a)

To determine the forecasted demand function substitute the values of consumer income and price of related goods I the demand function:

Q=1000−100P+0.2M−500P_R=1000−100P+0.2(30000)−500(5)=4500−100PQ=1000−100P+0.2M−500P_R=1000−100P+0.2(30000)−500(5)=4500−100P

b)

The inverse demand function is:

P=4500−Q100P=45−0.01QP=4500−Q100P=45−0.01Q

c)

The total revenue function and marginal revenue functions are:

TR=PQ=45Q−0.01Q2MR=∂TR∂Q=45−0.02QTR=PQ=45Q−0.01Q2MR=∂TR∂Q=45−0.02Q

d)

The TVC and MC functions are:

TVC=AVC×Q=(40−0.08Q+0.0001Q2)Q=40Q−0.08Q2+0.0001Q3MC=∂TVC∂Q=40−0.16Q+0.0003Q2TVC=AVC×Q=(40−0.08Q+0.0001Q2)Q=40Q−0.08Q2+0.0001Q3MC=∂TVC∂Q=40−0.16Q+0.0003Q2

To maximize profits, the firm should produce at the point where the MR is equal to the MC:

45−0.02Q=40−0.16Q+0.0003Q20.0003Q2−0.14Q−45=0Q=6.8≈745−0.02Q=40−0.16Q+0.0003Q20.0003Q2−0.14Q−45=0Q=6.8≈7

e)

The price that the firm should set to maximize profits is:

P=45−0.01Q=45−0.01(7)=44.93≈45P=45−0.01Q=45−0.01(7)=44.93≈45

f)

For a firm to stay open in the short run, the price should be more than AVC:

AVC<P40−0.08Q+0.0001Q2<45−0.01Q0.0001Q2−0.07Q−5<0Q<750AVC<P40−0.08Q+0.0001Q2<45−0.01Q0.0001Q2−0.07Q−5<0Q<750

So, a firm should produce less than 750 units to stay in the market and it is producing only 7 units.

g)

The total fixed cost is $5000 and the TVC is:

TVC=40Q−0.08Q2+0.0001Q3=40(7)−0.08(7)2+0.0001(7)3=280+3.92+0.0343=283.95TVC=40Q−0.08Q2+0.0001Q3=40(7)−0.08(7)2+0.0001(7)3=280+3.92+0.0343=283.95

So, the total cost is:

TC=TVC+TFC=283.95+5000=5283.95TC=TVC+TFC=283.95+5000=5283.95

The total revenue is:

TR=45Q−0.01Q2=45(7)−0.01(7)2=315−0.49=314.51TR=45Q−0.01Q2=45(7)−0.01(7)2=315−0.49=314.51

The profit or loss is:

π=TR−TC=314.51−5283.95=4969.44π=TR−TC=314.51−5283.95=4969.44

So, the firm attains a loss.