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Given production function $f(L,K)=16L^\frac{1}{4}K^\frac{3}{4}$, where each unit of labour costs £50 and each unit of capital costs £100 and you have a budget of £500,000. Find the number of units of labour to maximise production.

We were given this question in a maths class, so I am slightly unsure of my solution, as follows:

We have $Y = 16L^\frac{1}{4}K^\frac{3}{4} + \lambda(500,000-50L-100K)$

$\frac{\partial Y}{\partial L}=4L^\frac{-3}{4}K^\frac{3}{4}-50\lambda =0$

$\frac{\partial Y}{\partial K}=12L^\frac{1}{4}K^\frac{-1}{4}-100\lambda =0$

$\frac{\partial Y}{\partial \lambda}=500,000 - 50L - 100K =0$

Then, $\frac{(\frac{\partial Y}{\partial L})}{(\frac{\partial Y}{\partial K})} = \frac{4L^\frac{-3}{4}K^\frac{3}{4}}{12L^\frac{1}{4}K^\frac{-1}{4}} = \frac{50 \lambda}{100 \lambda} $

$\implies \frac{K}{3L}=\frac{1}{2} \implies k = \frac{3}{2}L$

Then inputting into $\frac{\partial y}{\partial \lambda}, 500,000 = 50L + 100(\frac{3}{2}L) = 200L$

$\implies L = 2,500$

Therefore, at a budget of £500,000, 2,500 units of labour maximises production?

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3 Answers 3

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If your aim is to maximize the production then your approach is correct.

But if the aim is to find the optimal number of units of labor, then you should solve it as a profit maximization problem with a budget constraint.

Then the problem should be as following

Maximize the profit such that the total cost spent on production does not exceed the budget $$ Max_{L,K}\ Profit(K,L)=P(16L^\frac{1}{4}K^\frac{3}{4}) - 50L - 100K $$ $$st. 50L + 100K \le 500000$$

Then the Lagrange function would be: $$ \mathcal{L}(K,L,\lambda) = (P(16L^\frac{1}{4}K^\frac{3}{4}) - 50L - 100K) + \lambda(500,000-50L-100K)$$

FOC:

$\frac{\partial \mathcal{L}}{\partial L}=4PL^\frac{-3}{4}K^\frac{3}{4} - 50-50\lambda =0$

$\frac{\partial \mathcal{L}}{\partial K}=12PL^\frac{1}{4}K^\frac{-1}{4} - 100-100\lambda =0$

$\frac{\partial \mathcal{L}}{\partial \lambda}=500,000 - 50L - 100K =0$

Solving the system of FOC equations will yield $L = 2500, K = 3750$ I used Wolfram to get the solution, but you can solve that using the approach you used in the question.

Wolfram code:

Z = Lambda

Solve[{4P (L^(-3/4))(K^(3/4)) - 50 - 50Z==0, 12P (L^(1/4))(K^(-1/4)) - 100 - 100Z==0, 500000-50L-100K==0},{L,K,Z}]
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    $\begingroup$ Very clear, thank you! $\endgroup$ Nov 23, 2021 at 21:46
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    $\begingroup$ @Macosso: I have two points. First, "total cost spent on production does not exceed the budget" means $50L+100K\le 500\,000$, i.e., the opposite inequality sign. The second one is a technical remark. In fact, there is no need to use the method of Lagrange multiplier. Modifying the budget condition to $L=10^4-2K$ enables one to reduce the number of variables in $F:=16P(LK^3)^{1/4}-50L-100K$. Then $F$ becomes $16P(10^4K^3-2K^4)^{1/4}$. Since $x^{1/4}$ is increasing, it suffices to maximize the polynomial $10^4K^3-2K^4$. Of course, this yields identical result as in your post. $\endgroup$
    – Marian G.
    Nov 28, 2021 at 21:54
  • $\begingroup$ Yes, you are righr for point 1 and 2. 1. The constraint should should be $\le$ the sign has been updated. 2. You dont have to use the Lagrange methods when there are only 2 variables, you can just solve for for one variable in th budget constraint and plug it to the production function. And this reduces the amount of variables. But we tend to use more Lagrange method because it is more powerful, when we have multiple variables etc. In this specific case I used the Lagrange method because the question was asked with initial solutions thar involved Lagrange. $\endgroup$
    – Macosso
    Nov 29, 2021 at 7:33
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    $\begingroup$ @MarianG. it would be great to add your point 2 as an additional answer, it would be very helpful for people looking for different ways of solving this kind of problems. $\endgroup$
    – Macosso
    Nov 29, 2021 at 14:07
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    $\begingroup$ @Macosso The solution you obtained is independent of the price $P$ which cannot be correct. If $P$ is small enough, consider some value of $P$ that is close enough to $0$, in that case hiring 2500 units of labor and 3750 units of capital will lead to losses. A better choice in that case is to hire 0 units of labor and 0 units of capital since that would lead to 0 profits (which is better than loss). $\endgroup$
    – Amit
    Jun 29 at 5:21
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In this thread, two problems are discussed:

