The cost function is given as:

C(y) = VC(y) + FC(y)

Where C(y) stands the cost function

VC(y) stands the variable cost.

FC(y) stands the fixed cost

y stands the output level produced by a firm.

Short run Average Cost is the cost of output per unit which is as given as cost per unit..AC(y)=TC(y)y=VC(y)y+FC(y)y..AC(y)=TC(y)y=VC(y)y+FC(y)y.

Short run Average Variable Cost is the variable cost incurred on per unit of output.

AVC(y)=VC(y)yAVC(y)=VC(y)y

Short run Average Fixed Cost is the fixed cost incurred on per unit of output.

AFC(y)=FCyAFC(y)=FCy

(a)As Average cost is the summation of Average fixed cost and average variable cost.

Since average cost and average variable cost curves are expressed as U-shaped due to the Law of Variable Proportions. As Law states that there will be an increase in cost at decreasing rate due to Increasing return to factor at the beginning of production, so Average cost as well as variable cost curves will be falling expressed as curving downward. But after some level of output, costs will start increasing at increasing rate due to Diminishing return to factor. As a result, both of the cost curves will start rising expressed as curving upward.

These cost curves are basically driven by marginal cost curve which shows the change in cost due to change in output. Marginal cost curves behaves independently as it changes itself at any change in output whereas the average cost curves and average variable cost curves move slowly due to spreading effect of output. For example; as output will increase then that output level will spread the cost curves more.

Marginal cost curve passing from Average cost curve and Average variable cost curve at the minimum point because marginal cost curve moves faster as compared to both of those curves. As marginal cost curve falls, then average curves also starts falling but when marginal cost curve starts rising, then average cost curves are still falling because it takes time to change its curvature. Hence, in this condition, there is only the minimum point of average cost curves where both of the curves would equate.

After concluding this, the following relationship is established between average cost and marginal cost:

? When MC is lesser than AC, then AC will be falling.

? When MC is equal to AC, then MC curve will pass through the minimum point of AC.

? When MC is greater than AC, then AC will be rising.

Since, Average Variable cost curves behaves same as average cost curve because the total variable cost curve and total cost curves behave same due to the constant level of output. Despite of the same behaviour of these curves, the minimum point of average variable cost curve comes before the minimum point of average cost curve Since average cost is the sum total of average fixed cost and average variable cost so there is some pressure of falling average fixed cost on average cost curve whereas; average variable cost curves moves smoothly.

(b)Average cost is sum total of Average Fixed Cost and Average Variable cost. Due to the law of variable proportions, Average Variable cost falls and then rises whereas; Average Fixed cost falls with the rise in production due to constant level of Total Fixed Cost at each level of output.

Average fixed cost falls but always remains positive which acts as a restricting factor to Average cost.

AC=AFC+AVCAC=AFC+AVC

Whereas AVC behaves same as AVC but AFC > 0 which always leads to excess of AC over AVC.

That?s why it is written as

yAC>yAVCp

please see the attached file for the complete solution.