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Consider a firm producing with the following technology:

\begin{equation} Y = AL^{\alpha}K^{\beta} \end{equation}

Assuming that factors are paid their marginal contribution to output, it can be shown that the optimal factor choice for this firm is:

\begin{equation}\label{eq:k-l ratio} \frac{L^*}{K^*} = \frac{\alpha}{\beta}\frac{r}{w} \end{equation}

(this comes from finding each factor's payment, and combining them)

Thus, the furthest we can go in terms of characterising the equilibrium in this economy/firm relates to the optimal capital-labour ratio. In effect, nothing can be said about the level of inputs and outputs, $L^*$, $K^*$, or $Y^*$.

This is not the case if there is just one input. For instance, if $Y=AL^\alpha$, then

$$L^*=\left(\alpha A\frac{p}{w}\right)^{1-\alpha}$$

The reason is not related to the degree of returns to scale, as this is not constrained. It is not related to elasticity of substitution either, as in the case of a CES, it is also the capital-labour ratio which is solved for. Here, the solution is

$$ \frac{L^*}{K^*} = \left(\frac{\alpha}{1-\alpha}\frac{r}{w}\right)^\sigma $$

So, the question is, why can we not pin down the level of optimal inputs and output? Is this because we have treated factor prices as exogenous, so we are missing the supply side of each factor market?

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    $\begingroup$ If by "perfection competition" you also mean that the firm is in a perfectly competitive industry, then the zero-profit condition $pAL^\alpha K^\beta-rK-wL=0$ (where $p$ is the price for the good produced) should help you pin down both quantities. $\endgroup$ – Herr K. Mar 30 '17 at 20:01
  • $\begingroup$ It seems it doesn't. You have $pY -rK -wL=0$. Divide all by $L$ and you get a function of labour productiity and capital-labour ratio. Both are solely a function of parameters. Nothing can be discovered. $\endgroup$ – luchonacho Mar 30 '17 at 20:10
  • $\begingroup$ Note that as @denesp answers shows, we can obtain the levels of optimal inputs for a single firm. But not the aggregate levels over the whole economy, where prices become endogenous and then we do need the supply side. $\endgroup$ – Alecos Papadopoulos Mar 30 '17 at 21:16
  • $\begingroup$ @AlecosPapadopoulos I think you mean the demand side. $\endgroup$ – Giskard Mar 30 '17 at 21:55
  • $\begingroup$ @denesp Well, I had in mind the supply side for factors of production, but in the aggregate context, indeed, we need also the aggregate demand side for the price level. $\endgroup$ – Alecos Papadopoulos Mar 30 '17 at 22:38
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Thus, the furthest we can go in terms of characterising the equilibrium in this economy/firm relates to the optimal capital-labour ratio. In effect, nothing can be said about the level of inputs and outputs, $L^*$, $K^*$, or $Y^*$.

I don't think this is true. You combined two equations into one, and thereby lost information. The optimization problem is

\begin{equation} \max_{K,L} p\cdot AL^{\alpha}K^{\beta} - w \cdot L - r \cdot K \end{equation} Taking derivatives w.r.t. $L$ and $K$ yields two first order conditions and the only two unknowns are $L$ and $K$. $$ \alpha \cdot p\cdot AL^{\alpha-1}K^{\beta} = w $$ $$ \beta \cdot p\cdot AL^{\alpha}K^{\beta-1} = r. $$ As you noted this implies the cost minimization equation of $$ \frac{L^*}{K^*} = \frac{\alpha}{\beta}\frac{r}{w}, $$ but why go that way? You could get $$ L = \left(\frac{r}{\beta \cdot p\cdot A K^{\beta-1}}\right)^{\frac{1}{\alpha}} $$ and plug this into the first equation, yielding $$ \alpha \cdot p\cdot A\left(\frac{r}{\beta \cdot p\cdot A K^{\beta-1}}\right)^{\frac{\alpha-1}{\alpha}}K^{\beta} = w. $$ And then $$ K^{\frac{\alpha\beta - (\alpha-1)(\beta-1)}{\alpha}} = K^{\frac{\alpha + \beta - 1}{\alpha}} = \frac{w}{\alpha \cdot p\cdot A\left(\frac{r}{\beta \cdot p\cdot A}\right)^{\frac{\alpha-1}{\alpha}}}. $$ Clearly this is not very pretty but it is very solvable.

Note: I did not go into second order conditions but this is only a maximum if $\alpha + \beta < 1$.

