4.5 Putting together the Learning Curve and the Cost Function

Objective: make assumptions that incorporate the learning curve into the cost function as a special case.

• Recall that the learning curve equation can be written as
  
  $$\ln(c_t) = \ln(c_1) + \alpha_c \ln(n_t) + u_t.$$
  
  

• And the cost equation as
  
  $$\ln(C) = \ln(k) + (1/r) \ln(y) + (\alpha_1/r) \ln(p_1) + (\alpha_2/r) \ln(p_2) + (\alpha_3/r) \ln(p_3) + v_t.$$
  
  

Then the question becomes can we put restrictions and assumptions on the cost function so that the learning curve is a special case?

Here's how it goes.

• Define the state of knowledge A_t as $A_t = n_t^{-\alpha_c}$.
• Assume that effects of the input prices are captured by a GNP deflator, ie
  
  $$GNPD_t = (\alpha_1/r) \ln_(p_1) + (\alpha_2/r) \ln_(p_2) + (\alpha_3/r) \ln_(p_3).$$
  
  

This leads to a simpler equation:

$$\ln(C_t') = \ln(k') + (\alpha_c/r) \ln(n_t) + 1/r \ln(y_t) + u_t.$$

Here C_t' is a real total cost because it as been adjusted by the GNP deflator.

Finally move to unit real costs rather than total real costs and you obtain

$$\ln(c_t) = \ln(k') + (\alpha_c/r) \ln(n_t) + ((1 - r)/r) \ln(y_t) + u_t,$$

which for r = 1 is the learning curve model.

How much sense does the previous equation make?

It says that the log of your average real cost at time t depends on two things. 1, how much you have produced up to time t which surrogates for how much knowledge you have and 2, how much you produce at time t as denoted by y_t. If you produce more and your returns to scale are greater than 1 $(r \> 1)$ then your average cost should decrease - which makes sense.