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Consider the Solow model with technological progress. Suppose the production function is given by $F(K(t),A(t)L(t))$. The variables used are standard.

Questions:

  1. Assuming the output price is normalized to $1$, is the maximization problem $\displaystyle \max_{K,L} [F(K,AL) - wL - rK]$ or $\displaystyle \max_{K,AL} [F(K,AL) - wAL - rK]$?
  2. Is the labour income $wL$ or $wAL$?
  3. Further, is wage rate $w = \frac{\partial F(K,AL)}{\partial L}$ or $w = \frac{\partial F(K,AL)}{\partial (AL)}$?

I believe the rental rate is $r = \frac{\partial F(K,AL)}{\partial K}$ but if that's not the case, please correct me.

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The common usage is probably

  1. $\displaystyle \max_{K,L} [F(K,AL) - wL - rK]$

  2. $wL$

  3. $w = \frac{\partial F(K,AL)}{\partial L}$

BUT

you could get the other answers just by redefining the unit of measurement of labor. The above answers are valid for "$w$ is the wage for a unit of labor", the other answers are valid for "$w$ is the wage for a unit of effective labor", so labor unit where technological progress is taken into account. The effective unit is equivalent to $1/A$ conventional units of labor. It's like saying "using machine X, a worker can now perform the work of 10 normal workers"; this new worker will probably still file 8 hour work days (conventional unit) and not 80 hour work days (effective unit).


Another technical aside: there is no difference between $$ \max_L F( A\cdot L) $$ and $$ \max_{AL} F( A\cdot L) $$ if $A$ is constant.

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  • $\begingroup$ I'm not assuming $A$ to be constant. It's a function of time $(t)$, so something like $A(t) = A_0(1+\epsilon)^t$ may also be possible. I think I should have written the notation more clearly. $\endgroup$ Apr 12 at 5:30
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    $\begingroup$ Yes, I understand that, but in your maximization problem it is constant, right? You maximize in each period? $\endgroup$
    – Giskard
    Apr 12 at 5:32
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    $\begingroup$ To put it differently: the value of $A$ is not affected by your choice of the decision parameters $K$ and $L$? (Technological progress is exogenous.) $\endgroup$
    – Giskard
    Apr 12 at 5:33
  • $\begingroup$ Yes, that's correct. It's only a function of time and not labour or capital. I now see what you mean. Thank you. $\endgroup$ Apr 12 at 5:35
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    $\begingroup$ The reason I posted this is because our instructor told us that the growth of capital income and labour income are same in the steady state (assuming constant technological and population growth). While calculating that, he took labour income as $wAL$ and its growth as $\frac{\dot{\overline{wAL}}}{wAL}$. That's what led to my questions. He eventually showed that they're both are equal to $\frac{\dot{A}}{A} + \frac{\dot{L}}{L}$. (Thought I'll write the context.) $\endgroup$ Apr 12 at 5:36

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