Algebraic derivation of PPF

Our production functions were:

Qx = 5 Lx so that Lx = (1/5) Qx

Qy = 10 Ly so that Ly = (1/10) Qy

and our total labor force available was: Lt = 50

Now, the basic constraint is that the sum of labor in the X industry and in the Y industry must equal the total amount of labor available; algebraically, this means that:

Lt = Lx + Ly

substitute for Lt the value of 50 workers, and substitute for Lx and Ly the expressions found above, namely Lx = (1/5) Qx and Ly = (1/10) Qy, and we will have:

50 = (1/5) Qx + (1/10) Qy

or, multiplying both sides of the equation by 10

500 = 2 Qx + Qy

or

Qy = 500 - 2 Qx

the slope of -2 is clearly seen in this form of the equation, as is the Y intercept of 500. To find the X-intercept, we could set Qy equal to zero and solve the equation 0 = 500 - 2 Qx or Qx = 250.

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