1. Profits and Wages per Efficiency Units of Labor

2. Labor and Rental Capital Market Equilibrium w=MPL R=MPK

3. Profits Profit= Y-wL-RK In competitive equilibrium: Profit=0 Why? -- Profit<0  better to shut down production -- Profit>0  double profit by doubling inputs and hence production -- * There is another way to argue that Profit=0 using what is called “Euler’s theorem”, see the last 3 slides of this set.

4. From Zero-Profits to Wages per Efficiency Unit of Labor Profit=0=Y-wL-RK wL=Y-RK w=Y/L-R(K/L) w/A=Y/(LA)-R(K/LA)

5. Euler’s Theorem in Economics Euler’s theorem states that for a constant returns to scale function F(K,L) it is true that Y=F(K,L)=MPK*K+MPL*L 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 Euler’s 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 b>0. 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 Euler’s 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 MPK*K+MPL*L (you have to use the product rule of differentiation here!). So we get F(K,L)=MPK*K+MPL*L.