solutions we must solve the the full Kuhn-Tucker Lagrangian problem:

KT (z, λ0, λ1, λ2) = −w · z+ λ0(z1 + z2 − q) + λ1x1 + λ2x2

We are then given the following �rst order conditions from KT :

∂KT∂z1

= −w1 + λ0 + λ1 = 0 ∂KT∂λ1

= z1 ≥ 0; λ1z1 = 0

∂KT∂z2

= −w2 + λ0 + λ2 = 0 ∂KT∂λ2

= z2 ≥ 0; λ2z2 = 0

∂KT∂λ0

= z1 + z2 − q = 0

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I again used the code prepared by: Mariana Abibe, Lois Hales, Dominka �uczak, Matthew Maycock, Asish Subedi

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We now consider the following cases:

Case 1: λ0 > 0;λi = 0; zi > 0

For this case, we have the conditions above become

∂KT∂z1

= −w1 + λ0 = 0 ∂KT∂λ1

= z1 > 0; λ1z1 = 0

∂KT∂z2

= −w2 + λ0 = 0 ∂KT∂λ2

= z2 > 0; λ2z2 = 0

∂KT∂λ0

= z1 + z2 − q = 0

and we have that w1 = λ0 = w2, and so when w1 = w2 we have that the two factors of input cost the same; since theyare treated symmetrically with respect to the production function and cost the same amount we have that they can besubstituted for one another without concern. Thus, we want any amount of z1 and z2 that adds to q (as long as zi > 0given our assumptions at least); notably we can write z2 = q − z1.

Case 2: λ0 > 0;λ1 = 0, λ2 > 0; z1 > 0, z2 = 0

Here we have the conditions:

∂KT∂z1

= −w1 + λ0 = 0 ∂KT∂λ1

= z1 > 0; λ1z1 = 0

∂KT∂z2

= −w2 + λ0 + λ2 = 0 ∂KT∂λ2

= z2 = 0; λ2z2 = 0

∂KT∂λ0

= z1 − q = 0

Since w1 = λ0 then −w2 + w1 + λ2 = 0 and w2 = w1 + λ0, and so w1 < w2. Thus factor one is the cheaper factor, whichis why we are not buying any of factor two, and z1 = q.

Case 3: λ0 > 0;λ1 > 0;λ2 = 0; z1 = 0, z2 > 0

Finally, we arrive with the conditions:

∂KT∂z1

= −w1 + λ0 + λ1 = 0 ∂KT∂λ1

= z1 = 0; λ1z1 = 0

∂KT∂z2

= −w2 + λ0 = 0 ∂KT∂λ2

= z2 > 0; λ2z2 = 0

∂KT∂λ0

= z2 − q = 0

and so w2 = λ0, meaning that w1 = w2 + λ1, or that w1 > w2 and we see that z2 = q. This gives us the conditional factordemands as:

z(w, q) =

〈q, 0〉 : w1 < w2

〈z1, q − z1〉 : w1 = w2, 0 < z1 < q〈0, q〉 : w1 > w2

Of note is that in the w1 = w2 case, we can have 0 ≤ z1 ≤ q, as this doesn't actually change the problem � the strictinequalities are only from our Kuhn-Tucker assumptions.

3. The cost function is C(w, q) = w1z1(w, q) + w2z2(w, q).

• If w1 < w2 then z = 〈q, 0〉, and so C(w, q) = w1q + w20 = w1q.

• If w1 = w2 then z = 〈z1, q − z1〉, hence C(w, q) = w1z1 + w2(q − z1) = w1z1 + w1(q − z1) = w1q (and likewise,C(w, q) = w2q).

• If w1 > w2 then z = 〈0, q〉 and thus C(w, q) = w10 + w2q = w2q.

• Thus, we have that C(w, q) = qmini wi.

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4. For w1 = w2 = 1 the cost function is C(w, q) = q and this is drawn as:

#3. Consider a production function q = f(z1, z2) = z0.31 z0.22

[(a)]

1. Find the maximum level of inputs that maximize pro�ts at wages = 〈w1, w2〉 = 〈15, 10〉 and the price of the �nal outputp = 150.

