a. Assuming the market to be a perfectly competitive market,

MC=4+2QMC=4+2Q

In a perfect competition, the marginal cost is equal to the price of the firm.

So when Q = 4

MC=4+(2∗4)MC=4+(2∗4)

MC=4+8MC=4+8

MC=12MC=12

Since, Price = MC

So , P = $12

b. Average Cost = Total Cost/ Q

Given:

TC=20+4Q+Q2TC=20+4Q+Q2

Dividing Q on both sides of the TC equation we have,

TC/Q=20/Q+4+QTC/Q=20/Q+4+Q

AC=20/Q+4+QAC=20/Q+4+Q

Average Variable Cost (AVC) = Variable cost/ Q

The variable cost is the component of total cost which changes with the level of output.

In this case VC will be,

VC=4Q+Q2VC=4Q+Q2

Dividing Q on both sides of VC equation we have,

VC/Q=4+QVC/Q=4+Q

AVC=4+QAVC=4+Q

If we consider Q = 4 then,

AC=20/Q+4+QAC=20/Q+4+Q

AC=20/4+4+4AC=20/4+4+4

AC=5+8AC=5+8

AC=$13AC=$13

and,

AVC=4+QAVC=4+Q

AVC=4+4AVC=4+4

AVC=$8AVC=$8

C. From the above derivation of price, average costs and average variable costs we see that, although the average costs of the firm ($13) is greater than its average revenue ($12), but the firm is still able to earn more than its average variable costs ($8). So even when the firm is sustaining losses in production of alarm clocks, it should still remain in the market in short run.

In the long run the price will settle at the lowest point of AC curve. At this point the price will be equal to lowest point of AC.

The lowest point of AC is obtained when the slope of the AC curve becomes zero.

AC=20/Q+4+QAC=20/Q+4+Q

Differentiating the above equation with respect to Q we have,

dAC/dQ=−20/Q2+1dAC/dQ=−20/Q2+1

Equating the above equation with 0

0=−20/Q2+10=−20/Q2+1

20/Q2=120/Q2=1

Q2=20Q2=20

Q=(20)1/2Q=(20)1/2

Q=4.47Q=4.47

Putting the value of Q = 4.47 in AC we have,

AC=20/4.47+4+4.47AC=20/4.47+4+4.47

AC=4.47+4+4.47AC=4.47+4+4.47

AC=$12.94AC=$12.94

So the long run equilibrium price will be equal to P = AC = $12.94

From the above analysis we see that, the long run price in the market increases. This implies that the firm should remain in the market in the long run, as the price of the alarm clock is likely to be increased.

d. Given the current price level and the price level that will remain in the future, investing in a new machinery will lead to increase in average costs of production as the fixed cost component increases. This will lead to greater losses for the firm. Even though the firm's shut down point will not change, yet making an investment when the firm is already incurring losses is not a viable alternative.

Costs:

Costs are the monetary expenditure incurred in production of goods and services in the economy. The costs of the firms are of the following types: Total costs, variable costs, fixed costs, marginal costs, and average costs.