Important Cobb Do

7/24/2019 Important Cobb Do

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CobbDouglas Production Functions

1 mathematical tricks

the derivative ofx with respect to x is x1

xx =x+ (for any and )

1x =x

ifm = nB, then n = m1/B

2 the production function

A production function y =f(x1, x2) is a CobbDouglasproduction functionif it can be written in the form

y= Axa1xb2 (1)

whereA, a and b are positive constants.So the partial derivatives of a CobbDouglas production function are :

MP1 = y

x1=aAxa11 x

b2 (2)

MP2= y

x2=bAxa1x

b12 (3)

The absolute value of the slope of an isoquant is the technical rate ofsubstitution, orT RS. ThisT RSequalsM P1/MP2 so that (2) and (3) implythat

T RS=MP1MP2

=ax1bx2

(4)

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Equations (2) and (3) imply that the CobbDouglas technology is mono-

tonic, since both partial derivatives are positive. Equation (4) demonstratesthe technology is convex, since the (absolute value) of the T RS falls as x1increases andx2 decreases.

3 returns to scale

Suppose that all inputs are scaled up by some factor t. The new level ofoutput is

f(tx1, tx2) =A(tx1)a(tx2)

b =ta+bAxa1xb2 (5)

Notice from equation (5) thatf(tx1, tx2) =ta+bf(x1, x2). We haveincreasing

returns to scale iff(tx1, tx2) > tf(x1, x2) whenever t >1. So here we haveincreasing returns to scale ifta+b > t, which is the same thing as a + b >1.

Similarly, decreasing returns to scale arise iff(tx1, tx2)< tf(x1, x2) when-evert >1. Here decreasing returns to scale occur ifta+b < t, or a + b


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The expression in square brackets in (9) and (10) is just y . So

apy= w1x1 (11)

bpy= w2x2 (12)

As well, equations (11) and (12) imply that

x2= b

a

w1w2

x1 (13)

Plugging in for x2 from (13) into (7) yields

paAxa11 [

b

a

w1w2 x

1]b

=w1 (14)

which can be written

pAa1bbbwb11 wb2 x

a+b11 = 1 (15)

orx1ab1 =pAa

1bbbwb11 wb2 (16)

Taking both sides to the power 1/(1 a b) yields

x1 = A

1/(1ab)

a

(1b)/(1ab)

b

b/(1ab)

w

(1b)/(1ab)

1 w

b/(1ab)

2 p

1/(1ab)

(17)which is the profitmaximizing firms demand for input #1, as a function ofthe prices of the 2 inputs, and of the price of the output.

Substituting for x 1 from (17) into (13) (and some simplifying) givesthe demand for input #2,

x2 = A1/(1ab)aa/(1ab)b(1a)/(1ab)w

a/(1ab)1 w

(1a)/(1ab)2 p

1/(1ab)

(18)Equation (11) implies that y = w1x1ap , which (from (17) implies that

y= A1/(1ab)

aa/(1ab)

bb/(1ab)

wa/(1ab)1 w

b/(1ab)2 p

(a+b)/(1ab)

(19)

which is the equation for the perfectly competitive firms supply curve : itsprofitmaximizing quantity of output, as a function of the price p of theoutput (and of the prices of the inputs).

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5 profit maximization and returns to scale

Since a and b are both positive, equation (19) indicates that the quantitysupplied of output y, by the profitmaximizing perfectly competitive firm,will be increasing in the price p of its output, and decreasing in the pricesw1 andw2 of its inputs, if and only if

1 a b >0

Now the condition for the production technology to exhibit decreasing returnsto scale was a + b 0.

So as long as the technology exhibits decreasing returns to scale, then

the firms supply curve slopes up, and its quantity of output is a decreasingfunction of all input prices. Equations (17) and (18) show that the (un-conditional) demand for each input is decreasing in the price of that input,provided that the technology exhibits decreasing returns to scale.

[If a+ b = 1, so that there are constant returns to scale, then equa-tions (17), (18) and (19) are meaningless, since 1/(1 a b) = 1/0 = . Ifa + b >1, so that there are increasing returns to scale, then it turns out thatequations (7) and (8) dont actually define a maximum. So section 4makessense only if we have decreasing returns to scale.]

6 cost minimizationThe cost minimization problem (chapter 20) takes the target level of outputy as given, along with the unit prices w1 andw2 of the inputs.

The firstorder conditions for cost minimization imply that M P1/MP2 =w1/w2, which (from equation (4)) here implies

a

b

x2x1

=w1w2

(20)

which is actually the same thing as condition (13) above.The firm must actually meet its target level of output, so that it must

choose input levels x1 and x2 such that Axa1xb2= y. Using (13) to substitutefor x2, this requirement becomes

y= Axa1[b

a

w1w2

x1]b (21)

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or

y= Aab

bb

wb

1wb

2 xa+b

1 (22)or

xa+b1 =A1abbbwb1 w

b2y (23)

Taking both sides of (23) to the power 1/(a + b),

x1 = A1/(a+b)ab/(a+b)bb/(a+b)w

b/(a+b)1 w

b/(a+b)2 y

1/(a+b) (24)

which is the conditional input demand function for input #1. Substitutingfrom (24) into (13), the conditional input demand for input #2 is

x2 = A1/(a+b)aa/(a+b)ba/(a+b)w

a/(a+b)1 w

a/(a+b)2 y

1/(a+b) (25)

Equations (24) and (25) imply that

w1x1= A1/(a+b)ab/(a+b)bb/(a+b)w

a/(a+b)1 w

b/(a+b)2 y

1/(a+b) (26)

andw2x2 = A

1/(a+b)aa/(a+b)ba/(a+b)wa/(a+b)1 w

b/(a+b)2 y

1/(a+b) (27)

so that the total cost of producing y units in the cheapest possible way is

C(w1, w2, y) =w1x1+ w2x2 = Bwa/(a+b)1 w

b/(a+b)2 y

1/(a+b) (28)

where the constant B is defined as

B A1/(a+b)([a

b]b/(a+b) + [

b

a]a/(a+b)) (29)

7 returns to scale again

The average cost is

AC=C(w1, w2, y

y =

Bwa/(a+b)1 w

b/(a+b)2 y

1/(a+b)

y (30)

But y1/(a+b)/y= y1/(a+b)1 =y(1ab)/(a+b) so that

AC=Bwa/(a+b)1 w

b/(a+b)2 y

(a+b1)/(a+b) (31)

The exponent (1ab)/(a + b) ony in equation (31) is positive ifa + b 1 and zero if a+ b = 1. This exponent being posiitvemeans that the average cost is increasing with the target output level y .

So the average cost decreases with output if and only ifa+b >1. Thisresult confirms the results of section 3 : another definition of increasingreturns to scale is that the average cost decline with the quantity of output.

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