2. Find the level of output with the input choices found in part (a).

3. What are the pro�ts at that output level?

[a:]

1. The pro�t maximization problem asks that we maximize the di�erence of revenues and costs, and so we are trying to solve:

π(z) = maxz

[(150z0.31 z0.22 )− (15z1 + 10z2)

]We note that for any positive amount of output, z1 6= 0 and z2 6= 0; so unless the 0 vector is maximal, we will not have acorner solution. To �nd the maximum of the above function, we note that we have no constraints so we merely search forcritical points. Taking partial derivatives we �nd

∂π

∂z1= 45

z0.22

z0.71

− 15 = 0

∂π

∂z2= 30

z0.31

z0.82

− 10 = 0

And so we can solve for z1 and z2:

z2 =z3.51

35

z0.31 = z0.82 /3 =

(z3.51

35

)0.8/

3 =z2.81

35

z2.51 = 35

z1 = 9

And if z1 = 9 then z2 = (93.5)/(35) = ((32)3.5)/35 = 37/35 = 9. Is this a maximum, though? We �rst examine Hessianmatrix of our pro�t function:

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π(z) =

− 31.50z0.22

z1.71

9z0.71 z0.82

9z0.71 z0.82

− 24z0.31

z1.82

∣∣∣∣∣∣∣∣∣∣z=〈9,9〉

=

− 7

613

13 − 8

9

And since (−7)(−1)1 > 0 and ((−7/6)(−8/9) − (1/3)(1/3))(−1)2 = 50/54 > 0, we have that the Hessian matrix of πevaluated at 9, 9 is negative de�nite and so 〈9, 9〉 is a maximum point.

2. The level of output is q = f(z) = 90.390.2 = 90.5 = 3

3. The pro�ts at this level are π(z) = 150(90.390.2)− (15(9) + 10(9)) = 225

#4. Given the production function

q = f(z) =

L∏l=1

zαll

with∑Ll=1 αl = 1:

[(a)]

1. What can you say about the returns to scale?

2. Derive the conditional factor demands.

3. Derive the cost function C(q,w) and draw it for w1 = w2 = 1.

4. Derive the average and marginal cost functions.

5. With such a function, can you determine the pro�t maximizing level of output? Why?

[a:]

1. The returns to scale are constant. This can be seen by the fact that

f(βz) =

L∏l=1

(βzl)αl =

L∏l=1

(βαlzαll ) =

(L∏l=1

βαl

)(L∏l=1

zαll

)= β

∑Ll=1 αif(z) = βf(z)

2. For the conditional factor demands we note that if we have a positive level of output then zl > 0 for all l, and so we haveno corner solutions. Our Lagrangian for the cost minimization function is thus of the form:

L(z, λ) = −w · z+ λ

(L∏l=1

zαll − q

)

And so our �rst order conditions are:

∂L

∂zi= −wi + αi

∏Ll=1 z

αll

zi= 0

∂L

∂λ=

L∏l=1

zαll − q = 0

and so taking the ratio of two �rst order conditions with regards to z1 and zi we have:

wiw1

=

(αi

∏Ll=1 z

αll

zi

)/(α1

∏Ll=1 z

αll

z1

)=αiz1α1zi

zi =αiw1z1α1wi

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And so our constraint with regards to λ becomes

q =

L∏l=1

zαll =

L∏l=1

(αlw1z1α1wl

)αl=

L∏l=1

((w1z1α1

)αl (αlwl

)αl)=

L∏l=1

(w1z1α1

)αl L∏l=1

(αlwl

)αl=

(w1z1α1

)∑Ll=1 αl L∏

l=1

(αlwl

)αl=

w1z1α1

L∏l=1

(αlwl

)αl

And so we have that

z1 = qα1

w1

L∏l=1

(wlαl

)αland by symmetry we have zi = q

αiwi

L∏l=1

(wlαl

)αl3. The cost function is C(w, q) = w · z(w, q) which, given the above, is:

C(w, q) =

L∑i=1

qαi

L∏l=1

(wlαl

)αl= q

L∏l=1

(wlαl

)αl L∑i=1

αi = q

L∏l=1

(wlαl

)αlFor the two factor case with w1 = w2 = 1 the cost function becomes

C(w, q) =

(w1

α1

)α1(w2

α2

)α2

q =q

αα11 (1− α1)1−α1

Where we have dropped subscripts since α2 = 1 − α1. The graph of this cost function will then be a line with a slope1/(αα(1− α)1−α):