Problem 1: \begin{eqnarray*} \max_{l, k} & \ 16 l^{\frac{1}{4}}k^{\frac{3}{4}} \\ \text{s.t. } & 50l + 100k \leq 500,000 \\ \text{and } & l\geq 0, k\geq 0\end{eqnarray*} Solution to this problem satisfy:

$\text{MRTS} = \dfrac{\text{MP}_L}{\text{MP}_K}=\dfrac{k}{3l} = \dfrac{50}{100}$ and $50l+100k = 500,000$

Solving the system, we get $(l, k) = (2500,3750)$

Problem 2: \begin{eqnarray*} \max_{l, k} & \ p(16 l^{\frac{1}{4}}k^{\frac{3}{4}}) -50l-100k \\ \text{s.t. } & 50l + 100k \leq 500,000 \\ \text{and } & l\geq 0, k\geq 0\end{eqnarray*}

To solve the above problem, we can first solve the cost-minimization problem of the firm.

\begin{eqnarray*} \min_{l, k} & 50l+100k \\ \text{s.t. } & 16 l^{\frac{1}{4}}k^{\frac{3}{4}} \geq q \\ \text{and } & l\geq 0, k\geq 0\end{eqnarray*}

Solution to this problem satisfy:

$\text{MRTS} = \dfrac{\text{MP}_L}{\text{MP}_K}=\dfrac{k}{3l} = \dfrac{50}{100}$ and $16 l^{\frac{1}{4}}k^{\frac{3}{4}} = q$

Solving the system, we get the conditional labor and capital demand as: $(l^c, k^c)(q) = \left(\left(\frac{2}{3}\right)^\frac{3}{4} \frac{q}{16}, \left(\frac{3}{2}\right)^\frac{1}{4} \frac{q}{16}\right)$

and the cost function is $C(q) = \dfrac{50 \left(\frac{2}{3}\right)^\frac{3}{4} + 100 \left(\frac{3}{2}\right)^\frac{1}{4}}{16}q$

Let's define $c = \dfrac{50 \left(\frac{2}{3}\right)^\frac{3}{4} + 100 \left(\frac{3}{2}\right)^\frac{1}{4}}{16}$

Therefore, $C(q) = cq$.

Now we can re-write the above profit maximization problem as: \begin{eqnarray*} \max_{q} & \ pq -cq \\ \text{s.t. } & cq \leq 500,000 \\ \text{and } & q \geq 0\end{eqnarray*}

Solving this problem, gives the supply: \begin{eqnarray*} q^s(p) \in \begin{cases} \left\{\frac{500,000}{c}\right\} & \text{if } p > c \\ \left[0,\frac{500,000}{c}\right] & \text{if } p = c \\ \{0\} & \text{if } p < c \end{cases} \end{eqnarray*}

Substituting the supply in conditional labor demand gives the labor demand as: \begin{eqnarray*} l^d(p) \in \begin{cases} \left\{2500\right\} & \text{if } p > c \\ \left[0,2500\right] & \text{if } p = c \\ \{0\} & \text{if } p < c \end{cases} \end{eqnarray*} and capital demand is $k^d(p)=\frac{3}{2}l^d(p)$

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Based on Macosso's comment, I'll present another way for finding the local maximum of the function under consideration such that the budget constraint is met. Instead of Lagrange's multiplier, we investigate the parametric system of functions and reduce the original maximization problem to a simple maximization of polynomial values.

