1) A firm has carefully estimated its production function to be: Q = K0

1) Let the initial number of capital units and labor units be 1 each.

So, the output is:

Q=K0.4L0.6Q=(1)0.4(1)0.6=1unitQ=K0.4L0.6Q=(1)0.4(1)0.6=1unit

Now, both the number of capital units and labor units has increased by 10%. So, the output is:

Q=K0.4L0.6Q=(1.1)0.4(1.1)0.6=1.1unitsQ=K0.4L0.6Q=(1.1)0.4(1.1)0.6=1.1units

The percentage increase in output is:

%ChangeinQ=1.1−11×100=10%%ChangeinQ=1.1−11×100=10%

Since, the increase in the labor and capital is same as the increase in output, so the firm faces constant returns to scale.

2) Let the number of lawns to be taken care of be Q.

The equation for accounting profit is:

πA=TR−TC=PQ−(2600+1000+4Q+2Q+2400)=30Q−6000−6Q=24Q−6000πA=TR−TC=PQ−(2600+1000+4Q+2Q+2400)=30Q−6000−6Q=24Q−6000

The equation for economic profit is:

πE=πA−ImplicitCost=24Q−6000−0=24Q−6000πE=πA−ImplicitCost=24Q−6000−0=24Q−6000

If the students have to service 200 lawns, then they should not launch the business because the profits earned are negative.

π=24Q−6000=4800−6000=−1200π=24Q−6000=4800−6000=−1200

3) (a)The total cost (TC) of the firm is:

TC=AC×Q=2000+60Q+0.2Q2TC=AC×Q=2000+60Q+0.2Q2

The variable cost is:

TVC=0.2Q2+60QTVC=0.2Q2+60Q

The marginal cost of the firm is:

MC=∂TVC∂Q=0.4Q+60MC=∂TVC∂Q=0.4Q+60

The marginal cost of producing the tenth unit of output is:

MC=0.4Q+60=0.4(10)+60=64MC=0.4Q+60=0.4(10)+60=64

(b) The level of output that minimizes average total cost is:

∂AC∂Q=−2000Q2+0.2=0⇒Q=100∂AC∂Q=−2000Q2+0.2=0⇒Q=100

The second order condition is:

∂2AC∂Q2=4000Q3=40001003=0.004>0∂2AC∂Q2=4000Q3=40001003=0.004>0

So, the AC is minimized at 100 units.