Problem

1 Technology sets

For each part, sketch a technology set for a 1-input, 1-output technology satisfying all of the listed properties:

1. Increasing returns to scale, allows free disposal and shutdown.
2. Neither increasing nor decreasing returns to scale.
3. Convex, allows free disposal but riot shutdown.
4. Allows a free lunch but riot free disposal.

2 Returns to scale

1. Write down a production function (with any number of inputs) which exhibits increas-ing returns to scale, but which isn't homogeneous of any degree. (Hint: Pick a concave, increasing function which isn't a power function.)

2. Write down a production function (with any number of inputs) which exhibits decreas-ing returns to scale, but which isn't homogeneous of any degree. (Hint: Pick a convex, increasing function which isn't a power function.)

For each of these parts, you must prove both that your function isn't homogeneous (a single numerical example is fine), and that it exhibits the right sort of returns to scale.