Profits and Wages per Efficiency Units of Labor

1
Profits and Wages per Efficiency Units of Labor
2
Labor and Rental Capital Market Equilibrium

• wMPL
• RMPK

3
Profits

• Profit Y-wL-RK
• In competitive equilibrium Profit0
• Why?
• -- Profitlt0 ? better to shut down production
• -- Profitgt0 ? double profit by doubling inputs
  and hence production
• -- There is another way to argue that Profit0
  using what is called Eulers theorem, see the
  last 3 slides of this set.

4
From Zero-Profits to Wages per Efficiency Unit of
Labor

• Profit0Y-wL-RK
• wLY-RK
• wY/L-R(K/L)
• w/AY/(LA)-R(K/LA)

5
Eulers Theorem in Economics

• Eulers theorem states that for a constant
  returns to scale function F(K,L) it is true that
• YF(K,L)MPKKMPLL
• Just substitute w and R for the marginal products
  and use the definition of profit and you will see
  that this theorem implies
• ? in a competitive equilibrium profit is zero
  (if production is subject to constant returns to
  scale)

6
Proof Eulers Theorem?

• Actually not that difficult. If you understand
  the concept of constant returns to scale AND the
  concept of a mathematical identity.
• Constant returns to scale
• bF(K,L)F(bK,bL)
• for all bgt0.
• Because the equation holds for all positive
  numbers b, it is an identity. This just means
  that the left-hand side of the equation and the
  right-hand side ARE EXACTLY THE SAME FOR ALL b
  (not just for one value of b).

7
Proof Eulers theorem?

• To put it differently, mathematical identity
  means that bF(K,L) as a function of b and
  F(bK,bL) as a function of b are functions that
  lie on top of each other.
• Because the function lie on top of each other
  they have the same values and the same slopes.
  And the slope of bF(K,L) is F(K,L) while the
  slope of F(bK,bL) is MPKKMPLL (you have to
  use the product rule of differentiation here!).
  So we get F(K,L)MPKKMPLL.