Cobb–Douglas Production Functions

1

mathematical tricks
• the derivative of αxβ with respect to x is αβxβ−1 • xα xβ = xα+β (for any α and β) •
1 xα

= x−α

• if m = nB , then n = m1/B

2

the production function

A production function y = f (x1 , x2 ) is a Cobb–Douglas production function if it can be written in the form y = Axa xb 1 2 (1)

where A, a and b are positive constants. So the partial derivatives of a Cobb–Douglas production function are : M P1 = M P2 = ∂y = aAxa−1 xb 1 2 ∂x1 (2)

∂y = bAxa xb−1 (3) 1 2 ∂x2 The absolute value of the slope of an isoquant is the technical rate of substitution, or T RS. This T RS equals M P1 /M P2 so that (2) and (3) imply that T RS = M P1 ax1 = M P2 bx2 1 (4)

Equations (2) and (3) imply that the Cobb–Douglas technology is monotonic, since both partial derivatives are positive. Equation (4) demonstrates the technology is convex, since the (absolute value) of the T RS falls as x1 increases and x2 decreases.

3

returns to scale

Suppose that all inputs are scaled up by some factor t. The new level of output is (5) f (tx1 , tx2 ) = A(tx1 )a (tx2 )b = ta+b Axa xb 1 2 Notice from equation (5) that f (tx1 , tx2 ) = ta+b f (x1 , x2 ). We have increasing returns to scale if f (tx1 , tx2 ) > tf (x1 , x2 ) whenever t > 1. So here we have increasing returns to scale if ta+b > t, which is the same thing as a + b > 1. Similarly, decreasing returns to scale arise if f (tx1 , tx2 ) < tf (x1 , x2 ) whenever t > 1. Here decreasing returns to scale occur if ta+b < t, or a + b < 1.

4

profit maximization π = pf (x1 , x2 ) − w1 x1 − w2 x2 (6)

A firm in perfect competition seeks to maximize its profit

∂f ∂f This profit maximum is achieved when p ∂x1 = w1 and ∂x2 = w2 . From equations (2) and (3), with a Cobb–Douglas production function, profit is maximized if paAxa−1 xb = w1 (7) 1 2

pbAxa xb−1 = w2 1 2

(8)

Equations (7) and (8) describe the input choices which maximize profit, but they do constitute two equations in two unknowns. Some substitutions are needed to derive the actual profit–maximizing choices of x1 and x2 as explicit functions of w1 , w2 and p. If both sides of (7) are multiplied by x1 , and both sides of (8) are multiplied by x2 , then (9) pa[Axa xb ] = w1 x1 1 2 pb[Axa xb ] = w2 x2 1 2 2 (10)

The expression in square brackets in (9) and (10) is just y. So apy = w1 x1 bpy = w2 x2 As well, equations (11) and (12) imply that x2 = b w1 x1 a w2 (13) (11) (12)

Plugging in for x2 from (13) into (7) yields paAxa−1 [ 1 which can be written b−1 −b pAa1−b bb w1 w2 xa+b−1 = 1 1

b w1 b x1 ] = w 1 a w2

(14)

(15)

or b−1 −b x1−a−b = pAa1−b bb w1 w2 1

(16)

Taking both sides to the power 1/(1 − a − b) yields x1 = A1/(1−a−b) a(1−b)/(1−a−b) bb/(1−a−b) w1
−(1−b)/(1−a−b)

w2

−b/(1−a−b) 1/(1−a−b)

p

(17) which is the profit–maximizing firm’s demand for input #1, as a function of the prices of the 2 inputs, and of the price of the output. Substituting for x − 1 from (17) into (13) (and some simplifying) gives the demand for input #2, x2 = A1/(1−a−b) aa/(1−a−b) b(1−a)/(1−a−b) w1 Equation (11) implies that y = w1 x1 , ap −a/(1−a−b)

w2

−(1−a)/(1−a−b) 1/(1−a−b)

p

(18) which (from (17) implies that w2
−b/(1−a−b) (a+b)/(1−a−b)

y = A1/(1−a−b) aa/(1−a−b) bb/(1−a−b) w1

−a/(1−a−b)

p

(19)

which is the equation for the perfectly competitive firm’s supply curve : its profit–maximizing quantity of output, as a function of the price p of the output (and of the prices of the inputs). 3

