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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Arellano and Bond (1991) or Blundell and Bond (1998) in R estimation

I have the following problem with the Arellano and Bond (1991) or Blundell and Bond (1998) estimators in R using the plm package. I receive the following problem when trying to run the needed ...
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43 views

Empirically estimating TFP

Suppose we assume that production function has a Cobb-Douglass form: $$Y=A\times K^\alpha\times L^\beta,$$ where $Y$ is output (GDP), $A$ is Total Factor Productivity and $L$ is labor. By log-...
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Are prices part of total factor productivity?

I am trying to understand how production is related to income/profit and where do prices enter. Suppose there is a single firm with a Cobb-Douglas production technology: $$Y=AK^{\alpha}L^{\beta}$$ ...
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prodest package in R, TFP growth estimation

I have a question regarding the results and its transformations received from prodest package estimation. So, the dataset is available here https://drive.google.com/file/d/1aedWYABus1fQjKWxkOmYOmxv-...
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1answer
33 views

TFP in R via estprod package

I want to calculate the TFP by using the estprod function (I use R 4.0.2). As far as I understood the only way to calculate the TFP is manual following this logic . ...
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Which diagnostic tests are suitable for Measuring Total Factor Productivity (TFP)?

I did the TFP estimation with Levinsohn Petrin (in R) but I have doubts about the diagnostic tests e.g. I know that the estimation of autocorrelation in residuals has to be done (it would be very ...
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69 views

How was the Cobb Douglas function derived?

In economics and econometrics, the Cobb–Douglas production function is a particular functional form of the production function, widely used to represent the technological relationship between the ...
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TFP estimation in R by using prodest and estprod packages

I am writing my Bachelor Thesis and I really need help with the TFP estimation. so far I have a dataset with log values of Value added (va), Labour (l), Capital (k), and Materials (m). The initial ...
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Returns to Scale Microeconomics

Are there any production function $f(x_1,\ldots,x_n)$ that is having decreasing returns to scale, given that the marginal product in every input $i$ in the function $f$ is constant?
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CES production function with non constant returns to scale

In the equation \begin{equation} Y=\left[ aK^{\frac{\sigma -1}{\sigma }}+\left( 1-a\right) L^{\frac{\sigma -1% }{\sigma }}\right] ^{\frac{\mu \sigma }{\sigma -1}} \label{ces_pf} \end{equation} if $\...
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106 views

Total factor productivity (TFP) estimation in R

I'm trying to calculate TFP with the estprod package in R because this package allows me to calculate with Gross-Output (like with levinsohn petrin) but I have ...
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Explaining negative “cyclic” unemployment with a production function

First of all, here is a related post. Unfortunately, it does not answer my question. I have heard that there is an understanding of negative unemployment in economic theory. Obviously, absolute ...
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how to calculate Leontief demand functions from first order conditions of a CES function when sigma tends to 0?

This question is NOT about how to approximate a CES function to a leontief function. Knowing that: $i= good (\begin{array}{*{20}{c}} {1}&{or}&{2} \end{array})$ $j= firm (\begin{array}{*{20}{c}}...
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Capital in terms of labor

I have a question that asks to find $\frac{\partial K}{\partial L} $ from $Q=cL^aK^b$, when $Q$ and $c$ are constants. It lists 4 answer choices but I’m just not sure how to approach it. Implicit ...
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1answer
64 views

Interpretation of $x c '(x)$

Consider a cost function that is continuous, differentiable and (possibly) convex: $c(x):\mathbb{R}^+\to \mathbb{R}$. I was wondering if there is a "common" way to interpret the expression: $...
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The shift of Marginal Cost Curve

In standard microeconomics textbooks they usually assume that the cost curve consists of just two variables which are Capital and Labor ( I'm talking about this equation: TC = rK + wL) So when we ...
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Parameter value for a CES production function

Consider a firm with the following CES production function, which utilizes only two production factors (capital and labor) whose prices are, respectively, $r > 0$ and $w > 0$: $$ y = \gamma \...
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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≥...
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Revenues and cost functions

Let's assume that there is a firm that produces a single good, $q=f(x)$, where $x$ is a single input. The firm can sell it on the market at a price $p$. It's production cost is given by a cost ...
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Percentage of workforce change, with two PPFs

A hypothetical economy produces two goods, X and Y. The performance (yield) for every worker is steady, and every worker for Y can produce 10 units of product. If $L_x+L_Y=200 $ (meaning that the ...
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1answer
94 views

Derivation of the elasticity of substitution of a general production function with labor-augmenting technological progress

I am following and trying to fully understand a famous and interesting work of Bentolila and Saint-paul (2003). They try to explain movements of the factor's share in terms of a relationship between ...
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Negative elasticity of substitution in a CES production function

I have empirically estimated the elasticity of substitution parameter in the following model: $$Y_t=[(A_1L_tK_{t})^{\rho} +(A_2M_{t})^{\rho}]^\frac{1}{\rho} $$ here, $Y_t$ is output, $A_i$ is a ...
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When do we have diminishing marginal returns to labor?

