Questions tagged [production-function]

A function whose value is the produced quantity associated with a given vector of factor inputs. The production function represents the technology available to the firm.

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How can I obtain Leontief and Cobb-Douglas production function from CES function?

In most Microeconomics textbooks it is mentioned that the Constant Elasticity of Substitution (CES) production function, $$Q=\gamma[a K^{-\rho} +(1-a) L^{-\rho} ]^{-\frac{1}{\rho}}$$ (where the ...
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Solow Model: Steady State v Balanced Growth Path

Okay, so I'm having real problems distinguishing between the Steady State concept and the balanced growth path in this model: $$ Y = K^\beta (AL)^{1-\beta} $$ I have been asked to derive the steady ...
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Why is the Cobb-Douglas production function so popular?

As relatively novice quantitative analyst/ Cost analyst, Ive been asked to estimate the level of a given organizations productivity more than once, and then forecast for the next couple of periods. ...
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CES: Production function: Elasticity of substitution $\sigma = 1/(1 + \rho)$

I have to prove that $\sigma = 1/(1 + \rho)$ for the CES production function: \begin{align} q = (l^\rho + k^\rho)^\frac{1}{\rho} \end{align} I found out that I need to solve the following equation: \...
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CES Production Function with $\rho>1$

In using CES production functions of the form $f(x_1,x_2)=(x_1^\rho+x_2^\rho)^{1/\rho} $, we always assume that $\rho\leq1$. Why do we make that assumption? I understand that if $\rho>1$, the ...
Sher Afghan's user avatar
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Concave production function implies convex cost function

Let's assume we have an increasing production function $f:\mathbb{R^+} \to \mathbb{R^+}$ Now, assume this production function is concave and that the price of input z is fixed (this is a single-input ...
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How to derive firm's cost function from production function?

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 ...
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Deriving the translog production function

Ive been having difficulty deriving the translog production function defined as: $$\ln y=\alpha_0+\sum_{i=1}^n\alpha_i \ln x_i+\frac{1}{2}\sum_{i=1}^n\sum_{j=1}^n\ \beta_{ij}\ln x_i\ln x_j $$ I know ...
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How to get this Production Function in Growth Rates

I'm struggling to understand how Khan & Reinhart (1990) go from the next production function. $$y=A f(K,L,Z)$$ Where $y$ is the production of the economy, $A$ is a variable which contains the ...
John M. Riveros's user avatar
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CES production function estimation

Introduction There are different ways of estimating the parameters of a production function. For example, single-equation and system equation techniques are both possible. Another difference among ...
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Replicate Blundell and Bond (2000) results using R

I want to replicate Blundell and Bond (2000) Table III in R. I'm using the function pgmm from package plm, which (apparently) ...
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Is it true that $\frac{dL}{dq}=1/\frac{\partial q}{\partial L}$?

Marginal costs MC is defined as $MC=\frac{dC}{dq}$. Taking into account that $C=wL+rK$, $$MC=\frac{dC}{dq}=w\frac{dL}{dq}+r\frac{dK}{dq}$$ Recall that marginal product of labor $MP_{L}=\frac{\...
ji borrob's user avatar
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How does inflation impact the welfare of the economy?

When the government causes inflation through printing money, the individuals who saved their money in the bank are poorer. Is there a way to determine how different inflation rates impact the welfare ...
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List of production functions that satisfy the Inada conditions

It is known that in the class of CES production functions, only the Cobb-Douglas production function satisfies the Inada conditions. Which other functions satisfy the Inada conditions?
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Elasticity of Substitution of CRS Production Function

Suppose that $F(\cdot)$ has CRS in $K$ and $L$, the elasticity of Substitution is $\sigma_{K L} \equiv F_{L} F_{K} / F F_{L K}$. I once derived this equation but I remember that it takes me quite ...
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Doraszelski and Jaumandreu (2018) Intuition

Doraszelski and Jaumandreu (2018) estimate a CES production function with two forms of productivity shocks (1) labor augmenting and (2) Hicks neutral. They claim that the increase in labor augmenting ...
Michael Gmeiner's user avatar
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list of exotic production functions?