Our aim is to maximize the profit function $$F(K,L) :=P\cdot16\cdot L^{\frac14}\cdot K^{\frac34}-(50L+100K)$$ with the budget condition $50L+100K\le5\cdot 10^5$ with $K,L\ge 0$. This condition can be rewritten as $50L+100K=c$, where $c$ is arbitrary fixed parameter with $0\le c\le5\cdot10^5$. Consequently, this yields $L+2K=c/50$, i.e., $L=c/50-2K$. This enables us to reduce the number of the variables in the function $F$ defined above. We obtain $$ F(K,L)\biggl |_{L=\frac c{50}-2K} =\underbrace{16\cdot P\cdot\sqrt[4]{\left(\frac c{50}-2K\right)\cdot K^3}-c}_{\varphi(K)}. $$ Remeber that $c$ is a fixed parameter. Hence, to find the local extrema of $\varphi(K)$, one can neglect the constant additive parameter $c$ and investigate the function $$ 16\cdot P\cdot\sqrt[4]{\left(\frac c{50}-2K\right)\cdot K^3}. $$ Nevertheless, taking into account that $\sqrt[4]x$ is strictly increasing, we deduce that it preserves the monotonicity and, hence, the local extrema and their type (min., max.). Therefore, the functions $$ f(K):=\left(\frac c{50}-2K\right)\cdot K^3 $$ and $\varphi(K)$ attain their local extrema at the same $K$-coordinate(s). Expanding the last polynomial expression implies $$ \begin{align*} f(K) &=\frac c{50}\cdot K^3-2K^4,\\[4mm] f'(K) &=\frac{3c}{50}\cdot K^2-8K^3 =K^2\cdot\underbrace{\left(\frac{3c}{50}-8K\right)}_{\text{zero point: $K=\frac{3c}{400}$}}. \end{align*} $$ Since the derivative $f'(K)$ is positive in the left neighbourhood of $K=3c/400$ and negative in the right one, it follows that $f(K)$ attains its local maximum at this point.

Let us now evaluate the value of this maximum: $$ \begin{align*} \varphi\left(\frac{3c}{400}\right) &=16\cdot P\cdot\sqrt[4]{\left(\frac c{50}-\frac{3c}{200}\right )\cdot\left(\frac{3c}{400}\right)^3}-c\\[1em] &=P\cdot c\cdot\frac{\sqrt[4]{54}}{25}-c\\[1em] &=c\cdot\left(P\cdot\frac{\sqrt[4]{54}}{25}-1\right). \end{align*} $$

Since we are searching for $$ \max_{\substack{K,L\ge 0\\50L+100K\le 5\cdot 10^5}}F(K,L), $$ it finally suffices to find the maximum of $\varphi(3c/400)$ with $0\le c\le 5\cdot 10^5$. We easily obtain that $$ \begin{align*} \max_{\substack{K,L\ge 0\\50L+100K\le 5\cdot 10^5}}F(K,L) &=\max_{0\le c\le 5\cdot10^5}\varphi\left(\frac{3c}{400}\right)\\[1em] &=\max_{0\le c\le 5\cdot10^5}c\cdot\left(P\cdot\frac{\sqrt[4]{54}}{25}-1\right)\\[1.5em] &=\begin{cases} 5\cdot 10^5\cdot\left(P\cdot\frac{\sqrt[4]{54}}{25}-1\right)& \text{for $P>\frac{25}{\sqrt[4]{54}}$},\\[1em] 0& \text{for $0<P\le\frac{25}{\sqrt[4]{54}}$}. \end{cases} \end{align*} $$

Of course, to localise the extreme (optimum), we can write (regarding the first case $P>25/\sqrt[4]{54}$) $$ \begin{align*} K_0&=\frac{3c}{400}\Biggl|_{c=5\cdot 10^5}=3750,\\[1em] L_0&=\frac c{50}-2K_0\Biggl|_{c=5\cdot 10^5}=2500. \end{align*} $$

Remark: I think it was not obvious to consider the value $c=5\cdot 10^5$ right from the beginning.

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