Question #63502

You are given the following cost functions:

TC = 100 + 60Q − 3 Q2 + 0.1 Q3

TC = 100 + 60Q + 3 Q2

TC = 100 + 60Q

a) Compute the average variable cost, average cost, and marginal cost for each

function.

b) In each case, indicate the point at which diminishing returns occur. Also indicate

the point of maximum cost efficiency (i.e., the point of minimum average cost).

Page 5 of 7

c) For each function, discuss the relationship between marginal cost and average

variable cost and between marginal cost and average cost. Also discuss the

relationship between average variable cost and average cost.

TC = 100 + 60Q − 3 Q2 + 0.1 Q3

TC = 100 + 60Q + 3 Q2

TC = 100 + 60Q

a) Compute the average variable cost, average cost, and marginal cost for each

function.

b) In each case, indicate the point at which diminishing returns occur. Also indicate

the point of maximum cost efficiency (i.e., the point of minimum average cost).

Page 5 of 7

c) For each function, discuss the relationship between marginal cost and average

variable cost and between marginal cost and average cost. Also discuss the

relationship between average variable cost and average cost.

Expert's answer

TC1 = 100 + 60Q − 3 Q2 + 0.1 Q3

TC2 = 100 + 60Q + 3 Q2

TC3 = 100 + 60Q

a) The average variable cost functions are:

AVC1 = (TC1 - FC1)/Q = (60Q − 3Q^2 + 0.1Q^3)/Q = 60 − 3Q + 0.1Q^2,

AVC2 = (60Q + 3Q^2)/Q = 60 + 3Q,

AVC3 = 60Q/Q = 60,

The average cost functions are:

ATC1 = TC1/Q = (100 + 60Q − 3Q^2 + 0.1Q^3)/Q = 100/Q + 60 − 3Q + 0.1Q^2,

ATC2 = (100 + 60Q + 3Q^2)/Q = 100/Q + 60 + 3Q,

ATC3 = (100 + 60Q)/Q = 100/Q + 60.

The marginal cost functions are:

MC1 = TC1' = 60 − 6Q + 0.3Q^2,

MC2 = 60 + 6Q,

MC3 = 60.

b) The diminishing returns occur at the point at which MC starts to rise (MC = min and MC' = 0).

For TC1 it is MC1' = 0.6Q - 6 = 0, Q1 = 10 units.

For TC2 it is MC2' = 6 =/= 0, so diminishing returns are from the beginning (Q = 0).

For TC3 it is MC3' = 0, so there is no diminishing returns, but there are constant returns.

The points of maximum cost efficiency (ATC = min) are:

For TC1 it is ATC1' = -100/Q^2 - 3 + 0.2Q = 0,

For TC2 it is ATC2' = -100/Q^2 + 3 = 0, 3Q^2 = 100, Q2 = 5.77 units,

For TC3 it is ATC3' = -100/Q^2 = 0, there is no roots, so there is no point of maximum cost efficiency.

c) The relationship between marginal cost and average variable cost and between marginal cost and average cost are different in different cases. The

relationship between average variable cost and average cost is very close, because they are almost the same.

TC2 = 100 + 60Q + 3 Q2

TC3 = 100 + 60Q

a) The average variable cost functions are:

AVC1 = (TC1 - FC1)/Q = (60Q − 3Q^2 + 0.1Q^3)/Q = 60 − 3Q + 0.1Q^2,

AVC2 = (60Q + 3Q^2)/Q = 60 + 3Q,

AVC3 = 60Q/Q = 60,

The average cost functions are:

ATC1 = TC1/Q = (100 + 60Q − 3Q^2 + 0.1Q^3)/Q = 100/Q + 60 − 3Q + 0.1Q^2,

ATC2 = (100 + 60Q + 3Q^2)/Q = 100/Q + 60 + 3Q,

ATC3 = (100 + 60Q)/Q = 100/Q + 60.

The marginal cost functions are:

MC1 = TC1' = 60 − 6Q + 0.3Q^2,

MC2 = 60 + 6Q,

MC3 = 60.

b) The diminishing returns occur at the point at which MC starts to rise (MC = min and MC' = 0).

For TC1 it is MC1' = 0.6Q - 6 = 0, Q1 = 10 units.

For TC2 it is MC2' = 6 =/= 0, so diminishing returns are from the beginning (Q = 0).

For TC3 it is MC3' = 0, so there is no diminishing returns, but there are constant returns.

The points of maximum cost efficiency (ATC = min) are:

For TC1 it is ATC1' = -100/Q^2 - 3 + 0.2Q = 0,

For TC2 it is ATC2' = -100/Q^2 + 3 = 0, 3Q^2 = 100, Q2 = 5.77 units,

For TC3 it is ATC3' = -100/Q^2 = 0, there is no roots, so there is no point of maximum cost efficiency.

c) The relationship between marginal cost and average variable cost and between marginal cost and average cost are different in different cases. The

relationship between average variable cost and average cost is very close, because they are almost the same.

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