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  • $\begingroup$ Actually, this seems to be possible only if there are no CRS. Otherwise, $\alpha +\beta -1=0$, and so the capital factor disappears from the equation. $\endgroup$ – luchonacho Mar 31 '17 at 7:21
  • $\begingroup$ @luchonacho It is a well-known proposition that if profit maximum exists for a CRS production function the maximal attainable profit is zero. Profits cannot be positive, as then you could increase profits by doubling inputs and outputs. Similarly profits cannot be negative because you could increase profits by halving inputs and outputs. And if profits are zero for a certain input pair ($K,L$) profits will also be zero for $(tK,tL)$ for all positive $t$. I also made a small edit to the answer as the final formula indeed does not work for this case. $\endgroup$ – Giskard Mar 31 '17 at 7:48
  • $\begingroup$ @denesp so, in effect, is it impossible to know optimal levels under CRS? If so, you should add this to the answer, to reflect than only under IRS or DRS levels can be pinned down. $\endgroup$ – luchonacho Mar 31 '17 at 7:54
  • $\begingroup$ @luchonacho CRS is a special case (and is not at all mentioned in your question). You can ask a separate question on CRS or look for the proposition I just mentioned on the site, I am sure I have seen it before. To be clear: Even for CRS it is not impossible to pinpoint optimal levels. Depending on the parameters either 1) arbitrarily large profits are possible and there is no optimum. 2) $(K,L) = (0,0)$ is the only optimum. 3) For all $K \geq 0$ there is an $L$ such that $(K,L)$ is optimal. $\endgroup$ – Giskard Mar 31 '17 at 7:59
  • $\begingroup$ @denesp If my question is "why this is not possible?", and the correct answer is "this is possible under A but not under B", I would say that an answer stating "this is possible" is incomplete (if not incorrect). Can you please confirm that under CRS you can indeed find the levels? It seems impossible to me. As I pointed out earlier, capital disappears in your final equation. $\endgroup$ – luchonacho Mar 31 '17 at 8:04
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Ok, after several analysis of @denesp's answer, and our subsequent chat discussion, I think I got the answer I was looking for.

As @denesp rightly pointed out, the optimisation problem yields two FOCs. These are:

\begin{equation} \frac{w}{p} = \alpha AL^{\alpha-1}K^{\beta} \end{equation} \begin{equation} \frac{r}{p} = \beta AL^{\alpha}K^{\beta-1} \end{equation}

We can rearrange each of them, to the form $L=f(K)$. These are, respectively:

$$ L=\left(\frac{\alpha Ap}{w}\right)^{\frac{1}{1-\alpha}}K^{\frac{\beta}{1-\alpha}} $$

$$ L=\left(\frac{r}{\beta Ap}\right)^{\frac{1}{\alpha}}K^{\frac{1-\beta}{\alpha}} $$

Now, for a given parameterisation (with non-trivial $p>0$), we can plot this functions in the $\{K,L\}$ space. Additionally, we can the optimal capital-labour ratio, which comes from equalising the MRTS to the MRS. As I showed in the question, this optimal relationship is given by:

$$ L^*=K^*\left(\frac{r}{w}\frac{\alpha}{\beta}\right) $$

So, we can now plot the three aforementioned functions:

enter image description here

First, notice that there is a trivial equilibrium at $K^*=L^*=0$. Second, there is another equilibrium with positive optimal inputs and production. Naturally, this point crosses the optimal capital-labour ratio. Importantly, the level of capital and labour can be known. For example, capital is given by the formula shown in @denesp's answer.

Which equilibrium does the firm choose? As @denesp rightly noticed, the non-trivial equilibrium is preferred only if there are decreasing returns to scale ($\alpha+\beta<1$), as a positive production yield positive profits. This can be confirmed using the SOCs, or calculating optimal profits (see below). In the case of increasing returns to scale ($\alpha+\beta>1$), firms would rather not produce, as any positive level of production yield losses.

Yet, the most interesting case - and that which motivated my question - is that of constant returns to scale. It is easy to see in the graph above that under CRS, the two labour functions become linear. Do they have the same slope than the optimal capital/labour ratio? This depends on whether we are thinking from a partial or general equilibrium approach:

  • Partial equilibrium: For a exogenous value of $p$, the two lines are likely to have a different slope than the capital/labour ratio. The three lines start from the origin, as shown below:

enter image description here

To find the equilibrium however, the graph is not entirely helpful, because the optimal choice depends on how $p$ compares with the equilibrium price $p^*$, mentioned below. If $p>p^*$, a firm will want to produce infinity, or as high as possible. Conversely, if $p<p^*$, firms will no produce at all as any positive production leads to losses.