4. The average cost function is C(w, q)/q and the marginal cost function is ∂C∂q , and these are:

AC(w, q) =C(w, q)

q=

L∏l=1

(wlαl

)αl=∂C

∂q(w, q) =MC(w, q)

5. Due to CRS of this particular Cobb-Douglas function, p = MC, any output is pro�t maximizing. If p 6= MC no pro�tmaximizer exists.

#5. For the Leontief function q = f(z) = min{. . . , zi/ai, . . .} where al > 0, ∀l∈L and L is the number of inputs:[(a)]

1. Draw a few isoquants for L = 2. Geographically determine the optimal combination of outputs for z1 and z2.

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2. For L = 2 derive the conditional factor demands and the cost function.

3. What can you say about the returns to scale?

4. Generalize your result in (b) for L > 2.

[a:]

1. The below image shows the various isoquants. We take a = a1 and b = a2 for simplicity in labeling.

We can see that the optimal combination of outputs given inputs will be at the `corners' of the isoquants � here, when youincrease one of the factors, you do not increase the quantity of output.

2. Our output in this case is q = f(z) = min{z1/a1, z2/a2}. If q = zi/ai is then zi(w, q) = qai (here we take zi/ai as theminimum from before). For zj 6=i we don't want more zj than necessary and so zj/aj = zi/ai but then we have thatzj(w, q) = qaj . Thus we have z1(w, q) = qa1 and z2(w, q) = qa2.

The cost function C(w, q) = z1w1 + z2w2 = qw1a1 + qw2a2 = q(w1a1 + w2a2).

3. This function has constant returns to scale. Note that f(αz) = min{. . . , αzi/ai, . . .}. If zi/ai < zj/aj then multiplyingboth sides by α does not change the inequality since we assume α > 0, and so the same product is the minimal product,and so the minimum is scaled the same as the input.

4. Since we have for both the �rst good we pick as the minimum has conditional factor demand of zi(w, q) = qwiai and thatzj 6=i(w, q) = qwjaj we have that zi(w, q) = qwiai for all zi. The cost function would be C(w, q) =

∑ni=1 wizi(w, q) =

q∑ni=1 wiai.

• NOTE: that ai is in fact the unit factor requirement as in our discussion of the small economy model.

#6. Find the conditional factor demands and the cost function for the CES production function:

q = f(z1, z2) = (zρ1 + zρ2)1/ρ

[a:]

1. We do not have to worry about corner solutions, because if we compute marginal productivity of each zi for ρ < 1 whichare the standard values for a CES function, then at 0 this marginal productivity is in�nite. Hence, the �rm at �nite pricesof factors will never choose zi = 0. Therefore, we do not have any corner solutions. Our conditional factor demand problemis thus just a normal cost minimization Lagrangian problem:

L(z, λ) = −w · z+ λ((zρ1 + zρ2)1/ρ − q)

And this gives us the �rst order conditions:

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∂L

∂z1= −w1 + λ(zρ1 + zρ2)

ρ−1ρ zρ−11 = 0

∂L

∂z2= −w2 + λ(zρ1 + zρ2)

ρ−1ρ zρ−12 = 0

∂L

∂λ= (zρ1 + zρ2)

1/ρ − q = 0

and we have that:

w1

w2=zρ−11

zρ−12

=⇒ z1 = z2

(w1

w2

) 1ρ−1

=⇒ qρ =

(z2

(w1

w2

) 1ρ−1

)ρ+ zρ2 = zρ2

(1 +

(w1

w2

) ρρ−1

)= zρ2

(w

ρρ−1

1 + wρρ−1

2

wρρ−1

2

)and so

z2 =qw

1ρ−1

2(w

ρρ−1

1 + wρρ−1

2

)1/ρ and by symmetry z1 =qw

1ρ−1

1(w

ρρ−1

1 + wρρ−1

2

)1/ρwe sometimes introduce W =

(w

ρρ−1

1 + wρρ−1

2

) ρ−1ρ

z2 =qw

1ρ−1

2

W1ρ−1

and by symmetry z1 =qw

1ρ−1

1

W1ρ−1

2. To �nd the cost function, we add the cost of each factor:

C(w, q) = w1qw

1ρ−1

1

W1ρ−1

+ w2qw

1ρ−1

2

W1ρ−1

= qw

ρρ−11 +w

ρρ−12

W1ρ−1

= qWρρ−1

W1ρ−1

= qW

So W is in fact the CES unit cost function (this is something that is actually worth remembering for later)