Standard production functions are Cobb-Douglas, CES, Leontief. The most exotic production function I have seen is the Ethier production function. I am wondering whether there is a book/list of ...
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Estimating elasticity of substitution in nested CES functions

I have aggregate data on $L_t, K_t$ and $X_t$, and want to estimate elasticity of substitution parameters, $\gamma$ and $\sigma$ for these factors. Assuming the production function takes the following ...
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Is a production function bilinear?

I believe the following is the multiplicative property of bilinearity: $$ Y=F(K,AL) $$ $$ c_1 F(K,AL) = F(c_1 K, AL) $$ $$ c_2 F(K,AL) = F(K, c_2 AL) $$ But when we have multiplied through the ...
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How do we estimate production functions?

In a standard economics education we learn about production functions, indicating an output as a function of a given input of capital and labour. An average model looks like this: (1) $F(L,K)=L^{...
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CobbDouglas: Constant marginal costs and constant returns to scale

A company has a production function: $$y=x_1^{\alpha}x_2^{1-\alpha}$$ where $0<\alpha<1$. Factor input 1 costs $w_1> 0$ and factor input 2 costs $w_2> 0$. The company wants to minimize its ...
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What is the returns to scale of the production function q = min {K, L^(1/2)}?

I learned that when there is decreasing returns to scale, the average cost is always increasing. But the professor told us today that the other way around might not always be true. So if average cost ...
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Why labour, capital, and output levels cannot be pinned down in perfect competition?

Consider a firm producing with the following technology: \begin{equation} Y = AL^{\alpha}K^{\beta} \end{equation} Assuming that factors are paid their marginal contribution to output, it can be ...
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Transformation Function

In Mas-Colell microeconomics textbook I have found that profit maximization problem (as well as many further optimization tasks) could be represented with application of some transformation function (...
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Lessons From Successfully small island economies

What economic and development lessons/strategies can developing Caribbean countries learn from successfully small island nations like Singapore?
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Error while estimating a production function with prodest package in R

I have an unbalanced panel with 15063 firms between 2012 and 2018. I am using this code to estimate the production function with Levinsohn & Petrin method: ...
Jorge Paredes's user avatar
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Cobb-Douglas Production Function - Finding units of labour to maximise production

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 ...
James Burton's user avatar
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Mathematical derivation of the Production Possibility Frontier

What are the mathematical basics of production possibility frontier? How can I derivate it? Can I have an example for it?
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CES v. Leontief Aggregator in Production

Consider a production process with two distinct capital types such that there is a capital aggregator. You could view $k_v$ as a more versatile capital (e.g. can be converted into many different ...
Frank Swanton's user avatar
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Is Artificial Intelligence a completely new (and underestimated) production factor?

Accenture research defines Artificial Intelligence as completely new production factor along labour and capital. Is such notion acceptable by economists? As far as I understand then every asset of AI ...
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Examples of how economists come up with a production function for a firm?

I am a young man studying economics on his own. In every microeconomics text I find, they teach you what a production function is, but they never show examples (or practice problems) on how one comes ...
Copyright's user avatar
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Determining cost from production function, wage, and rental rate

So I have a production function $Q=2K + 20L^{1/2}$ and I suppose the wage is $w=5$ and the rental rate is $r=9$. I want to find the long-run cost of production, which I know is constrained by $\frac{...
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Cobb Douglas Production: Identification issues for technical change

It's well know that under a Cobb Douglas production function, capital and labor augmenting technical progress cannot be individually identified. Accordingly, people usually assume Hicks or Harrod ...
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Walras Law in a production economy with fixed costs

Consider a price taking firm with fixed costs $fc \geq 0$: \begin{align*} \Pi &= \max_{n^D} \left\{ P_c F(n^D) - w\times n^D - fc \right\} \end{align*} A representative household owns this firm:...
Albert Zevelev's user avatar
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Notation of a Cobb-Douglas function printed in 1989

I am trying to understand a paper written back in 1989 about long run population growth. It seems like the PDF is a scanned image of the paper. The notation for the function is on page 11 of the pdf (...
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The Econometrics of the Stone Geary Production function

The Stone-Gerry production function generally takes a form of: $$f(x_{i})=\prod_{i=1}^n(x_{i}-a_{i})^{\gamma_i}$$ where $x_i$ represents all inputs in the model, $a_i$ is a constant representing the ...
EconJohn's user avatar
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How does a positive capital shock cause an increase in land price?