When we have a production function exhibiting constant returns to scale, with only labor and technology, why don't we have any diminishing marginal returns to labor?
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110 views

Production Possibility Frontier for 3 goods

Is it possible to construct a Production Possibility Frontier (PPF) for 3 goods? Would this require a 3D graph and a part of a sphere, in one quadrant (as opposed to the usual 2D plot, with a part of ...
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CES aggregator for intermediates with heterogeneous productivity

I'm following Atkeson, Burstein (2019 JPE), and cannot understand the aggregation result. There is measure $M(z)$ of firms with productivity $z$, with production function $$ y(z) = z k(z)^\alpha l(z)^...
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Optimization problem of a Cobb-Douglas function with 3 inputs

A perfectly competitive firm uses 3 inputs to manufacture a certain product according to the following Cobb-Douglas production function: $$ Q = A L_1^{\alpha_1} L_2^{\alpha_2} L_3^{\alpha_3} $$ ...
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comparison of micro production functions

There are many different production function estimation methods, relevant for micro and firm data. For example Olley-Pakes, Levinsohn-Petrin, Ackerberg et al., Wooldridge etc. But does anyone know of ...
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133 views

Deriving long-run cost functions from production function

Suppose that I have a production function $(aK + bL)^3$ in a perfect competition where a and b are constants. I am confused on how to obtain the long-run cost function from this production function ...
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Combining functions that satisfy the Inada conditions

Suppose $f: \mathbb{R} \to \mathbb{R}$ and $h: \mathbb{R}^n \to \mathbb{R}$ are functions that satisfy the Inada conditions, and also $$ \forall i: \lim_{x_i \to \infty} h(\mathbf{x}) = \infty. $$ ...
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Kuhn-Tucker conditions in linear cost minimization

Suppose we have the production function $f: \mathbb{R}^{2} \to \mathbb{R}$ given by $$ f(x,y) = ax + by $$ and input prices $p_{1}$ and $p_{2}$, and we want to minimize the cost function $p_{1}x_{1} ...
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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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In a production function, the technology can vary across the firms and times. But the “power” must be same for the same industry?

Let $F_t=A_tK_t^\alpha L_t^{1-\alpha}$ be the production function with two parameters. In regression, we know the firm level $F_t$, $K_t$, and $L_t$. We want to estimate $A_t$ and $\alpha$. I've ...
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Too much of a good thing - examples of production (or utility) functions that are initially increase in a factor and then decrease

I'm attempting to build a spatial model that estimates labour distributions after climate change impacts agricultural production. The key point to note is that some rainfall is good, and some ...
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Production for multiplant firm?

A multi-plant firm will never produce at a plant with an increasing marginal cost when they own another plant with a decreasing marginal cost. Is this true? My reasoning behind this is that firms ...
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When will the reserves be exhausted?

A rough estimate of the oil and gas reserves in some country at the beginning of 2010 was 15 billion tons. production that year was approximately 250 million tons. when will the reserves be exhausted ...
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256 views

How to prove that profit function is convex in price (with smaller price)?

According to this site, if output price increases from $p*$ to $p'$ and factor prices remain constant, then a new production bundle chosen must yield at least the same amount of profits as the old ...
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93 views

How to prove that a concave production imply that the input requirement sets are convex?

According to page 7 of this slide, "A convex production set Y implies that the associated input requirement set V(y) is convex". How can one go about proving it?
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Log differentiation of aggregate production function [closed]

Suppose we have an aggregate production function $Y = A F(K, L)$. I'm following some slides which then states that we if log differentiate we get the following: $\frac{\dot{Y}}{Y} = \frac{\dot{A}}{A} +...
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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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1answer
121 views

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 ...
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Production technology of $y=2x$

I have a given product $y$ that is produced by the input $x$ in the following relation: $2x=y$. In our example, we are given the unit price of $x$ is $16$. Find the unit cost of $y$. The answer is $8$....
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Why is the price of capital ‘r’ ? (From Cost function)

according to the Cost formula in microeconomics class, Total Fixed Cost is represented as “rK” (K as in unchanging, fixed K) now my economics teacher tells me this ‘r’ is the interest rate at which ...
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Can a CES parameter be negative?

I estimated the CES function (https://en.wikipedia.org/wiki/Constant_elasticity_of_substitution) using national accounts data for France. I got (using notation in wikipedia): a = 1.19 r = -0.48 so ...
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Continuity of Prices for Constant Elasticity of Substitution Production Functions

Imagine I have a CES production function $$ Y_{\sigma} = Z [ \sum_{i=1}^N \alpha_{i} X_i^{\frac{\sigma}{1-\sigma}}]^{\frac{1-\sigma}{\sigma}}$$ I know that as $\sigma \to 1$, the corresponding ...
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Production function involving profit maximisation

​Hi, I don't get how the answer of d is deduced in this question because I don't think I made any mistakes in my calculation and have also used all the information given. After knowing L is 800, I ...
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119 views

MRTS question involving production function [closed]

My work out shows constant MRTS and also increasing returns to scale. I thought the answer was C as I only found increasing marginal products of labour and capital. I really don't see how the answer ...
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421 views

CES production function profit and supply function

I need to derive the profit function for the following CES function: $$ f(z) = (\sqrt{z_{1}^{\rho} + z_{2}^{\rho}})^{1/ \rho}$$ where $\rho \leq 1$. This is the answer that I am supposed to be getting:...
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191 views

Decreasing and increasing returns to scale question

Hi, I have deduced that this function exhibit increasing returns to scale but I am not sure how to verify part d. My answer doesn't show that there is decreasing returns to scale but I can't be sure d ...
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1answer
263 views

Keynesian-cross analysis [closed]

I have a question from my textbook which is: Using the Keynesian-cross analysis, assume that the consumption function is given by C = 100 + 0.6(Y – T). If planned investment is 100 and T is 100, then ...
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Relationship between Elasticity of substitution of sectoral outputs and elasticity of substitution of inputs

There are two sectors Y1 and Y2. Composite output is given by CES form - Each sector employs Capital and Labor in combination through Cobb-Douglas Production Technology. The paper mentions that ...