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Homework answers / question archive / 1) A firm has carefully estimated its production function to be: Q = K0
1) A firm has carefully estimated its production function to be: Q = K0.4L 0.6 , where Q = units of output, K = units of capital, and L = units of labor. If both capital and labor were increases by 10%, how much would output increase (in percentage terms)? What sort of returns to scale does the firm face? Explain.
2) Three college students consider the option of forming a lawn care and landscaping business during their summer vacation. They estimate the following costs:Insurance $2,600, Equipment Rental Fees $1,000, Fuel and Supplies $4 per lawn ,Miscellaneous Expenses $2 per lawn. Their projected revenue depends on the number of lawns serviced. The price per job is $30. The going wage for a typical unskilled college student is about $2,400 for the summer. Derive equations for total accounting profit and total economic profit. Should the students launch the business if they expect to service about 200 lawn jobs during the summer?
3)A firm's average cost (AC) is described by the equation: AC = 2,000/Q + 60 + 0.2Q.
(a) Determine the equation for the firm's marginal cost (MC). What is the MC of producing the tenth unit of output?
(b)Compute the level of output that minimizes average total cost (AC)?
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.
