<p>What you are proposing is not profit maximising. Consider a deviation from you proposal where the firm takes out one less unit of the market loan. Then the marginal loss in output is <span class="math-container">$r_i$</span> while the marginal saving in costs is <span class="math-container">$r_m &gt; r_i$</span>. As such, the firm can strictly increase its profits by reducing the amount of market loans.</p>
<p>The problem for the firm is the following:
<span class="math-container">$$
\max F(k_m + k_b) - r_b k_b - r_m k_m \\\text{ subject to } k_b \le k_0, k_b \ge 0, k_m \ge 0
$$</span>
where <span class="math-container">$k_m$</span> is the amount of market loans and <span class="math-container">$k_b$</span> is the amount of subsidized loans (which is capped at <span class="math-container">$k_0$</span>).</p>
<p>The Langrangian is
<span class="math-container">$$
L = F(k_m + k_b) - r_b k_b - r_m k_m - \lambda(k_0 - k_b) + \mu_b k_b + \mu_m k_m
$$</span>
The Kuhn-Tucker first order conditions read
<span class="math-container">\begin{align*}
&amp;F'(k_m + k_b)  +\lambda + \mu_b  = r_b,\\
&amp;F'(k_m + k_b) + \mu_m = r_m\\
&amp;\lambda(k_0 - k_b) = 0,\\
&amp;\mu_b k_b = 0,\\
&amp;\mu_m k_m = 0
\end{align*}</span>
$$</p>
<p>If <span class="math-container">$\lambda = \mu_b &gt; 0$</span> then <span class="math-container">$k_b = 0$</span> and <span class="math-container">$k_b = k_0$</span>, which is impossible.</p>
<p>There are several cases to consider:</p>
<ol>
<li><p><span class="math-container">$\lambda = 0, \lambda_b &gt; 0, \mu_m &gt; 0$</span>.
In this case we have that <span class="math-container">$k_b = 0$</span> and <span class="math-container">$k_m = 0$</span>. This can only be if
<span class="math-container">$$
F'(0) + \mu_b = r_b \text{ and } F'(0) + \mu_m = r_m,
$$</span>
So <span class="math-container">$F'(0) &lt; r_b &lt; r_m$</span>. In this case, it is simply too costly to borrow anything.</p>
</li>
<li><p><span class="math-container">$\lambda = 0$</span>, <span class="math-container">$\mu_b &gt; 0$</span> and <span class="math-container">$\mu_m = 0$</span>. In this case, <span class="math-container">$k_b = 0$</span>. Then
<span class="math-container">$$
F'(k_m) + \mu_b = r_b \text{ and } F'(k_m) = r_m
$$</span>
But then <span class="math-container">$r_m = F'(k_m) &lt; r_b$</span>, which is impossible. So this case is ruled out.</p>
</li>
<li><p><span class="math-container">$\lambda &gt;0$</span>, <span class="math-container">$\mu_b = 0$</span>, <span class="math-container">$\mu_m &gt; 0$</span>. Here <span class="math-container">$k_m$</span> is equal to zero, <span class="math-container">$k_b = k_0$</span> and <span class="math-container">$k_m = 0$</span>. Then
<span class="math-container">$$
F'(k_0) + \lambda = r_b \text{ and } F'(k_0) + \mu_m = r_m.
$$</span>
In this case, the firm will take up the entire subsidized loan. However, taking up additional amount of market loan would be too costly as <span class="math-container">$r_m &gt; F'(k_0)$</span>. Here increasing <span class="math-container">$k_0$</span> will increase profits as <span class="math-container">$\lambda &gt; 0$</span> and increasing <span class="math-container">$k_0$</span> will also increase total output <span class="math-container">$F(k_0)$</span>.</p>
</li>
<li><p><span class="math-container">$\lambda &gt; 0$</span>, <span class="math-container">$\mu_b = 0$</span>, <span class="math-container">$\mu_m = 0$</span>. In this case we have that <span class="math-container">$k_b = k_0$</span> and
<span class="math-container">$$
F'(k_0 + k_m) + \lambda = r_b \text{ and } F'(k_0 + k_m) = r_m.
$$</span>
This is the case in the figure. The firm will take up the entire amount <span class="math-container">$k_0$</span> and some additional loans <span class="math-container">$k_m$</span>. Now, notice that if <span class="math-container">$k_0$</span> increases (and we remain in case 4) we still have that <span class="math-container">$F'(k_0 + k_m) = r_m$</span>. So the total amount of capital <span class="math-container">$k_0 + k_m$</span> should remain constant. This means that the increase in <span class="math-container">$k_0$</span> is entirely compensated with an equal decrease in <span class="math-container">$k_m$</span>. The profits will increase <span class="math-container">$(\lambda &gt; 0)$</span> but the total amount of capital and the total output <span class="math-container">$F(k_0 + k_m)$</span> will not change.</p>
</li>
<li><p><span class="math-container">$\lambda = 0, \mu_b = 0, \mu_m = 0$</span>. In this case,
<span class="math-container">$$
F'(k_b + k_m) = r_b \text{ and } F'(k_b + k_m) = r_m,
$$</span>
which is impossible as <span class="math-container">$r_b &lt; r_m$</span>.</p>
</li>
<li><p><span class="math-container">$\lambda = 0, \mu_b = 0$</span> and <span class="math-container">$\mu_m &gt; 0$</span>. In this case, <span class="math-container">$k_m = 0$</span>, so
<span class="math-container">$$
F'(k_b) = r_b \text{ and } F'(k_b) + \mu_m = r_m,
$$</span>
In this setting the firm will only take up subsidized loans up to the point where <span class="math-container">$F'(k_b) = r_b$</span>. Increasing <span class="math-container">$k_0$</span> will have no effect as the constraint <span class="math-container">$k_b \le k_0$</span> is not binding at the optimum.</p>
</li>
</ol>

