<p><strong>tl;dr:</strong></p>
<p>The statement must have some mistake, if the capital-labor ratio is 1.5 then 30 mill dollar reduction if capital use should result in 20 mill decrease in activity on labor market, but if the capital-labor ratio would be <span class="math-container">$\approx 0.66$</span> then the statement would be completely correct and 30 million reduction of lending would yield 45 million decrease in economic activity. This is because if <span class="math-container">$K/L=0.66$</span> and we know that <span class="math-container">$K=-30$</span> we also automatically know that <span class="math-container">$-30/L=0.66 \implies L\approx -45$</span>.</p>
<p>Thus if we will assume the author made a typo and they meant to say that capital labor ratio <span class="math-container">$K/L=0.66$</span> not <span class="math-container">$1.5$</span>, the statement would make sense.</p>
<p>This is because capital and labor in the economy, generally speaking, cannot be just used separately but has to be combined or 'mixed' in certain proportions when we try to maximize economic output or other variables like profits.</p>
<p>Note you seem to be assuming that firms will take that 30 mill lending to pay wages of workers, but that is simply wrong assumption. Rather firms will use that lending to buy 30 million worth of capital, and then in addition to that 30 million worth of capital they will have to hire workers and pay them additional 45 million in wages (the total spending on capital and labor in the scenario above is 75 mil). The money to pay back loans for capital and to pay wages will come from firm's  revenues that are not modeled in the simple scenario above.</p>
<p><strong>Longer Explanation</strong></p>
<p>Production typically requires use of both capital and labor jointly (there can be some exception e.g. live reading of a book for small audience etc.). For example, quite often used production function is Cobb-Douglass production function <span class="math-container">$Y=AK^{\alpha}L^{1-\alpha}$</span>, with <span class="math-container">$0&lt;\alpha&lt;1$</span>.</p>
<p>Note if firms would only use <span class="math-container">$K$</span> and no labor <span class="math-container">$L=0$</span> their output would be zero no mater how much capital they throw at the problem. Next note since alpha  <span class="math-container">$0&lt;\alpha&lt;1$</span> there are diminishing returns to using either <span class="math-container">$K$</span> or <span class="math-container">$L$</span>. As a consequence, it would not make sense to use just <span class="math-container">$1$</span> of labor and spend all other resources on <span class="math-container">$K$</span>, or vice versa.</p>
<p>To illustrate this consider case where firm with its  budget can afford at max 100 units of either labor or capital so its budget constraint is <span class="math-container">$L+K=100$</span>. You can see that if the firm would use <span class="math-container">$1$</span> of labor and <span class="math-container">$99$</span> units of capital the output (assuming <span class="math-container">$A=1$</span> and <span class="math-container">$\alpha=0.4$</span> would be:</p>
<p><span class="math-container">$$Y=99^{0.4}1^{0.6}  \approx 6.28$$</span></p>
<p>However, if we would use more balanced mix of capital and labor (e.g. 50/50) we would get higher output:</p>
<p><span class="math-container">$$Y=50^{0.4}50^{0.6} = 50$$</span></p>
<p>As a consequence given the production function there will always be some optimal capital-labor ratio <span class="math-container">$\frac{K^*}{L^*}$</span> that will maximize firm's output. For example, in the case above we can find that optimal ratio by maximizing the output with respect to the budget constraint using following Lagrangian:</p>
<p><span class="math-container">$$\mathcal{L}=K^{0.4}L^{0.6} - \lambda ( K+L-100)$$</span></p>
<p>By setting the first order conditions to zero, and using the constraint the above problem yields following optimal quantities of <span class="math-container">$K^*=40$</span> and <span class="math-container">$L^*=60$</span>. Indeed you can verify that with thee quantities the production is truly maximized as it can't get larger:</p>
<p><span class="math-container">$$Y=40^{0.4}60^{0.4} \approx 51.02$$</span></p>
<p>with capital-labor ratio in the optimum being <span class="math-container">$\frac{K^*}{L^*}=\frac{40}{60}=0.66$</span>. Hence firm that would want to maximize output would in this case would chose capital  so that the capital-labor ratio is 0.66.</p>
<p>Consequently, in simple model such as the one presented above, if we know that firm uses 30 million units of capital it will use <span class="math-container">$30/L=0.66\implies L= 45$</span> units of labor. The same applies to changes in the use of capital since in simple model such as the one above the <span class="math-container">$K/L$</span> ratio is fixed so as long as firms want to maximize output (which then translates to firm income). Hence from any change to capital use you can infer change in the labor use.</p>

