Question

The variable costs associated with a certain process are GHS0.65 per item. The fixed costs per

day have been calculated as GHS250 with special costs estimated as GHS0.02Q², where Q is the
output level produced (ie the number of items produced ).
a. Derive the average cost of production function.

b. Calculate the output level at which average cost is minimised.

c. Calculate the average cost of production.

d. What limitation is placed on Q?
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Answer 1

a. Average cost equation

`\bar c=0.02Q+0.65+250Q^(-1)`

b. Q = 112 units

c. Average cost (Q = 112 units) = GHS 5.12

d. Usually, the actual production capacity.

Explanation:

We will use $ as the currency symbol for GHS.

a) We have:

- Variable costs: 0.65Q

- Fixed costs: 250

- Special costs: 0.02Q^2

Then we can write the total cost equation as:

`C(Q)=0.65Q+250+0.02Q^2`

The average cost function can be calculated dividing the total cost equation by Q. Then, we have:

`\bar c=(C(Q))/(Q)=(0.02Q^2+0.65Q+250)/(Q)=0.02Q+0.65+250Q^(-1)`

b) The output level that minimize the average cost can be calculated deriving the average cost equation and making it equal to zero:

`(d\bar c)/(dQ)=0.02+(-1)250Q^(-2)=0.02-250Q^(-2)=0\\\\\\(250)/(Q^2)=0.02\\\\\\Q^2=250/0.02=12,500\\\\Q=√(12,500)\approx 112`

The output level that minimizes cost is Q=112 units.

c. The average cost of production for the output level of 112 is:

`\bar c=0.02Q+0.65+250Q^(-1)\\\\\bar c(112)=0.02*112+0.65+250/112\\\\\bar c(112)=2.24+0.65+2.23\\\\\bar c(112)=5.12`

d. The limitation is not specified, by this models have a range of Q values where it is valid. These range is usually dependent on the scale of the production. The factory will have a maximum level of production, and over this level, the cost equation is different as other investments are needed.