<p>This is an example of a perfectly competitive market with constant returns to scale. The representative firm in question is, therefore, a price-taker in both the output and input markets. You can see this above by the (implicit) price of the output good being 1 and <span class="math-container">$r$</span> and <span class="math-container">$w$</span> being measured in units of the final good. The firm does not decide which price to set. In that setup, the firm profit must be 0 in the optimum. Euler's Theorem proves that the factor payments will fully exhaust output if the production function features constant returns to scale, i.e. is homogeneous of degree 1. For any other input choice than hiring labor and capital up to the point where their marginal product is equal to their rental rates, the firm will make losses.</p>
<p>If the firm had market power, it would make economic profits.</p>
<p>You are right that the firm is in principle indifferent between producing and not producing such that the firm size is indeterminate. That's why in macroeconomic models you have to pair the above setup with a market clearing condition for capital and labor, i.e. all factors available in the economy are actually demanded.</p>
<p>EDIT: To clarify in response to shgwani2804's answer, it is important to keep in mind that the question above refers to a representative firm in a general equilibrium macroeconomic model. Thus, the rental rates for capital and labor cannot be arbitrary values but rather need to adjust to clear the labor and capital markets. That (among other things) ensures that the rental rates equal the marginal products. In that case, profits will indeed be 0. Off-equilibrium, a maximum may not even exist.</p>

CC BY-SA 4.0
2024-10-29T01:29:33.097Z
So, in my tractor example, would it be correct to say "if $Y(K,L) - wL - rK > 0$ then some other firm could do exactly the same thing but sell the stuff they produce for a lower price and they would get all the business. Therefore, $Y(K,L) - wL - rK$ must be zero"?
Flounderer
CC BY-SA 4.0
2024-10-29T06:25:41.617Z
Yes, and in the process the additional factor demand would also drive up the factor prices to reach 0 profit.
jpfeifer
CC BY-SA 4.0
2024-10-28T09:39:19.373Z
2024-10-28T14:51:07.120Z
2024-10-28T14:51:07.120Z
jpfeifer
<p>As has already been stated in the comments, the statement is conditional on the existence of a maximum.</p>
<p>Concretely, say <span class="math-container">$\alpha = 0.5, \ A = 3, \ r = 1, \ w = 1$</span>, then <span class="math-container">$f(1,1) = 1$</span></p>
<p>If profit is non-zero at any <span class="math-container">$(K,L) \in \mathbb{R}^2_{++}$</span>, then the firm can always increase its profits by employing additional labour and capital (since the given profit function is homogeneous of degree 1). So why stop at <span class="math-container">$(1,1)$</span>? It's better to produce <span class="math-container">$(2,2)$</span>, and even better to produce <span class="math-container">$(3,3)$</span>, ad infinitum.</p>
<p>Moreover, your textbook implicitly assumes that the price of this commodity in the market is equal to 1. Of course, the price depends on demand as well as supply. In this case, if price is 1, then the markets may not clear - there may be no solution to the firm's optimisation problem (i.e. no maximum exists), so there is no finite level of output (and therefore input) for which the profit is maximised. In fact, only market clearing price for the given technology (and an appropriate demand curve) is <span class="math-container">$P = \frac{r+w}{A}$</span></p>
<p>Your confusion emerges from a misunderstanding of what constitutes a firm and a market in the economic sense. The traditional business and accounting terminology does not apply to economic thinking in most cases. A classic example of this would be the fact that economic models do not distinguish between debt and equity - both are treated simply as 'capital', since they are employed in the production of goods and services.</p>
<p><strong>EDIT:</strong> Expanding on the last paragraph...</p>
<p>I have failed to find a good reference, but I will try to give you a basic sense of it here. A firm is characterised by its technology (the production function) and its objective (typically to maximise profit). These two attributes provide a complete description of the firm at this level of analysis, of <em>any</em> kind of firm - whether it's a small family run business, or a multinational corporation.</p>
<p>Capital is treated as an abstract commodity, an input, that can be purchased in a market (hence the name 'capital markets'). That may seem absurd to an accountant, but for an economist, if you've invested your own money into your business, you've foregone (at the very least) interest that you could have earned if you had deposited this money with a bank. If you are working full time on your own business, you have foregone the salary you could have earned in a job. This is the notion of opportunity cost, and such costs are accounted for within the cost of capital and labour.</p>
<p>All of this follows from the definition of the firm. If the firm makes 0 profit at every possible level of output, the firm has no preference to produce nothing - it is, in fact, indifferent to produce any level of output. This error in reasoning comes from conflating the firm with its management/owners. The implicit costs of capital, effort etc. of the entrepreneur/owners are simply included in costs of production for a firm. 0 profits for the firm does not mean 0 return on the capital invested by the owners of capital (whether the entrepreneur or lenders). That is a classic misunderstanding of what profit stands for in economics vs. accounting.</p>

