8

Since the exponents add to one the production function has constant returns to scale, which means that, given factor prices, total cost is linear, which means that it's derivative (= marginal cost) is contant. If you change the exponent 1-alpha to beta where alpha+beta < 1, there will be decreasing returns to scale (but still homotheticity) and you will ...


7

No function that is homogeneous of degree one, is at the same time strictly concave in its arguments. If the function is differentiable (or non-differentiable at a finite number of points), then the Hessian of a linear homogeneous function is singular. So if you want to end up with a unit cost function that is strictly concave, you have to drop at the same ...


5

Partial answer: for simplicity let $P_c =1$. The budget constraint: $c= wn + \Pi$ Simplify (plug in $\Pi$): $c= F(n)- fc$ Goods clearing: $c = F(n)$ The household's budget constraint is inconsistent w/ goods market clearing. The firm pays a fixed cost that doesn't go to anyone. In a "true GE model" all payments have to go to someone in the economy. ...


5

Given $$Y = Af(K,L,Z)$$ it follows that $$\dot Y = \dot A f + A\frac{\partial f}{\partial K}\dot K + A \frac{\partial f}{\partial L} \dot L + A \frac{\partial f}{\partial Z} \dot Z,$$ where dotted expression are time derivatives and dividing with $Y$ it follows $$\frac{\dot Y}{Y} = \frac{\dot A}{A} + \left[A\frac{\partial f}{\partial K}\right]\frac{\dot K}{...


5

As Bertrand pointed out, strict-concavity will necessarily fail along any rays through the origin. But one can have strict concavity for normalized price systems. So let $f:\mathbb{R}^n_+\to\mathbb{R}_+$ be a production function. We let $\Delta^n_{++}$ be the set of all points in $\mathbb{R}^n$ with all coordinates being strictly positive and summing to one. ...


4

I am not sure if it is intuitive but this is because because CRS function is homogenous of degree 1. Full derivation: First, general formula for any arbitrary elasticity of substitution between $L$ and $K$* is given by (see Sydsæter et al. EMEA pp 430): $$\sigma_{L,K} = \frac{-F_K'F_L'(xF_K'+ yF_L')}{xy \left( (F_L')^2F_{KK}^{''} - 2 F_K'F_L'F_{KL}'' + (F_K'...


4

This is something you can figure out yourself. If you have not yet tried, I encourage you to do so. Draw any set $H \in \mathbb{R}^n $. Select any point $x$ of set. Is the set of all points $y$ "under" this point $x$, that is $$ \left\{y\in \mathbb{R}^n | y << x \right\}, $$ a subset of $H$? If no, free disposal is violated. If yes, remove ...


4

Output $z$ is given as $z = x + y$ where $x=min(a,2b)$ and $y = max(3c,4d)$. So assume that you want $x=12$ then $a=12$ AND $b=6$. Since this part of the production delivers only the minimum of serveral production processes $a$ and $2b$ the producer must get insure that ALL subproductionprocesses $a$ and $2b$ deliver the minimum. The producer therefore has ...


4

To understand what is the issue here, try dutifully to examine all possible sub-cases in the production function. The production function is $$Q_0 = \min\{K_0, L_0^{1/2}\}$$. Consider cases A. $K_0 < L_0^{1/2}$ Here $Q_0 = K_0$. Consider $Q_{\lambda }\equiv \min\{\lambda K_0, \lambda^{1/2}L_0^{1/2}\},\;\;\; \lambda>1.$ Subcase A1. If $\lambda K_0\leq \...


4

You have a Leontief production function and in optimum you will always have $K=\sqrt{L}=q_1$. Now increase both inputs by factor $k>1$ and you arrive at $k*K > \sqrt{kL}=q_2$ where the first inequality follows from $k>\sqrt{k}$ for $k>1$ and the second ineqality from the fact that only the minimum matters in your production function. Therefore ...


4

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_{...


3

I'm not a macroeconomist (so definitely not used to looking at those types of equations in that setting), but I think the intuition is that the estimated substitutability of labor and materials is <1 (see page 1048). As labor becomes more productive, you need relatively fewer units of it and shift resources towards the other (non-augmented) inputs. ...


3

Production functions are defined without specific values for parameters, so they all could if you impose that the logical parameter implies a negative return. For example, consider a Cobb-Douglas production function of capital and labor, $Y=\beta_0 K^{\beta_k}L^{\beta_l}\omega \varepsilon$ where $\omega$ denotes firm-observed productivity and $\varepsilon$ ...


3

Start from the first order conditions: $$ \begin{align*} &F_k = r + (1- p)\\ &F_L = w\\ \end{align*} $$ Now totally differentiate with respect to $p$: $$ \begin{align*} &F_{KK} dK + F_{KL} dL = - dp\\ &F_{LK} dK + F_{LL} dL = \frac{\partial w}{\partial p} dp\\ \end{align*} $$ This is equivalent to: $$ \begin{align*} &dK = \frac{- dp - F_{...


3

This text is basically about the use of assets. They can be sold once generating some cash or they can be put to use generating cash. The latter can be done multiple times (which is what the repeatability phrase is about), and in the long run should generate more cash than just selling it. When writing that the latter is the event of economic consequence, ...


3

It is generally true that a profit-maximizing firm with a constant-returns to scale technology can produce a positive output only if the profit is zero. Output prices are pinned down by the zero-profit condition. It follows that the profit-maximizing output level is completely indeterminate at equilibrium prices. However, equilibrium output is not ...