CC BY-SA 4.0
2021-10-31T15:40:25.200Z
Your right, I made a typo. Thank you for your answer, it is exactly what I was looking for.
MathStudent
CC BY-SA 4.0
2021-10-31T09:44:16.700Z
2021-10-31T09:51:14.440Z
2021-10-31T09:51:14.440Z
1muflon1
<p>The statement is not necessarily accurate, and not just because of the numerical mistake explained in the first paragraph of 1muflon1's answer.  Here are some reasons:</p>
<ol>
<li>The capital-labour ratio over the past 30 years may not be a good predictor of the capital-labour ratio, at the margin, in the businesses whose lending is reduced.  Perhaps those businesses happen to be in sectors whose capital-labour ratios are either well above average (eg steel production) or well below average (eg office-based professional services).  Perhaps there have been recent changes in relative prices which encourage profit-maximising businesses to adopt either more capital-intensive or more labour-intensive techniques.</li>
<li>The lending might not all have been used to pay for capital. Some might have been used to pay for training of labour (which could be considered investment in human capital, but it is unlikely that measurements of the capital-labour ratio over the past 30 years included human capital within capital).</li>
<li>Measurement of capital is not straightforward.  Issues to be considered include: a) the effect of changing interest rates on the present value of capital items with a long life; b) the basis for depreciating capital equipment as it ages; c) the treatment of capital equipment which, though still in good working order, has become technologically obsolete as a result of innovation (eg fax machines being superseded by email and scanners).  Hence past data on capital-labour ratios, based on whatever conventions were used in measuring capital, may not be a reliable guide to the present.</li>
<li>The effect of capital investment on labour can be expected to be spread over many years, and not to be uniform. Some capital equipment will have a longer life than others, resulting in differing patterns of effect on employment.  There are also likely to be differences between the labour needed to install and commission equipment (and to produce it if this is done locally) on the one hand and to work with it once it is up and running on the other.  Thus the effect on labour of a reduction in lending to businesses is unlikely to be at a constant rate.</li>
</ol>

CC BY-SA 4.0
2021-10-31T14:24:55.483Z
2021-10-31T20:19:55.437Z
2021-10-31T20:19:55.437Z
Adam Bailey
<p>I do not have much training in macroeconomics, so I want to know if this statement is accurate:</p>
<p>&quot;A \$30M dollar decrease in lending to businesses leads to a $45M decrease in aggregate labor market activity. Since the capital to labor ratio over the past 30 years is 1.5, these quantities are consistent with each other.&quot;</p>
<p>Don't worry about whether or not the numbers are correct, that is not what I am wondering, just the statement theoretically. For a bit more context, I am dealing with a local labor market analysis where I have a shock to credit in the local area and am just trying to see if the numbers make sense in a back of the envelope calculation sense. I am an undergraduate, so this is the first time I am doing this sort of analysis. Part of me thinks they do make sense, since business lending is a form of capital, but the other more micro part of me wonders how it is possible that a dollar of capital can translate to 1.5 dollars more activity in the labor market, since the businesses can't pay more than that dollar directly to the workers. It could also be that this kind of comparison fundamentally does not make sense.</p>

<macroeconomics><labor-economics>
A Simple Question about Labor Capital Ratios
MathStudent
# A Simple Question about Labor Capital Ratios

(Asked by: username_0 on 2021-10-31)

I do not have much training in macroeconomics, so I want to know if this statement is accurate:

"A \$30M dollar decrease in lending to businesses leads to a $45M decrease in aggregate labor market activity. Since the capital to labor ratio over the past 30 years is 1.5, these quantities are consistent with each other."

Don't worry about whether or not the numbers are correct, that is not what I am wondering, just the statement theoretically. For a bit more context, I am dealing with a local labor market analysis where I have a shock to credit in the local area and am just trying to see if the numbers make sense in a back of the envelope calculation sense. I am an undergraduate, so this is the first time I am doing this sort of analysis. Part of me thinks they do make sense, since business lending is a form of capital, but the other more micro part of me wonders how it is possible that a dollar of capital can translate to 1.5 dollars more activity in the labor market, since the businesses can't pay more than that dollar directly to the workers. It could also be that this kind of comparison fundamentally does not make sense.

**Tags:** macroeconomics, labor-economics

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## 2 answers

### Answer by username_1 on 2021-10-31. Score: 2 | Accepted answer

**tl;dr:**

The statement must have some mistake, if the capital-labor ratio is 1.5 then 30 mill dollar reduction if capital use should result in 20 mill decrease in activity on labor market, but if the capital-labor ratio would be $\approx 0.66$ then the statement would be completely correct and 30 million reduction of lending would yield 45 million decrease in economic activity. This is because if $K/L=0.66$ and we know that $K=-30$ we also automatically know that $-30/L=0.66 \implies L\approx -45$.