CC BY-SA 4.0
2024-10-28T14:57:07.847Z
In general, this answer is correct. But the problem set linked by @Flounderer refers to a general equilibrium Solow model. Here, the factor prices are not arbitrary values as in the answer but rather need to be equal to their corresponding marginal products, which would correspond to $r=1.5$ and $w=1.5$.  In that case a maximum exists with 0 profits.
jpfeifer
CC BY-SA 4.0
2024-10-28T16:06:47.317Z
@jpfeifer I took the above numbers for concreteness to show a single firm's behaviour in the market for a single commodity. You are right, I have overlooked the analysis specific to the Solow model in this answer. Thank you for pointing that out.
shgwani2804
CC BY-SA 4.0
2024-10-29T08:26:21.603Z
Your last paragraph seems to be spot on, but can you recommend a good reference for learning what "firm" and "market" really mean in an economic sense?
Flounderer
CC BY-SA 4.0
2024-10-29T10:32:46.630Z
@Flounderer I'll update the answer to reflect my understanding
shgwani2804
CC BY-SA 4.0
2024-10-28T13:15:25.743Z
2024-10-29T10:48:59.497Z
2024-10-29T10:48:59.497Z
shgwani2804
<p>I'm looking at some notes by Edmond and the model solutions <a href="https://pages.stern.nyu.edu/%7Ecedmond/ge07pt/ps1_answers.pdf" rel="nofollow noreferrer">here</a>. Suppose a firm pays a rate <span class="math-container">$r$</span> for capital and <span class="math-container">$w$</span> for labour and its production is given by <span class="math-container">$Y = AK^\alpha L^{1-\alpha}$</span>. Then its profits are
<span class="math-container">$$f(K,L) = AK^\alpha L^{1-\alpha} - rK - wL.$$</span>
The question asks the reader to maximise this via differentiation. OK, no problem. But then it turns out the maximum, if it exists, is zero, because in fact <span class="math-container">$f(K,L)$</span> is homogeneous in <span class="math-container">$K$</span> and <span class="math-container">$L$</span>, so if a global maximum exists, <span class="math-container">$f(K,L)$</span> <a href="https://math.stackexchange.com/questions/3865605/max-value-of-a-homogeneous-function">must be zero there</a>. The explanation given is:</p>
<blockquote>
<p>Clearly, this implies that profits are zero. Is this silly? We hear
talk of positive profits all the time. To some extent, this is just a
matter of differences in terminology. To an economist, the accounting
concept known as ‘corporate profits’ is really just a payment to the
owners of capital and so in this example should be treated as part of
rK. As you might recall from Firms and Markets, in economics the term
‘profit’ is often reserved for discussing (say) the ability of a
monopolist to exploit its market power. Since this is a perfectly
competitive market, it’s not surprising that economic profits are
zero.</p>
</blockquote>
<p>I don't buy this explanation at all. In most cases, if you plug reasonable numbers in for <span class="math-container">$A$</span>, <span class="math-container">$r$</span> and <span class="math-container">$w$</span>, there are going to be some values of <span class="math-container">$K$</span> and <span class="math-container">$L$</span> which make <span class="math-container">$f(K,L)$</span> positive, so I don't think that <span class="math-container">$f(K,L)$</span> has a global maximum at all. But the rest of the problem set is based on reasoning about how <span class="math-container">$Y$</span> behaves at a global maximum, so I am confused about how any further calculations make sense.</p>
<p>Also, I don't understand what is meant by &quot;this is a perfectly competitive market&quot;. There is no market here. We are just looking at a model of a single firm which hires, for example, <span class="math-container">$L$</span> farmers for <span class="math-container">$w$</span> dollars each and <span class="math-container">$K$</span> tractors for <span class="math-container">$r$</span> dollars each, and outputs <span class="math-container">$Y(K,L)$</span> dollars worth of stuff. If the maximum possible profit is zero, the firm might as well hire no farmers and no tractors at all. What am I misunderstanding here?</p>

