Mathematics

# Production Possibilities Frontier

A business produces two products, with 200 labour hours spread between each. Per unit of product 1 requires 4 labour hours. Per unit of product 2 requires 2.5 labour hours. Assuming full employment, the equation is 4x + 2.5y = 200. Here, x represents product 1 and y product 2.

Question: (i) Identify the slopes, (ii) interpret the meaning?

(i) As 200 labour hours are allocated to product x and y, there is a trade-off between x and y, you can either devote all your time to one or a combination between the two. Recall, y=mx+c, where m=slope, and c=constant. You must express the equation in this manner. Therefore, 4x + 2.5y = 200, as expressed above. Now solve for the equation:

=>  2.5 = -4x + 200      =>  cancel  to the right

y =>  -4/2.5 x + 200/2.5

y =>   -1.6x (mx) + 80 (c)

SLOPE = -1.6    ‘-‘ Value means an inverse relationship between x and y, therefore, the production of either product 1 and 2 is expressed in a downward sloping curve. If production of x increases by 1 unit, and the production of y decreases by 1 unit.

To find the intercepts, set x=0, or y=0.

X Intercept (y=0)

4x=200

x=200/4 = 50     =>  (50, 0)   => (x, y)

(ii) If all workers or resources are employed to produce product (x/1) and therefore production of y is zero. A maximum of 50 units of x can be produced.

Y Intercept (x=0)

2.5y= 200

y=200/2.5 = 80   =>  200/2.5/10   =>  200 x 10/25   => (0, 80) => (x, y)

(ii) If all workers or resources are employed to produce product (y/2) and, therefore, the production of x is zero. A maximum of 80 units of y can be produced.

Here is the production possibility frontier for this combination of products, “a graph that shows the combinations of output that the economy can possibly produce given the available factors of production and the available production technology” (Mankiw, 2015). Reference List

Mankiw, G. (2015). Principles of economics, 7th ed. United States: Cengage Learning.

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