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I recently learned how to solve the following type of problem using the method of Lagrangian multipliers:

Given a consumer with utility function $u(x,y)$, wealth $w$, prices $p =(p_x,p_y)$, budget constraint $w = xp_x + yp_y$. Find the demand function for $x$ and $y$. I also learned how to find cost functions by plugging these demand functions into my budget constraint and rewriting to give a cost function.

But those cases all involved having $w$ and somehow budgeting with respect to that budget.

Problem Setup:

Suppose a firm is a price-taker with market price for the good in question $p$. Suppose the firm has a production function $F(K,L)$. Derive the firms cost function.

My Question:

How should I approach this problem?

My Ideas / Thoughts Relevant for Solving Problem:

I am not given wealth $w$ although I suppose I could assume any firm who is purchasing has some budget. All previous problems I have done involved two goods, but this involves one. Perhaps I could therefore have a budget constraint $w = qp$ where $q$ is the quantity and $p$ is the price as given to the price-taking firm. But Lagrangian multplier problems I have seen involve two variables and this is just a function of one.

Remark:

The specific production function I am using is $F(K,L) = \sqrt{k} + \sqrt{l}$. I only include this in case it's relevant. I want to solve the problem myself.

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"I am not given wealth $w$ although I suppose I could assume any firm who is purchasing has some budget."

No. This is exactly where the fundamental microeconomic theory of the firm differs from the microeconomic Consumer Theory: the firm is not constrained by a budget. The reason is that this fundamental theory deals most and foremost with the "long-term" view, or even better, with the "planning view". So we assume that the amount necessary to cover expenses, will come from sales, since the firm won't enter production at a loss (remember also, this is a deterministic set-up, there is no uncertainty). Working capital considerations (the fact that usually first you have to actually pay expenses and then to actually collect revenues), does not enter the long-term view, justifiably, it is a short-phenomenon. Also, in the long-term or planning approach, there are no fixed costs, all factors are variable.

Now, the "cost-minimization" approach to solve the firm's optimization problem, is an alternative behavioral assumption to the profit-maximizing setup, and it is very relevant in many real-world cases: public utilities that exist mainly to satisfy demand, and their motive is not to maximize profits -rather they want to minimize cost for the given level output, as determined by demand, in the context of efficient use of the always scarce resources.

But also, the case of a price-taking firm that is too small compared to its market, is closer to a cost-minimizing behavior rather than profit maximizing, since the firm has not really control over its production (except downward by direct decision).

In both of the above cases, an exogenous variable appears: the level of output itself. So we solve the problem by treating the level of output as a "constant" or better, we solve it for any given level of output, and the solution we obtain has the level of output as one of its components.

So

$$\min_{K,L} C\equiv rK + wL \\ s.t. F(K,L) = \bar Q$$

with the Lagrangean

$$\Lambda = rK + wL +\lambda[\bar Q - F(K,L)]$$

The first order conditions are

$$r = \lambda F_K,\;\;\; w=\lambda F_L \tag{1}$$

which gives, at the optimum,

$$rK + wL = C = \lambda\big(F_KK + F_L L) \tag{2}$$

Now assume that the production function is homogeneous of some degree $h$ (not necessarily homogeneous of degree one, i.e. exhibiting "constant returns to scale", but homogeneous -and yours is, of degree $h=1/2$.). From Euler's theorem for homogeneous functions of degree $h$ we have that

$$F_KK + F_L L = hF(K,L) = h\bar Q \tag{3}$$

the last equality holding given the constraint of the initial problem. Inserting $(3)$ into $(2)$ we obtain

$$C = \lambda h \bar Q$$

The multiplier $\lambda$ is optimal marginal Cost, denote it $C'(\bar Q)$, so we arrive at

$$C = C'(\bar Q)\cdot (h\bar Q) \implies C'(\bar Q) + [(-1/h\bar Q)]\cdot C =0$$

This is a simple homogeneous differential equation with solution

$$C = A\cdot \exp\left\{-\int(-1/h\bar Q) {\rm d}\bar Q \right\} = A\cdot \exp\left\{(1/h)\ln \bar Q\right\}$$

$$\implies C^* = A\cdot (\bar Q)^{1/h} \tag{4}$$

for some constant $A >0$. To complete the solution, we need to express the object of interest, $C^*$, in terms of the exogenous entities: $r,w,\bar Q$. To do that derive the optimal marginal cost (which is equal to the multiplier)

$$(4) \implies \lambda^* = (1/h)A(\bar Q)^{1/h-1} \tag{5}$$

Inserting $(5)$ into the first-order conditions we have

$$r = (1/h)A(\bar Q)^{1/h-1} F_K,\;\;\; w=(1/h)A(\bar Q)^{1/h-1} F_L \tag{6}$$

It is time to use the specific functional form of the production function

$$F(K,L) = K^{1/2} + L^{1/2} \implies, F_K = \frac 12 K^{-1/2},\;\; F_L = \frac 12 L^{-1/2} \tag{7}$$

