Production Theory: Key Concepts and Applications

Posted on Dec 6, 2024 in Economy

Theoretical Support: Production Theory Exercises

1. Production with a Variable Input

The production function shows the maximum output for given inputs. In the short term, at least one input is fixed. Average product of labor (APL) is total product (TP) divided by labor units. Marginal product of labor (MPL) is the change in TP per unit change in labor.

2. Shapes of Product Curves

The shapes of the TP, APL, and MPL curves are related. APL is the slope of the line from the origin to a point on the TP curve. MPL is the slope of the TP curve. Typically, APL and MPL rise, reach a maximum, then fall. MPL becomes zero when TP is maximized and negative when TP declines. The declining part of the MPL curve illustrates the law of diminishing returns.

3. Stages of Production

We define three stages based on APL and MPL:

  • Stage I: From origin to maximum APL.
  • Stage II: From maximum APL to where MPL is zero.
  • Stage III: Where MPL is negative.

Rational producers operate in Stage II.

4. Production with Two Variable Inputs: Isoquants

In the long run, all inputs are variable. An isoquant shows combinations of labor (L) and capital (K) that yield the same output. Higher isoquants represent higher output.

5. Marginal Rate of Technical Substitution

The marginal rate of technical substitution (MRTSLK) is the amount of K that can be replaced by one unit of L, holding output constant. MRTSLK = MPL / MPK. MRTSLK diminishes along an isoquant.

6. Characteristics of Isoquants

Isoquants have three key properties:

  1. They have a negative slope.
  2. They are convex to the origin.
  3. They do not intersect.

7. The Isocost Line

An isocost line shows combinations of L and K that can be purchased with a given budget. The slope is -PL/PK, where PL is the price of labor and PK is the price of capital.

8. Producer Equilibrium

A producer is in equilibrium when output is maximized for a given budget, or equivalently, when the highest isoquant is reached given the isocost. This occurs where the isoquant is tangent to the isocost. At this point, MRTSLK = PL/PK, or MPL/PL = MPK/PK.

9. Expansion Path

If the budget changes while input prices remain constant, the isocost shifts parallel. The line connecting the tangency points of isoquants and isocosts is the expansion path.

10. Factor Substitution

If the price of a factor changes, the equilibrium is disturbed. Producers will substitute towards the relatively cheaper factor. The elasticity of technical substitution measures the responsiveness of the K/L ratio to changes in MRTSLK.

11. Returns to Scale

Returns to scale describe how output changes when all inputs are increased proportionally:

  • Constant returns to scale: Output increases proportionally.
  • Increasing returns to scale: Output increases more than proportionally.
  • Decreasing returns to scale: Output increases less than proportionally.