Lecture 12- Application - The Competitive Firm

Question 1

What is the total cost function (TC)?

Answer 1

TC = rK + wL

… Total cost = hourly rental priceunits of capital + hourly wage rateunits of labour

Question 2

What is the total revenue function (TR)?

Answer 2

TR = PQ

… Total revenue = price and quantity

Question 3

What is the profit function?

Answer 3

pi = TR - TC
… profit = PQ - rK - wL
… profit = pricequantity - hourly rental priceunits of capital - hourly wage rate*units of labour

Question 4

What must you remember about the output/quantity function?

Answer 4

It is a function that must include both K (capital) and L (labour) as both are used together to produce a certain quantity
… Q = (K,L)

Question 5

What are the assumptions we make about firms?

Answer 5

1) Profit maximisers
2) Perfectly competitive factor market- firm has no control of hourly rental price or hourly wage rate but can control number of units of capital and labour employed … w and r are constant- simply determined by market and accepted by firm
3) Perfectly competitive output market- firm has no control over price (p) … constant- simply determined by market and accepted by firm

Question 6

What must you remember when a firms output is given?

Answer 6

Total revenue (TR) will be constant 
This is because TR = pq
We assume that price is fixed when looking at firms and that the firm is a price taker- determined by market … p is constant 
The quantity is given and … it is also constant hence why TR is constant

Question 7

How can the firm achieve its objective when quantity is given?

Answer 7

The firms objective is to maximise profit
Profit (pi) = TR - TC
As TR is constant:
… Profit (pi) = c - TC
… a firm can achieve its objective to profit maximise by minimising total costs (TC)

Question 8

What does the following question tell you?

Find the lowest cost of producing 100 units of output if output is produced according to Q=5K^1/3*L^2/3 and r=2 and w=4

Answer 8

It tells you that output is fixed
The fact that the question is asking for the lowest cost means that at this point profits will be maximised as total revenue (TR) is constant as price is fixed in perfect competition and quantity/output has been provided
… if profit = total revenue - total costs and total revenue is constant then profit is maximised by finding the optimum (minimum in this case) point of total cost
You are also given values for r and w … you should be thinking of equations which include both

Question 9

How would you go about solving such a question?

Find the lowest cost of producing 100 units of output if output is produced according to Q=5K^1/3*L^2/3 and r=2 and w=4

Answer 9

1) TC (total cost) = wL + rK … TC = 2L + 4K- this becomes your objective function
It gives you a functional constraint with equation Q=5K^1/3L^2/3
2) But the fixed output is given at 100 … this is the constraint and you can make the equation 100 = 5K^1/3L^2/3
You now have both your objective function and functional constraint and can now form your Lagrange equation
3) Use V instead of L to denote your Lagrange function
4) … V = 2L + 4K - lambda(5K^1/3L^2/3 - 100)
5) partially differentiate V with respect to L, K and lambda
6) because of the powers it is better to divide the 2 partially differential equations with lambda in them
7) once this is done it’ll be easier to find values for L, K* and lambda*
8) Lastly you want to substitute the values found for K* and L* into the your objective (total cost) function to find TC*
9) This value of TC* will be your optimum (minimum in this case) point where profit for this firm will be maximised
10) DO it yourself and the value found for TC* should be 120- calculate yourself and see if you get the same- if not check the 3rd video for lecture 12 on blackboard