How to read this
The surface is the production function , a hill over the input plane; its contour lines are the isoquants. Every technology here is : shapes the isoquants, labels them.
- Substitution: walk along an isoquant. Its slope is , and says how strongly the input mix reacts to it.
- Scale: walk up a ray, all inputs times . The elasticity of scale is the percentage change in output when all inputs grow by 1%.
- One input: walk parallel to an axis. The slope is the marginal product .
Technology
Shape of the isoquants
Output along a ray
Mode
Walk along the isoquant : change the input mix, keep output fixed.
Walk along the ray: multiply both inputs by , keep the input mix fixed.
Keep fixed and change only : the slope of output is the marginal product .
The four shapes of Figure 10 in the notes:
Layers
Plot range
The production hill
isoquant on the hill and its shadow on the floor; the translucent plane is the height ; tangent at .
output along the ray (the vertical plane cuts the hill along it); isoquants ; dots at and .
output as grows with fixed (the vertical plane cuts the hill parallel to the axis); tangent with slope .
The hill from above: isoquants in the input space
The colours are the heights of the hill above, seen from the top; each boundary between two colours is an isoquant.
isoquant , tangent, slope , ray through .
isoquants ; the ray crosses them at evenly spaced points under constant returns, spreading out under decreasing returns and bunching up under increasing returns. The S-shaped law does both along the same ray.
the path and the isoquants it crosses at : equal steps in , and the labels show how much output each step adds.
How the input mix responds to the MRTS
Output along the ray
Total, marginal and average product of
Both axes are logarithmic, so the slope of the curve is the elasticity of substitution . Cobb-Douglas reference, slope 1.
against constant returns ; elasticity of scale on the right axis.
Top: total product with its tangent. Bottom: marginal product and average product .