CC BY-SA 4.0
2023-08-30T08:41:56.373Z
Thank you very much, very clear !
Ploit88
CC BY-SA 4.0
2023-08-30T08:00:31.530Z
2023-08-30T08:00:31.530Z
tdm
<p>I am reading <a href="https://economics.mit.edu/sites/default/files/publications/2014.04%20Do%20Firms%20want%20to%20borrow%20more.pdf" rel="nofollow noreferrer">Banerjee and Duflo's (2014)</a> seminal paper on credit constraints faced by firms in India.</p>
<p>In their theoretical model (see page 11, namely figure 1), they show that, if firms are credit not credit constrained, they will not increase output when offered subsidised credit.</p>
<p>This may be quite a stupid question, but I don't understand why firms would not invest at the market loans until point k(O) in figure 1 ; then, using the loans with lower interest rate, invest with bank (subidised) loans until point kb2. Why would we expect then to subsitute all market credit with bank (subsidized) credit before reaching that point ?</p>
<p>Thanks for any input !</p>

<development-economics><asymmetric-information><credit><credit-friction>
Banerjee and Duflo - Do Firms Want to Borrow More? - Theoretical model
Ploit88
# Banerjee and Duflo - Do Firms Want to Borrow More? - Theoretical model

(Asked by: username_0 on 2023-08-29)

I am reading [Banerjee and Duflo's (2014)](https://economics.mit.edu/sites/default/files/publications/2014.04%20Do%20Firms%20want%20to%20borrow%20more.pdf) seminal paper on credit constraints faced by firms in India.

In their theoretical model (see page 11, namely figure 1), they show that, if firms are credit not credit constrained, they will not increase output when offered subsidised credit.

This may be quite a stupid question, but I don't understand why firms would not invest at the market loans until point k(O) in figure 1 ; then, using the loans with lower interest rate, invest with bank (subidised) loans until point kb2. Why would we expect then to subsitute all market credit with bank (subsidized) credit before reaching that point ?

Thanks for any input !

**Tags:** development-economics, asymmetric-information, credit, credit-friction

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## 1 answers

### Answer by username_1 on 2023-08-30. Score: 0 | Accepted answer

What you are proposing is not profit maximising. Consider a deviation from you proposal where the firm takes out one less unit of the market loan. Then the marginal loss in output is $r\_i$ while the marginal saving in costs is $r\_m > r\_i$. As such, the firm can strictly increase its profits by reducing the amount of market loans.

