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Why Profit maximization implies that rate of return to capital equals the net marginal product of capital. Prove and give intuition why both are equal.

I also wonder that why it doesn't implies rate of return to capital equals the net marginal product of labor?

The highlight part and equation (6.32) is my concern

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closed as off-topic by The Almighty Bob, Lumi, cc7768, BKay, Jamzy Jun 25 '15 at 22:50

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  • $\begingroup$ Could you please define mathematically what you mean by rate of return to capital? I think what you state only holds w.r.t. the marginal rate of return. $\endgroup$ – Giskard Jun 24 '15 at 22:44
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    $\begingroup$ @AlecosPapadopoulos Honestly, i am almost graduating with math major from college and want to prepare for graduate economics course for MS of economics, but haven't get to study these two terminology in both introductory and intermediate course... Also, equation (6.32) look funky to me. $\endgroup$ – Victor Jun 25 '15 at 0:10
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    $\begingroup$ Oh, I see. Well I'll be happy to oblige and prepare an answer -but before preparing for macroeconomics, prepare for microeconomics, it would be my advice. $\endgroup$ – Alecos Papadopoulos Jun 25 '15 at 0:14
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    $\begingroup$ I'm voting to close this question as a low-quality question, as it contains pictures which are not there to illustrate your problem but as part of your question. I really like your question, however pictures are (a) not searchable, which defeats the whole point of a this SE, (b) are harder to read and (usually, but no in this case) (c) show a lack of effort. Could you add the equation to your question and just leave the chapter of the book as supplemental material? $\endgroup$ – The Almighty Bob Jun 25 '15 at 4:52
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    $\begingroup$ @Victor Please edit your question to remove the pictures so the material is searchable. Alecos Papadopoulos already answered your question, but it would be beneficial to the community to have this question formatted in an appropriate way. $\endgroup$ – cc7768 Jun 25 '15 at 14:28
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The production side is modeled as follows: There are $m=1,...,j$ identical firms (the number of firms is not necessarily equal to the number of workers of course), that operate in a perfectly competitive environment. This means that firms are price takers, both in the goods market and in the production input market, i.e. they take prices as given when they seek to attain their objective. It also means that markets "clear": in particular, prices adjust without frictions/delay so that all capital and all labor are employed. The firms solve a static (not intertemporal) problem: maximize profits period-by-period separately. Labor is provided totally inelastically, no labor-leisure choice here from the part of workers. Moreover each firm has a constant-returns to scale production function in capital and labor, i.e. the function is homogeneous of degree one.

Omitting the time subscript, the typical firm's production function is

$$F(K_j, L_j),\;\; j=1,...,m$$

and the objective of the firms is to maximize profits which are defined as the surplus of production over payments to labor $wL_j$, net payments to rented capital $rK_j$, and depreciation $\delta K_j$:

$$\max_{K_j, L_j} \pi = F(K_j, L_j)-wL_j-rK_j-\delta K_j$$

Note that these are "real" magnitudes in the economics sense of the word, i.e. that we have divided throughout by the price of output (we do not show it usually, we just say, "expressed in real terms").

Denote $\kappa_j \equiv K_j/L_j$, the capital-labor ratio at firm level. Due to the homogeneity of degree one we can re-write the maximization problem of the firm as

$$\max_{\kappa_j} \pi = L_j\cdot \big[F(\kappa_j, 1)-w-r\kappa_j-\delta \kappa_j\big]$$

Note that $L_j$ has become a multiplicative factor, so we can maximize only the term in brackets, and so only with respect to the capital-labor ratio. We also set $F(\kappa_j, 1) \equiv f(\kappa_j)$ to arrive at

$$\max_{\kappa_j} \pi = \max_{\kappa_j} \big[f(\kappa_j)-w-r\kappa_j-\delta \kappa_j\big]$$

The first order condition for a maximum is for the first derivative to be set equal to zero so

$$f'(\kappa_j) - r - \delta = 0 \implies f'(\kappa_j) - \delta = r$$

This is not yet the funky equation $(6.32)$,although it looks a lot like it, because the latter is expressed in terms of "per capita" capital $k\equiv K/N_1$, i.e. at the level of individuals/consumers/workers, not at firm level.

How do we arrive at $(6.32)$? Well, since we have assumed that all firms are identical, that labor is provided inellastically, and also that the markets for production inputs clear, we have that

$$mK_j = K \implies K_j = K/m,\;\;\; mL_j = N_1 \implies L_j = N_1/m $$

So

$$\kappa_j = \frac {K_j}{L_j} = \frac {K/m}{N_1/m} = K/N_1 \equiv k$$

and now we have obtained $(6.32)$.

Note how all the assumptions made have been used in order to arrive at this result.

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