I am trying to determine the exact theoretical mechanism for a positive capital shock to create an increase in land price. Assuming output is a function of land ($L$), labour ($N$) and capital ($K$) ...
Kelly's user avatar
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Why are production functions linear in technology?

Economists often assume a production function of the form $ Y = A f(K, L) $, where $Y$ is output, $K$ is capital, $L$ is labour and $A$ is technology. This form of production function can describe ...
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Decision over "max" production function:

I've been presented with the following problem: $$y=3(x_3)^{\frac13}(\max\{x_1,8x_2\})^{\frac13}$$ And the objective is to both maximize profit and minimize cost. First of all, if the problems are ...
mudcake's user avatar
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Deriving long-run cost function

I'm a bit unsure about how to derive a long-run cost function. Suppose my production function was $X(L, K)=L^a K^b$, where $a+b>1$. I'm thinking about doing the following, but I'm not sure it's ...
pril's user avatar
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1 answer
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Leontief function marginal product of labor/capital

Find marginal product of labor of a Leontief production function. For example, $$f(L, K) = min\{\frac{L}{a}, \frac{K}{b}\}$$ MY ATTEMPT Now as I understand it the marginal product of labor, $MP_L$ ...
user278039's user avatar
4 votes
1 answer
266 views

Constant Returns in a Production Function $\frac{Y}{L}=\left(\frac{K}{L}\right)^{\alpha}\left(\frac{R}{L}\right)^{\beta}$ ($R$ = Resource)

In his 1977 article (from which has developed a considerable literature on the Hartwick Rule for maintaining long-term constant consumption given depletion of non-renewable natural resources), ...
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What kind of production function would give a cubic-shape cost function?

I would like a production function that gives a cost function with the following shape: The figure was taken from "Microeconomic Theory: Basic Principles and Extensions, 12th edition", on ...
user141240's user avatar
4 votes
1 answer
176 views

Non CES Production Functions

I know CES production functions dominate economics, but I was curious, why? I've never seen a research paper or presentation utilize any form of a production function that is not CES. My question is ...
Michael Gmeiner's user avatar
4 votes
2 answers
325 views

Convenient S-shaped production function (i.e. with IRS and DRS) to derive a discontinuous demand for labor

Let say that a firm produces a commodity using only one input (i.e. Labor if we suppose to be in the very short run). Then we have a general production function of the following form $y=f(L)$, for $L≥...
Alessandro's user avatar
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1 answer
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Interpretation of the Cross Partials of the Cobb-Douglas

Consider a Cobb-Douglas Prod. Function $$Y=AL^{a}K^{1-\alpha}$$ This has the cross-partial: $$\frac{\partial^2 Y}{\partial K\partial L}=(1-\alpha)\alpha AL^{\alpha-1}K^{-\alpha}$$ Is the ...
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Understanding the Zellener-Revankar Production Function

I took out a book from my university library called Econometric Modelling with Time Series: Specification Estimation and Testing in an attempt to understand the importance of MLE in Econometrics. ...
EconJohn's user avatar
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Does the growth rate of a neoclassical production function converge as all input factors grow with constant, but different growth rates?

Suppose you have a neoclassical production function with N-inputs $F(x_t^1,...,x_t^N)$ All input factors grow in continuous time with constant, but not identical growth rates $g^j$. Assume $g^1 \leq ...
Thorsten L.'s user avatar
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1 answer
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stone geary production and Multiple equilibria in a simple solow model: Do complex roots mean anything?

I have been looking at a simple solow model with stone geary production technology and law of motion of capital specified as follows: $$f(k_t)=(k_t-\bar{k})^{0.5}$$ $$k_{t+1}=(1-\delta)k_t+sf(k_t)$$ $$...
EconJohn's user avatar
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The Household Production Function

I was wondering whether someone can explain the household production function (HPF). Specifically, the variant presented in Patanayak et al (2005). The paper uses the HPF to determine household ...
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