  • General equilibrium, or long run: consider the case of price above the equilibrium, $p>p^*$. Because of positive profits, more firms would like to enter the market. As such, an incumbent individual firm has the incentive to charge a lower price. This happens until the price is at the equilibrium $p^*$. This is the endogenous price which, under constant returns to scale, makes all firms indifferent to produce or not.

Conversely, if the initial price is below the equilibrium, no one produces. Therefore, there is an incentive for firms (consumers) to charge (accept) higher prices. This is until the price becomes $p^*$. Here, there are zero profits. In indifference (as always), we choose the non trivial equilibrium of positive production.

It can be shown that the equilibrium price is given by:

$$ p^* = A\left(\frac{r}{1-\alpha}\right)^{1-\alpha}\left(\frac{w}{\alpha}\right)^\alpha $$

(for example, for $A=w=r=1$ and $\alpha=0.5$, $p^*=2$)

But here is the key element of the answer. At this equilibrium price, the three linear functions shown in the above graph merge. Their slope is the same. They are indistinguishable. This means that, for the individual firm and in the general equilibrium, any level of capital and labour is optimal, as long as their ratio follows the optimal ratio defined earlier. In other words, under CRS, the level of inputs cannot be pinned down, because at any production level profits are zero. In consequence, how much to produce is irrelevant.

Notice that, by introducing the demand for this good, we could find out the aggregate demand for $Y$, which can help us to pin down the *aggregate** input level. Yet, the firm's input and output levels cannot be pinned down. In fact, firm size remains indeterminate. This is not even solved by introducing an arbitrary number of firms, unless assuming firms are identical.


A note on prices.

In the discussion with @denesp, the issue of the price level came up. For risk of misrepresenting @denesp's point, I will only show my conclusion on this. Namely, that in the general equilibrium, the sign of profits depend solely on the nature of returns to scale.

Consider the FOCs arising from the original problem. These are reproduced below:

\begin{equation} \frac{w}{p} = \alpha AL^{\alpha-1}K^{\beta} \end{equation} \begin{equation} \frac{r}{p} = \beta AL^{\alpha}K^{\beta-1} \end{equation}

Without loss of generality, these can be rewritten as:

$$ w=p\alpha \frac{Y}{L} $$

$$ r=p\beta \frac{Y}{K} $$

The cost function is given by

$$ C(K,L) = rK + wL $$

Replacing the two factor equations in the cost function leads to

$$ C(K,L) = p\alpha Y + p\beta Y = pY(\alpha+\beta)$$

Since income of the firm is $pY$, profits are:

$$ \pi(K,L) = pY - pY(\alpha+\beta) = pY\left(1-(\alpha+\beta)\right) $$

This is, the sign (and size) of profits depend on the degree of returns to scale. This is regardless of the price level (as long as it is positive, of course, as expected for a good).

To make the point clearer, the above equation for profits it's true in the general equilibrium. This is because by replacing $w$ and $r$ for the endogenous market values, I am assuming that the factor markets are always in equilibrium.

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  • $\begingroup$ I have several issues with this answer. (Which I have also pointed out in chat.) 1) It does not answer the question. The answer to the question is that the underlying assumption is false. 2) It treats the words optimum and equilibrium as if they were interchangeable, but they are not. 3) It assumes that the optimum always exists and is an interior point, but the first is not true in case of IRS and either the first or the second is not true in the generic CRS case. $\endgroup$ – Giskard Apr 1 '17 at 15:50
  • $\begingroup$ @denesp (see update in chat). For 1), this does answer the question to me. For 2), can you please make my error explicit, as it is not evident to me. For 3), I did say that interior solution does not exist for IRS. Check the paragraph starting with "Which equilibrium...". I added a correction about general equilibrium effects. Maybe that is what you had in mind for point 2). $\endgroup$ – luchonacho Apr 3 '17 at 15:16
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The optimal level of production is where the demand and supply curves intersect.

The cost of labor and capital make up the supply curve; the cost of production at different levels of output.

But the demand curve is the total number of products people are willing to buy at different price levels. The problem is the demand curve is unobservable. It exists, but it's unknown. A company can't just fluctuate prices all over the place to estimate the demand curve. It'll drive customers away.

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  • $\begingroup$ I don't see how this helps. Can you please be more specific on how is this relevant to the question? $\endgroup$ – luchonacho Mar 30 '17 at 20:12
  • $\begingroup$ All the formulas you list relate to calculating the supply curve. The supply curve then needs to be set equal to the demand curve in order to get the optimal level of inputs and outputs. However the demand curve is unobservable. Therefore the optimal level of inputs and outputs cannot be determined. $\endgroup$ – Matt Mar 30 '17 at 21:24

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