CC BY-SA 4.0
2024-10-27T11:31:58.130Z
It would probably be a good idea for you to give a numerical example of the "reasonable numbers" resulting in positive profit. Some people might find it hard to believe that you have managed to find a counterexample.
Jesper Hybel
CC BY-SA 4.0
2024-10-27T11:59:41.780Z
If $A > k + w$ then $f(1, 1) > 0$ for example. And $A$ is allowed to be anything, depending on the firm being modelled.
Flounderer
CC BY-SA 4.0
2024-10-27T13:13:48.860Z
The statement is: "the maximum, ***if it exists***, is zero". Of course the maximum need not exist.
Herr K.
<microeconomics><cobb-douglas>
Cobb-Douglas function and interpreting the maximum
Flounderer
# Cobb-Douglas function and interpreting the maximum

(Asked by: username_0 on 2024-10-27)

I'm looking at some notes by Edmond and the model solutions [here](https://pages.stern.nyu.edu/%7Ecedmond/ge07pt/ps1_answers.pdf). Suppose a firm pays a rate $r$ for capital and $w$ for labour and its production is given by $Y = AK^\alpha L^{1-\alpha}$. Then its profits are
$$f(K,L) = AK^\alpha L^{1-\alpha} - rK - wL.$$
The question asks the reader to maximise this via differentiation. OK, no problem. But then it turns out the maximum, if it exists, is zero, because in fact $f(K,L)$ is homogeneous in $K$ and $L$, so if a global maximum exists, $f(K,L)$ [must be zero there](https://math.stackexchange.com/questions/3865605/max-value-of-a-homogeneous-function). The explanation given is:

> Clearly, this implies that profits are zero. Is this silly? We hear
> talk of positive profits all the time. To some extent, this is just a
> matter of differences in terminology. To an economist, the accounting
> concept known as ‘corporate profits’ is really just a payment to the
> owners of capital and so in this example should be treated as part of
> rK. As you might recall from Firms and Markets, in economics the term
> ‘profit’ is often reserved for discussing (say) the ability of a
> monopolist to exploit its market power. Since this is a perfectly
> competitive market, it’s not surprising that economic profits are
> zero.

I don't buy this explanation at all. In most cases, if you plug reasonable numbers in for $A$, $r$ and $w$, there are going to be some values of $K$ and $L$ which make $f(K,L)$ positive, so I don't think that $f(K,L)$ has a global maximum at all. But the rest of the problem set is based on reasoning about how $Y$ behaves at a global maximum, so I am confused about how any further calculations make sense.

Also, I don't understand what is meant by "this is a perfectly competitive market". There is no market here. We are just looking at a model of a single firm which hires, for example, $L$ farmers for $w$ dollars each and $K$ tractors for $r$ dollars each, and outputs $Y(K,L)$ dollars worth of stuff. If the maximum possible profit is zero, the firm might as well hire no farmers and no tractors at all. What am I misunderstanding here?

**Tags:** microeconomics, cobb-douglas
> **Comments:**
> **username_1**: It would probably be a good idea for you to give a numerical example of the "reasonable numbers" resulting in positive profit. Some people might find it hard to believe that you have managed to find a counterexample.  
> **username_0**: If $A > k + w$ then $f(1, 1) > 0$ for example. And $A$ is allowed to be anything, depending on the firm being modelled.  
> **username_2**: The statement is: "the maximum, ***if it exists***, is zero". Of course the maximum need not exist.  

---

## 2 answers

### Answer by username_3 on 2024-10-28. Score: 2

This is an example of a perfectly competitive market with constant returns to scale. The representative firm in question is, therefore, a price-taker in both the output and input markets. You can see this above by the (implicit) price of the output good being 1 and $r$ and $w$ being measured in units of the final good. The firm does not decide which price to set. In that setup, the firm profit must be 0 in the optimum. Euler's Theorem proves that the factor payments will fully exhaust output if the production function features constant returns to scale, i.e. is homogeneous of degree 1. For any other input choice than hiring labor and capital up to the point where their marginal product is equal to their rental rates, the firm will make losses.

If the firm had market power, it would make economic profits.

