In Dhondt & Heylen (2009)*, they specify a certain production function.

Under some standard model assumptions such as perfect competition and so on, they calculate the optimal wage as

$$ w_t = \frac{\partial y_t}{\partial l_t} $$

and the optimal rent on capital as

$$ r_t = \frac{\partial y_t}{\partial k_t} $$

with $k_t $ the capital per capita and $l_t $ the amount of labour per capita and $y_t $ the output per capita.

Now what's the reason of this equality? If it because if you maximize for profit, that the maximum is reached in:

$$ 0 = \frac{\partial W}{\partial L} $$ $$ 0 = \frac{\partial W}{\partial K} $$

and that this must mean, as W = TO-TK, with TO total revenues and TK total costs, that

$$ \frac{\partial TO}{\partial L} = \frac{\partial TK}{\partial L} $$


$$ \frac{\partial TO}{\partial K} = \frac{\partial TK}{\partial K} $$


So in this case, it would be explained if you see ... as ... :

$$ \frac{\partial TK}{\partial K} = r_t $$


$$ \frac{\partial TO}{\partial K} = \frac{\partial y_t}{\partial k_t} $$


$$ \frac{\partial TK}{\partial K} = w_t $$


$$ \frac{\partial TO}{\partial L} = \frac{\partial y_t}{\partial l_t} $$

Is this correct?

*Employment and growth in Europe and the US—the role of fiscal policy composition


I hope this answers your question (if not please let me know!), but the reason they get their wage and rental rates:

$$ w_t = \frac{\partial y_t}{\partial l_t} \text{ and } r_t = \frac{\partial y_t}{\partial k_t} $$

(I assume) is for the same reason most papers use that equation. Consider the firm's problem, that they want to maximize profit, which is (total) revenue ($p_t \times y_t$) minus costs (which we can consider as the total cost of labor, which is $w_t \times l_t$, and capital, which is the rental cost of capital times capital used, $r_t \times k_t$). Therefore, we're trying to maximize:

$$\max_{l_t,k_t, w_t, r_t} \{ p_t y_t - ( w_t l_t + r_t k_t ) \}$$

If we normalize price of the output to $1$, we can solve for the optimal prices of labor and capital by solving the above problem. Taking the inputs one at a time, we get the first order conditions for labor:

$$0 = \frac{\partial y_t}{\partial l_t} - w_t$$

Which, of course, rearranges to

$$w_t = \frac{\partial y_t}{\partial l_t} $$

and for capital:

$$0 = \frac{\partial y_t}{\partial k_t} - r_t$$

which rearranges to:

$$ r_t = \frac{\partial y_t}{\partial k_t} $$

Of course, these are pretty standard derrivations across different fields of economics, so their exact reasoning might be slightly different, but this methodology should provide some of the clearest insight into why that's the case.

  • $\begingroup$ @babipsylon glad to help!! $\endgroup$ – AndrewC Jun 7 '18 at 18:30

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