Short Run Profit Maximisation Problem

Question 1

Profit formula

Answer 1

π = TR-TC

Question 2

Extended formula

Answer 2

π = p₁+y₁+…+pnyn - (w₁x₁ +…+wmxm)

For n amount of goods, and m inputs.

But we usuually focus on one good produced using 2 inputs…

Question 3

But we usuually focus on one good produced using 2 inputs…

What does the profit equation look like for 2 inputs

Answer 3

π = py - w₁x₁ -w₂x₂

Question 4

Short run profit maximisation - what is the final solution, and when should firms hire up until?

Answer 4

We assume quantity of factor 2 is fixed.
Max pf(x₁,xbar₂) - w₁x₁ - w₂xbar₂)

As the firm can only change the quantity of the first facotr, we find the first-order condition by differentiating with respect to X₁

We just get P(df/dx₁)-w₁=0 (pMP of 1 -w₁)
So therefore pMP₁=w₁
This says that price of the factor(w)=value of its marginal product (pMP)

E.g if the variable factor is labour, the firm hires up until pMPL=wage (or MPL=real wage (w/p)!!!)

Question 5

How do we show the short run profit maximisation graphically

Answer 5

Using production function and isoprofit lines

(NOT ISOQUANT OR ISOCOST)

Question 6

Isoprofit lines

Answer 6

Shows all the combinations of input X₁ (while holding X₂ fixed) and output Y that give the same level of profit.

Question 7

How to find the equation for isoprofit line

Answer 7

Rearrange the short run PM formula to make y the subject.

We get…
y = π/p + w₂/p xBAR₂ + w₁/p X₁

(π is fixed, since isoprofit)
Factor prices (w) and P is fixed in short run
Factor 2 is fixed (Xbar2)
Therefore the first 2 terms are constants, making them the intercept, and slope is w₁/p

Question 8

Draw diagram for SR PM.

Where is highest profit?

Answer 8

Production function is our constraint. Can’t produce above the production function to access higher profit.

Highest profit is where isoprofit meets tangent with production function.

Question 9

What if the price of the fixed factor changes? (w₂)

Answer 9

We obviously cannot change the quantity since fixed so we still have to pay for it.
Slope of the isoprofit line unchanged and equilibrium point stays the same.

However the line now represents a lower profit (since w₂ higher so costs more expensie, so π lower, or vice versa).

Question 10

What if price of unfixed factor w₁ changes? (Pg5)

2 things happen to graph

Answer 10

1. Slope changes since slope w₁/p

Higher w₁ makes isoprofit steeper. Firm uses less of factor 1, and produces less output.

2. Intercept changes.
   Since isocost is steeper, we are at a lower intercept, reflecting a lower level of profit at the new PM output.
   (producing less, and paying factor cost more)

So a higher wage means they will employ less labour, reduce output, and profit.

Question 11

What if P changes (2) (an increase)

Answer 11

Isoprofit flatter (since slope is w₁/p), so produce more by using more of factor 1 (since factor 2 is fixed)

Intercept higher, higher price incentivises to produce more, higher more workers (factor 1) meaning higher profit.