You are right that the firm is in principle indifferent between producing and not producing such that the firm size is indeterminate. That's why in macroeconomic models you have to pair the above setup with a market clearing condition for capital and labor, i.e. all factors available in the economy are actually demanded.

EDIT: To clarify in response to shgwani2804's answer, it is important to keep in mind that the question above refers to a representative firm in a general equilibrium macroeconomic model. Thus, the rental rates for capital and labor cannot be arbitrary values but rather need to adjust to clear the labor and capital markets. That (among other things) ensures that the rental rates equal the marginal products. In that case, profits will indeed be 0. Off-equilibrium, a maximum may not even exist.

> **Comments:**
> **username_0**: So, in my tractor example, would it be correct to say "if $Y(K,L) - wL - rK > 0$ then some other firm could do exactly the same thing but sell the stuff they produce for a lower price and they would get all the business. Therefore, $Y(K,L) - wL - rK$ must be zero"?  
> **username_3**: Yes, and in the process the additional factor demand would also drive up the factor prices to reach 0 profit.  

---

### Answer by username_4 on 2024-10-28. Score: 2

As has already been stated in the comments, the statement is conditional on the existence of a maximum.

Concretely, say $\alpha = 0.5, \ A = 3, \ r = 1, \ w = 1$, then $f(1,1) = 1$

If profit is non-zero at any $(K,L) \in \mathbb{R}^2\_{++}$, then the firm can always increase its profits by employing additional labour and capital (since the given profit function is homogeneous of degree 1). So why stop at $(1,1)$? It's better to produce $(2,2)$, and even better to produce $(3,3)$, ad infinitum.

Moreover, your textbook implicitly assumes that the price of this commodity in the market is equal to 1. Of course, the price depends on demand as well as supply. In this case, if price is 1, then the markets may not clear - there may be no solution to the firm's optimisation problem (i.e. no maximum exists), so there is no finite level of output (and therefore input) for which the profit is maximised. In fact, only market clearing price for the given technology (and an appropriate demand curve) is $P = \frac{r+w}{A}$

Your confusion emerges from a misunderstanding of what constitutes a firm and a market in the economic sense. The traditional business and accounting terminology does not apply to economic thinking in most cases. A classic example of this would be the fact that economic models do not distinguish between debt and equity - both are treated simply as 'capital', since they are employed in the production of goods and services.

**EDIT:** Expanding on the last paragraph...

I have failed to find a good reference, but I will try to give you a basic sense of it here. A firm is characterised by its technology (the production function) and its objective (typically to maximise profit). These two attributes provide a complete description of the firm at this level of analysis, of *any* kind of firm - whether it's a small family run business, or a multinational corporation.

Capital is treated as an abstract commodity, an input, that can be purchased in a market (hence the name 'capital markets'). That may seem absurd to an accountant, but for an economist, if you've invested your own money into your business, you've foregone (at the very least) interest that you could have earned if you had deposited this money with a bank. If you are working full time on your own business, you have foregone the salary you could have earned in a job. This is the notion of opportunity cost, and such costs are accounted for within the cost of capital and labour.

All of this follows from the definition of the firm. If the firm makes 0 profit at every possible level of output, the firm has no preference to produce nothing - it is, in fact, indifferent to produce any level of output. This error in reasoning comes from conflating the firm with its management/owners. The implicit costs of capital, effort etc. of the entrepreneur/owners are simply included in costs of production for a firm. 0 profits for the firm does not mean 0 return on the capital invested by the owners of capital (whether the entrepreneur or lenders). That is a classic misunderstanding of what profit stands for in economics vs. accounting.

> **Comments:**
> **username_3**: In general, this answer is correct. But the problem set linked by @Flounderer refers to a general equilibrium Solow model. Here, the factor prices are not arbitrary values as in the answer but rather need to be equal to their corresponding marginal products, which would correspond to $r=1.5$ and $w=1.5$.  In that case a maximum exists with 0 profits.  
> **username_4**: @jpfeifer I took the above numbers for concreteness to show a single firm's behaviour in the market for a single commodity. You are right, I have overlooked the analysis specific to the Solow model in this answer. Thank you for pointing that out.  
> **username_0**: Your last paragraph seems to be spot on, but can you recommend a good reference for learning what "firm" and "market" really mean in an economic sense?  
> **username_4**: @Flounderer I'll update the answer to reflect my understanding  

---