3

The Cobb Douglas production function with constants returns to scale $$y = \prod_i x_i^{\alpha_i} = A \prod_i \left(\frac{x_i}{\alpha_i}\right)^{\alpha_i} ,$$ where $A:= \prod_i \alpha_i^{\alpha_i}$ annoying constant. Cost minimization with perfect competition $$\min_x \ \ p^\top x\ \lvert \ y = \prod \left(\frac{x_i}{\alpha_i}\right)^{\alpha_i},$$ implies ...


3

3. How do we recover parameters from production function estimates (INCOMPLETE ANSWER - will be updated with how to do this in R once I have time to figure it out, or if somebody else knows...) Blundell and Bond aren't estimating the parameters of a Cobb-Douglas production function. They're estimates the parameters of a "dynamic (common factor) ...


3

I figured it out: The first-order condition of the cost minimization problem for, say, material inputs $m_{it}$ gives: $ \lambda \frac{\partial F}{\partial M} = P_M $ Where F is the production function, $P_M$ the material input prices. Multiply by $\frac{M}{F}$ and rearrange, $ \lambda = \frac{P_M M}{\beta_M F} $, where $\beta_M$ is the output elasticity ...


3

The uniqueness of z comes from the ND of $D^2f$ I believe. Let $z^*(p)$ be the solution of the problem at price $p$. Define the function, $G(p,z):=p \nabla f(z)-w$. From uniqueness we have $z=z^*(p)$ if and only if $G(p,z)=0$. That is, $G(p,z)=0$ is an implicit equation which tells us how $z^*(p)$ changes when $p$ does. The IFT gives us a way to find the ...


3

To find the production function, you can solve for $\frac{v}{w}$ in $ {l}_{c}$ and ${k}_{c}$ and set $\frac{v}{w}$ = $\frac{v}{w}$ then solve for $q$. This will yield $$ \frac{v}{w} = (\frac{4{l}_{c}}{3q})^{4} $$ $$ \frac{v}{w} = (\frac{4{k}_{c}}{q})^\frac{-4}{3} $$ Set both equations equal $$ (\frac{4{l}_{c}}{3q})^{4} = (\frac{4{k}_{c}}{q})^\frac{-4}{3} $$ ...


2

You could, for example take the function $f: \mathbb{R}^2_+ \times[0,1] \to \mathbb{R}$. $$ f(L,K, \rho) = L + K + (1-\rho) L K. $$ For $\rho = 1$, we have $f(L, K, 0) = L + K$ which is a production function with perfect substitutes. The marginal product (of say $L$) is given by: $$ \frac{\partial f(L, K, \rho)}{\partial x} = 1 + (1-\rho)K, $$ which equals $...


2

Well, I figured out that the error meant something about not fitting in the dataset. My dataset had 78221 obs. Everything was like data(chilean) the only difference had to be the id for each firm. So I did this: list <- data.frame(table(base$ruc)) # so here i got the string id for every firm list$idvar <- seq(1:nrow(list)) # here i put a number for ...


2

(1) why is imposing CES so important in our models? Because although its relatively quite general (relatively to some other widely used production functions like Cobb-Douglas - which is a special case of CES) it is still easy to estimate with parametric models and generally CES production functions are easy to work with (McFadden 1963). Until very recently ...


2

Touch wood that I did not make any mistakes. Consider the production function $X = (K + \alpha)(L + \beta)$. The elasticity of substitution is given by: $$ \frac{\partial \ln(K/L)}{\partial\ln(MP_L/MP_K)} = \frac{\partial(\ln(K) - \ln(L)}{\partial(\ln(K + \alpha)- \ln(L + \beta))} $$ Let's take the derivative of both numerator and denominator with respect to ...


2

Contrarily to utility functions, production functions are cardinal, and so they are not arbitrarily normalized. In the case of your CES production function, however, if we consider $$ F(x; A, a, \rho, \nu) = A \left( \sum_{i=1}^n a_i \, x_i^\rho \right)^{\nu/\rho}, $$ with $\sum_{i=1}^n a_i \neq 1$ it is easy via a simple reparameterization of $(A,a_i)$ to ...


2

Yes: $$ f(x_1,x_2)=-x_1x_2 $$ has constant marginal return in both inputs and decreasing return to scale by your definition. EDIT: Answer above assumes OP meant "constant" as constant w.r.t. the input whose marginal product we are talking about. For totally constant, see the the answer to the duplicate question.


2

As your cost function exhibits non constant first and second order derivative, a third degree polynomial is a good starting point. There are infinitely many production functions behind a cubic cost function. Below there is just one example, using a trick from Gorman, which consists to interpret a cost function whose expression is the sum of three (or four) ...


2

It can, production functions do not need to be linear in technology. Production functions of the form: $$F=AK^{\alpha}L^{1-\alpha}$$ are used because they are simple to work with, have some nice properties, and as the authors after which this function has its name (this is so called Cobb-Douglas production function), Cobb & Douglas (1928) this function ...


1

Turns out it was a typo, when deducing equation $2.15$ by substituting $2.3$ into $2.2$ the result should be $\frac{p_1}{p_2} < \frac{T-\tau_2B}{Ta_2}$ Credit to @dopplerdog from reddit for finding this out https://www.reddit.com/r/communism/comments/pnc2q6/economics_help_pls_question_on_a_passage_from/hcop5zx?utm_source=share&utm_medium=web2x&...


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