#7. Suppose the two-input production function has the corresponding cost function:

C(w, q) = (5w1 + 2w2)q2

[(a)]

1. Find the conditional factor demands (use Shephard's lemma).

2. Draw the cost function for w1 = w2 = 1.

3. Is the production function homothetic?

4. Find the pro�t function π(p,w) (you need to do the pro�t maximization problem using the cost function).

[a:]

1. Shephard's lemma states that the factor demands are the derivatives of cost with respect to wages. We have that:

〈z1(w1, w2, q), z2(w1, w2, q)〉 = ∇wC(w, q) = 〈q25, q22〉

2. The cost function for w1 = w2 = 1 is:

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3. The function will be homothetic if the rate change in one the cost factor is proportional to the other � i.e. if the ratios ofpartial derivatives (factor demands) are constant: 5q2/2q2 = 5/2 (or 2/5), and so the function is homothetic.

4. The function we are trying to �nd is the pro�t function π(p,w) that will be a function of p and w and not q. We have:π(p,w,q) = pq − C(w, q) = pq − (5w1 + 2w2)q

2. We need to maximize pro�ts over q and hence we can �nd the levelof output that yields maximum pro�ts and substitute it's value for q. Maximizing pro�ts means �nding the level ofoutput that maximizes pro�ts given wages and prices; and so we must �nd when the derivative of pro�t with respectto quantity of output is zero. This is the condition then that p − 2(5w1 + 2w2)q = 0 or q = p/[2(5w1 + 2w2)]. Settingγ = 5w1 + 2w2, the pro�t function becomes π(p,w) = p2/(2γ)− γp2/(4γ2) = 2p2/(4γ)− p2/(4γ) = p2/(4γ). Replacing γyields π(p,w) = p2/[4(5w1 + 2w2)].

#8. Suppose the two-input production function has the corresponding cost function:

C(w, q) = 2q2√w1w2

[a:]

1. Find conditional factor demands.

2. Draw the total cost function for w1 = w2 = 1.

3. Draw the marginal and average cost curves for w1 = w2 = 1.

4. Is the production function homothetic?

5. Find the pro�t function π(p,w).

6. Find the pro�t maximizing level of output q when w1 = w2 = 1 and the price of the �nal good is p = 8. Depict your�ndings on the graph you have drawn in (c). What is the level of pro�ts at that level of output?

[a:]

1. We use Shephard's lemma � and so the conditional demands are ∇wC =⟨q2√

w1

w2, q2√

w2

w1

⟩.

2. For w1 = w2 = 1 the total cost function is:

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3. With w1 = w2 = 1 the marginal costs are MC(q) = 4q while the average costs are AC(q) = 2q. In �gures:

4. The function will be homothetic if the ratio of partial derivatives of cost is constant � in this case, it is w1/w2 (or w2/w1)� so the function is homothetic.

5. In order to �nd the pro�t function in terms of just p and w we must �nd the level of output that maximizes pro�ts andassume that, given p and w the �rm will operate at that output. Pro�ts are π(p,w) = pq − C(w, q) = pq − 2q2

√w1w2.

Taking the derivative of this with respect to q we arrive at p−4√w1w2q and so q = p/(4

√w1w2). We then let γ = 2

√w1w2

so that q = p/(2γ) and the pro�t function becomes p2/(2γ)− γ(p/(2γ))2 which is equal to p2/(4γ) or p2/(8√w1w2).

6. Since the pro�t maximizing level of output is q = p/(4√w1w2), when we have w1 = w2 = 1 and p = 8 we arrive at

q = 8/4 = 2. The pro�ts at this given level of output and prices is 82/8 = 8.

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