Percentage of workforce change, with two PPFs

Question

A hypothetical economy produces two goods, X and Y. The performance (yield) for every worker is steady, and every worker for Y can produce 10 units of product. If $L_x+L_Y=200 $ (meaning that the total workforce is 200):

(My problem is in (d), everything else is just assisting so you get a better view of the problem)

-a) How many units can every X worker produce?

-b) Find the equation of the first production possibilities frontier (PPF)

-c) Find the equation of the second production possibilities frontier (PPF)

-d) If the technology of the production and the performance of workers for X and Y did not change, what was the change of the workforce between the first and second PPF?

My tries:

-a) If $L_Y=0 \to Y=0$ then $L_X=200$ and for $Y = 0 \to X=400$, so every worker for X can produce $\frac{400}{200}=2$ units

-b) For $y = λx+β \Rightarrow ... \Rightarrow y = -5x+2000$

-c) As the two equations are parallel, $y' = -5x + β' \Rightarrow ... \Rightarrow y' = -5x+2500$

-d) Now for d, where my problem is situated, my try is that the percentage change must equal = $\frac{2500-2000}{2000}100= 25$% but even if that's the correct answer I can't fully understand why this would be the case. What does the $β$ and $β'$ have to do with the workforce? I can't seem to be able to wrap my head around it. Can anyone help me?

Answer 1

The fact that you are getting a correct answer in D using your method is purely due to algebra and cancelling off of common terms.

Let us first see why the answer is actually 25% :

So why does a PPF shift? A PPF Shifts because of either a change in technology or change in resources. The question clearly states that there is no change in the technique of production or output per worker. This means the change in PPF is due to change in resource - which in this case is Labour.

Now in order to find % change in the labour force, consider that if the entire labour force is used in the production of the good Y ( which is what points on Y-axis denote )

Previous Production = 2000, New Production = 2500 [ from intercepts of your equations ]

Since we know that there has been no change in per-unit labour output hence it remains 10.

Thus Previous total labour force = 200, New total labour force = 2500/10 = 250 .

You can easily compute and see that percentage increase is 25%

Your method:

% change = $\frac{2500-2000}{2000}100= 25$

if you simply divide the numerator and denominator by 10 (which is output per work for y), you get what has been done and explained above.