5

profit maximization and returns to scale

Since a and b are both positive, equation (19) indicates that the quantity supplied of output y, by the profit–maximizing perfectly competitive firm, will be increasing in the price p of its output, and decreasing in the prices w1 and w2 of its inputs, if — and only if — 1−a−b>0 Now the condition for the production technology to exhibit decreasing returns to scale was a + b < 1, which is the same thing as 1 − a − b > 0. So as long as the technology exhibits decreasing returns to scale, then the firm’s supply curve slopes up, and its quantity of output is a decreasing function of all input prices. Equations (17) and (18) show that the (unconditional) 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 equations (17), (18) and (19) are meaningless, since 1/(1 − a − b) = 1/0 = ∞. If a + b > 1, so that there are increasing returns to scale, then it turns out that equations (7) and (8) don’t actually define a maximum. So section 4 makes sense only if we have decreasing returns to scale.]

6

cost minimization

The cost minimization problem (chapter 20) takes the target level of output y as given, along with the unit prices w1 and w2 of the inputs. The first–order conditions for cost minimization imply that M P1 /M P2 = w1 /w2 , which (from equation (4)) here implies w1 a x2 = b x1 w2 (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 Axa xb = y. Using (13) to substitute 1 2 for x2 , this requirement becomes y = Axa [ 1 b w1 b x1 ] a w2 (21)

4

or b −b y = Aa−b bb w1 w2 xa+b 1

(22) (23) b/(a+b) 1/(a+b)

or
−b b xa+b = A−1 ab b−b w1 w2 y 1

Taking both sides of (23) to the power 1/(a + b), x1 = A−1/(a+b) ab/(a+b) b−b/(a+b) w1
−b/(a+b)

w2

y

(24)

which is the conditional input demand function for input #1. Substituting from (24) into (13), the conditional input demand for input #2 is x2 = A−1/(a+b) a−a/(a+b) ba/(a+b) w1 Equations (24) and (25) imply that w1 x1 = A−1/(a+b) ab/(a+b) b−b/(a+b) w1 and w2 x2 = A−1/(a+b) a−a/(a+b) ba/(a+b) w1 a/(a+b) a/(a+b) a/(a+b)

w2

−a/(a+b) 1/(a+b)

y

(25)

w2

b/(a+b) 1/(a+b)

y

(26) (27) (28)

w2

b/(a+b) 1/(a+b)

y

so that the total cost of producing y units in the cheapest possible way is C(w1 , w2 , y) = w1 x1 + w2 x2 = Bw1 where the constant B is defined as a b B ≡ A−1/(a+b) ([ ]b/(a+b) + [ ]a/(a+b) ) b a (29) a/(a+b) w2

b/(a+b) 1/(a+b)

y

7

returns to scale again
C(w1 , w2 , y Bw1 AC = = y AC = Bw1 a/(a+b) a/(a+b)

The average cost is w2 y b/(a+b) 1/(a+b)

y

(30)

But y 1/(a+b) /y = y 1/(a+b)−1 = y (1−a−b)/(a+b) so that w2 b/(a+b) (a+b−1)/(a+b)

y

(31)

The exponent (1 − a − b)/(a + b) on y in equation (31) is positive if a + b < 1, negative if a + b > 1 and zero if a + b = 1. This exponent being posiitve means that the average cost is increasing with the target output level y. So the average cost decreases with output if and only if a + b > 1. This result confirms the results of section 3 : another definition of increasing returns to scale is that the average cost decline with the quantity of output. 5