Thus if we will assume the author made a typo and they meant to say that capital labor ratio $K/L=0.66$ not $1.5$, the statement would make sense.

This is because capital and labor in the economy, generally speaking, cannot be just used separately but has to be combined or 'mixed' in certain proportions when we try to maximize economic output or other variables like profits.

Note you seem to be assuming that firms will take that 30 mill lending to pay wages of workers, but that is simply wrong assumption. Rather firms will use that lending to buy 30 million worth of capital, and then in addition to that 30 million worth of capital they will have to hire workers and pay them additional 45 million in wages (the total spending on capital and labor in the scenario above is 75 mil). The money to pay back loans for capital and to pay wages will come from firm's revenues that are not modeled in the simple scenario above.

**Longer Explanation**

Production typically requires use of both capital and labor jointly (there can be some exception e.g. live reading of a book for small audience etc.). For example, quite often used production function is Cobb-Douglass production function $Y=AK^{\alpha}L^{1-\alpha}$, with $0<\alpha<1$.

Note if firms would only use $K$ and no labor $L=0$ their output would be zero no mater how much capital they throw at the problem. Next note since alpha $0<\alpha<1$ there are diminishing returns to using either $K$ or $L$. As a consequence, it would not make sense to use just $1$ of labor and spend all other resources on $K$, or vice versa.

To illustrate this consider case where firm with its budget can afford at max 100 units of either labor or capital so its budget constraint is $L+K=100$. You can see that if the firm would use $1$ of labor and $99$ units of capital the output (assuming $A=1$ and $\alpha=0.4$ would be:

$$Y=99^{0.4}1^{0.6} \approx 6.28$$

However, if we would use more balanced mix of capital and labor (e.g. 50/50) we would get higher output:

$$Y=50^{0.4}50^{0.6} = 50$$

As a consequence given the production function there will always be some optimal capital-labor ratio $\frac{K^\*}{L^\*}$ that will maximize firm's output. For example, in the case above we can find that optimal ratio by maximizing the output with respect to the budget constraint using following Lagrangian:

$$\mathcal{L}=K^{0.4}L^{0.6} - \lambda ( K+L-100)$$

By setting the first order conditions to zero, and using the constraint the above problem yields following optimal quantities of $K^\*=40$ and $L^\*=60$. Indeed you can verify that with thee quantities the production is truly maximized as it can't get larger:

$$Y=40^{0.4}60^{0.4} \approx 51.02$$

with capital-labor ratio in the optimum being $\frac{K^\*}{L^\*}=\frac{40}{60}=0.66$. Hence firm that would want to maximize output would in this case would chose capital so that the capital-labor ratio is 0.66.

Consequently, in simple model such as the one presented above, if we know that firm uses 30 million units of capital it will use $30/L=0.66\implies L= 45$ units of labor. The same applies to changes in the use of capital since in simple model such as the one above the $K/L$ ratio is fixed so as long as firms want to maximize output (which then translates to firm income). Hence from any change to capital use you can infer change in the labor use.

> **Comments:**
> **username_0**: Your right, I made a typo. Thank you for your answer, it is exactly what I was looking for.  

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### Answer by username_2 on 2021-10-31. Score: 2

The statement is not necessarily accurate, and not just because of the numerical mistake explained in the first paragraph of 1muflon1's answer. Here are some reasons:

1. The capital-labour ratio over the past 30 years may not be a good predictor of the capital-labour ratio, at the margin, in the businesses whose lending is reduced. Perhaps those businesses happen to be in sectors whose capital-labour ratios are either well above average (eg steel production) or well below average (eg office-based professional services). Perhaps there have been recent changes in relative prices which encourage profit-maximising businesses to adopt either more capital-intensive or more labour-intensive techniques.
2. The lending might not all have been used to pay for capital. Some might have been used to pay for training of labour (which could be considered investment in human capital, but it is unlikely that measurements of the capital-labour ratio over the past 30 years included human capital within capital).
3. Measurement of capital is not straightforward. Issues to be considered include: a) the effect of changing interest rates on the present value of capital items with a long life; b) the basis for depreciating capital equipment as it ages; c) the treatment of capital equipment which, though still in good working order, has become technologically obsolete as a result of innovation (eg fax machines being superseded by email and scanners). Hence past data on capital-labour ratios, based on whatever conventions were used in measuring capital, may not be a reliable guide to the present.
4. The effect of capital investment on labour can be expected to be spread over many years, and not to be uniform. Some capital equipment will have a longer life than others, resulting in differing patterns of effect on employment. There are also likely to be differences between the labour needed to install and commission equipment (and to produce it if this is done locally) on the one hand and to work with it once it is up and running on the other. Thus the effect on labour of a reduction in lending to businesses is unlikely to be at a constant rate.


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