Inerting $(7)$ into $(6)$ together with $h=1/2$ we obtain, after manipulation,

$$rK = \frac {A^2}{r}(\bar Q)^2,\;\; wL = \frac {A^2}{w}(\bar Q)^2 \tag{8}$$

Sum the two to obtain an alternative expression for the Cost function

$$rK+wL = C^* = A(\bar Q)^2\cdot \left[\frac Ar + \frac Aw\right] \tag{9}$$

But inserting $h=1/2$ into $(4)$, we also have that

$$C^* = A(\bar Q)^2 \tag{10}$$ So

$$ (9),(10) \implies A(\bar Q)^2\cdot \left[\frac Ar + \frac Aw\right] = A(\bar Q)^2$$

$$\implies \frac Ar + \frac Aw = 1 \implies A = \frac {wr}{w+r} \tag {11}$$

Inserting $(11)$ into $(4)$ we conclude obtaining

$$C^* = \frac {wr}{w+r}\cdot (\bar Q)^2 \tag{12}$$

Three things:
A) Verify that the second-order-conditions hold for all this to indeed lead to the optimal cost-function.

B) Solve the unconstrained profit-maximization problem with the same production function, normalizing the price of output to $p=1$ (i.e. treating the exogenous prices, $w,r$ as expressed in real terms), to verify that it will lead to a cost level that it is consistent with $(12)$.

C) If you are interested in the theory of the firm under a budget constraint, a related paper is Lee, H., & Chambers, R. G. (1986). Expenditure constraints and profit maximization in US agriculture. American Journal of Agricultural Economics, 68(4), 857-865.

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You don't need to have any knowledge of "wealth" here, because this "wealth" that is necessary to produce is exactly the value of the cost function, it is what you are trying to find.

Cost function is the minimal amount of expenditures necessary to produce a given amount of product given some prices. Suppose your the price of your final good is $p$, and prices of the inputs are $w_1$ and $w_2$. Then your problem is to find such $k$ and $l$ that you can produce given amount $x$ and spend as little as possible.

This can be formilized the following way: $$c(x, w_1, w_2) = \min\limits_{k,l}(w_1 k + w_2 l), s.t. F(k,l) = x$$

Now you should be able to solve it by applying Lagrange multiplier technique that you have learned.

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  • $\begingroup$ Thank you for answering. Why is the wealth the exact value of the cost function? Suppose I am a firm and I have wealth $w$. Based on your answer, it sounds like I would I spend all my wealth on production. Why? $\endgroup$ – Stan Shunpike Jan 16 '15 at 17:43
  • $\begingroup$ Well, it may be relevant for other questions, but here the firm desides how much wealth it needs to invest in capital and labor to produce some given amount of output, that's one of possible interpretation of cost function. It other contexts it may be reasonable to consider firm's wealth, for example if you want to know how much it can produce with given amount of money to spend, but it would be entirely unrelated problem. $\endgroup$ – Nikita Toropov Jan 16 '15 at 17:49
  • $\begingroup$ Also I may have failed to explain it, but cost function has nothing to do with actual behaviour of the firm. Cost function is a cpnvinient way of incorporating relevant information about production possibilities. In this sense wealth of the firm is nonexistent in basic microeconomic theory. It is generally assumed that firm is not limited in terms of liquidity and can always borrow money for operational needs. So only overall profit of the firm is studied and what we are looking here is the most efficient way of producing given quantity and all the imperfections are assumed away. $\endgroup$ – Nikita Toropov Jan 16 '15 at 18:02
  • $\begingroup$ Ahhh, that's very helpful. I didn't understand that at all about borrowing money. Question: is the amount produced fixed? I am looking at the following lecture notes faculty.ses.wsu.edu/Munoz/Teaching/EconS301_Fall2011/… page 3 and the quantity in the Lagrangian is fixed, ie $\bar{q}$. $\endgroup$ – Stan Shunpike Jan 16 '15 at 18:07
  • $\begingroup$ Yes, it is fixed when you are solving minimisation problem. Then you get a (possibly) different answer for each value of $q$ and the answer to the problem as a function of $q$ is called a cost function. $\endgroup$ – Nikita Toropov Jan 16 '15 at 18:12

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