Question

Give an example of conditions when the production possibilities frontier might not be concave.

Short answer

The conditions under which PPF might not concave are increasing returns to scale and constant returns to scale.

Step-by-step solution

Step 1. Example of convex production possibilities frontier

Assume that labor L=5 is a fixed labor input, and X and Y are two goods. The following table depicts the increasing return to scale.

Product X	Product Y	PPF
L	X	L	Y	X	Y
0	0	0	0	0	80
1	10	1	10	10	58
2	22	2	22	22	38
3	38	3	38	38	22
4	58	4	58	58	10
5	80	5	80	80	0

It is seen here that as more and more labor is employed 1-5, the quantities of good X and good Y kept increasing from 10-80. You need to note that the increase in output quantities with an increase in labor is not the same. In other words, the quantities of good X and good Yare increasing at an increasing rate.

The following is the production possibilities frontier under the increasing returns. The Y-axis depicts the good Y, and the X-axis depicts Good X.

Since the quantities of the goods X and Y are increasing at an increasing rate, the frontier's shape is convex and not concave.

Step 2. Example of straight-line production possibilities frontier

Here, the labor L=5, which is fixed, and X and Y are the goods. The table below depicts the constant returns to scale:

Product X	Product Y	PPF
L	X	L	Y	X	Y
0	0	0	0	0	50
1	10	1	10	10	40
2	20	2	20	20	30
3	30	3	30	30	20
4	40	4	40	40	10
5	50	5	50	50	0

It is seen here that as more and more labor is employed 1-5, the quantities of good X and good Y are increasing from 10-50 at a constant rate. That is, there is always a 10% increase.

The following is the production possibilities frontier under the constant returns. The Y-axis represents the good Y, and the X-axis represents Good X.

The quantities of goods X and Y are increasing at a constant rate. The frontier's shape is of a straight line and not concave.