The problem for the firm is the following:
$$
\max F(k\_m + k\_b) - r\_b k\_b - r\_m k\_m \\\text{ subject to } k\_b \le k\_0, k\_b \ge 0, k\_m \ge 0
$$
where $k\_m$ is the amount of market loans and $k\_b$ is the amount of subsidized loans (which is capped at $k\_0$).

The Langrangian is
$$
L = F(k\_m + k\_b) - r\_b k\_b - r\_m k\_m - \lambda(k\_0 - k\_b) + \mu\_b k\_b + \mu\_m k\_m
$$
The Kuhn-Tucker first order conditions read
\begin{align\*}
&F'(k\_m + k\_b) +\lambda + \mu\_b = r\_b,\\
&F'(k\_m + k\_b) + \mu\_m = r\_m\\
&\lambda(k\_0 - k\_b) = 0,\\
&\mu\_b k\_b = 0,\\
&\mu\_m k\_m = 0
\end{align\*}
$$

If $\lambda = \mu\_b > 0$ then $k\_b = 0$ and $k\_b = k\_0$, which is impossible.

There are several cases to consider:

1. $\lambda = 0, \lambda\_b > 0, \mu\_m > 0$.
   In this case we have that $k\_b = 0$ and $k\_m = 0$. This can only be if
   $$
   F'(0) + \mu\_b = r\_b \text{ and } F'(0) + \mu\_m = r\_m,
   $$
   So $F'(0) < r\_b < r\_m$. In this case, it is simply too costly to borrow anything.
2. $\lambda = 0$, $\mu\_b > 0$ and $\mu\_m = 0$. In this case, $k\_b = 0$. Then
   $$
   F'(k\_m) + \mu\_b = r\_b \text{ and } F'(k\_m) = r\_m
   $$
   But then $r\_m = F'(k\_m) < r\_b$, which is impossible. So this case is ruled out.
3. $\lambda >0$, $\mu\_b = 0$, $\mu\_m > 0$. Here $k\_m$ is equal to zero, $k\_b = k\_0$ and $k\_m = 0$. Then
   $$
   F'(k\_0) + \lambda = r\_b \text{ and } F'(k\_0) + \mu\_m = r\_m.
   $$
   In this case, the firm will take up the entire subsidized loan. However, taking up additional amount of market loan would be too costly as $r\_m > F'(k\_0)$. Here increasing $k\_0$ will increase profits as $\lambda > 0$ and increasing $k\_0$ will also increase total output $F(k\_0)$.
4. $\lambda > 0$, $\mu\_b = 0$, $\mu\_m = 0$. In this case we have that $k\_b = k\_0$ and
   $$
   F'(k\_0 + k\_m) + \lambda = r\_b \text{ and } F'(k\_0 + k\_m) = r\_m.
   $$
   This is the case in the figure. The firm will take up the entire amount $k\_0$ and some additional loans $k\_m$. Now, notice that if $k\_0$ increases (and we remain in case 4) we still have that $F'(k\_0 + k\_m) = r\_m$. So the total amount of capital $k\_0 + k\_m$ should remain constant. This means that the increase in $k\_0$ is entirely compensated with an equal decrease in $k\_m$. The profits will increase $(\lambda > 0)$ but the total amount of capital and the total output $F(k\_0 + k\_m)$ will not change.
5. $\lambda = 0, \mu\_b = 0, \mu\_m = 0$. In this case,
   $$
   F'(k\_b + k\_m) = r\_b \text{ and } F'(k\_b + k\_m) = r\_m,
   $$
   which is impossible as $r\_b < r\_m$.
6. $\lambda = 0, \mu\_b = 0$ and $\mu\_m > 0$. In this case, $k\_m = 0$, so
   $$
   F'(k\_b) = r\_b \text{ and } F'(k\_b) + \mu\_m = r\_m,
   $$
   In this setting the firm will only take up subsidized loans up to the point where $F'(k\_b) = r\_b$. Increasing $k\_0$ will have no effect as the constraint $k\_b \le k\_0$ is not binding at the optimum.

> **Comments:**
> **username_0**: Thank you very much, very clear !  

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