Question

Practice differentiating the following funtions in general and specific terms. Find first derivatives and second derivatives. Compare your solutions using the general functions with that of the specific functions. Provide an economic interpretation for each derivative along with its sign.

1. Utility functions:

a. U = f(x₁ , x₂ , x₃)

b. U = x₁  x₂ x₃

c. U = (x₁  x₂ x₃)²

2. Production functions:

a. Q = f(K, L)

b. Q = K^.5 L^.5

c. Q = aK^.1 L^.9

3. Profit functions:

a. ? = f(TR, TC)

b. ? = R(Q) – C(Q)  where R and C denote functional notation as we have used with f

c. ? = PQ – (A + bQ³)

d. ? = (15Q - .5Q²) – (40 + 50Q – 13.5Q² + Q³)

4. Find the marginal rate of substitution for each of the following utility functions.

a. U = x₁ x₂

b. U = x₁^.5  + x₂^.5

c. U = f(x_1, x₂)

Answer 1

1. Utility functions:

  U = f(x1 , x2 , x3)

  U/x1 = f ' (x1 , x2 , x3) =   f /x1  (derivate f w.r.t x1 only). This will give the marginal utility of consuming x1 commodity to the consumer.

U/x2 = f ' (x1 , x2 , x3) = f /x2 (derivate f w.r.t x2 only). This will give the marginal utility of consuming x2 commodity to the consumer.

U/x3 = f ' (x1 , x2 , x3) = f /x3 (derivate f w.r.t x3 only). This will give the marginal utility of consuming x3 commodity to the consumer.

Similarly second derivative is represented as:

²U/x1² = f "(x1 , x2 , x3) =   ²f /x1 ² ( Derivating the first derivative again w.r.t x1)

²U/x2² = f "(x1 , x2 , x3) =   ²f /x2 ² ( Derivating the first derivative again w.r.t x2)

²U/x3² = f "(x1 , x2 , x3) =   ²f /x3 ² ( Derivating the first derivative again w.r.t x3)

The equations above will show how much the marginal utility from consuming x1, x2 and x3 (respectively) changes as an additional unit of these commodities is consumed.

b. U = x1  x2 x3

U/x1 = x2 x3

U/x2 = x1 x3

  U/x3 = x1 x2

All the above derivatives are positive which means that Marginal utility from consuming x1, x2 and x3 is positive. Each additional consumption of these goods add to the total utility of the consumer.

Second derivatives:

²U/x1² = 0

²U/x2² = 0

²U/x3² = 0

Since the second derivatives are zero it means the change in marginal utility from consuming the three commodities will be zero. Marginal utility falls by the same amount every time additional units are consumed.

Cross derivatives:

  ²U/x1 x2 = x3

  ²U/x1 x3 = x2

  ²U/x3 x2 = x1

c. U = (x1  x2 x3)²

  U/x1 = (2x1 x2 x3 ). x1

  U/x2 = (2 x1 x2 x3). x2

U/x3 = (2 x1 x2 x3). x3

Second derivatives:

²U/x1² = (2x1 x2 x3 ) + x1 (2 x2 x3 )

²U/x2² = (2 x1 x2 x3) + x2 (2 x1 x3)

²U/x3² = (2 x1 x2 x3) + x3 (2x1 x2)

The Positive value in second derivative means that change in Marginal utility is more or it increases with every additional consumption.

2. Production functions:

a. Q = f(K, L)

Q/K = f '(K,L). This is also called marginal product of capital.

Q/L = f '(K,L). This is also called marginal product of Labor.

Second derivatives are shown as:

²Q/K² =  f "(K,L). This gives the value of change in marginal product of capital.

²Q/L² =  f "(K,L). This gives the value of change in marginal product of Labor.

b. Q = K.⁵ L⁵

Q/K = 5K⁴ L⁵

Q/L = 5L⁴ K⁵

Positive marginal product show that the capital and Labor add positive quantities to total output.

Second derivatives:

²Q/K² = 20K³ L⁵

²Q/L² = 20L³ K⁵

Positive change in marginal product once again show the positive contribution of capital and labor.

c. Q = aK.¹ L⁹

Q/K = a L⁹

Q/L = a K. 9L⁸

Second derivatives:

²Q/K² = 0

²Q/L² = a K. 72L⁷

Positive change in marginal product once again show the positive contribution of capital and labor.

Zero value shows no contribution

3. Profit functions:

a. ? = f(TR, TC)

?/Q =  f(MR, MC). This is because derivating TR with respect to Q gives us marginal revenue, MR. And derivating TC with respect to Q gives us marginal cost, MC.

²?/Q² =  f(MR, MC)

b. ? = R(Q) – C(Q)

   ?/Q =  R(Q)/Q -  C(Q)/Q

= MR - MC

²?/Q² = MR/Q - MC/Q

The second derivative is equal to rate of change in MR and MC of the firm.