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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Social Planner problem: two period

The production function is $F(K_t,N_t)=AK_t^\alpha N_t^{1-\alpha}$ and depreciation $(\delta)$ is equal to 1. The given preferences are as follows: $$U(c_1,l_1,c_2,l_2)=\gamma log(c_1)+(1-\gamma)log(...
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References on Cobb-Douglas Function

I am wondering why everybody uses Cobb-Douglas production functions? Nowadays they are so standard that they are just written down without further discussion. Does anybody know more about this and can ...
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Example of a (not quasi-linear) production function whose inputs are not perfect substitutes but are not asymptotic at the axes

I'm looking for an example of a family of production functions indexed by, say, rho, where the inputs become closer and closer to perfect substitutes as rho approaches 1, and yet, the marginal product ...
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Quasi-Linear Production and Inefficiency

For quasi-linear production of $F(K,L)= \sqrt{K}+L $. I know for the optimal efficiency to exist $q \geq \frac{w}{2r}$ where w is the cost labour and r is the cost of capital. My question is what ...
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How does aggregate production function shift by an increase in average and marginal productivity of labour?

Increasing productivity would mean increase in output for each given amount of labour employed, resulting in an upward shift of production function as shown below. However I'm unable to understand the ...
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How can I add a multidimensional panel to estimate TFP (year, id, region) in R?

I'm not sure about adding fixed effects for the variables year,id,region (using the estprod library) as in this paper using LP stimator. In the STATA forum ( #18 #...
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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:...
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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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29 views

How do I work with the growth accounting framework in a regression?

I am currently writing my master's thesis with the topic "The impact of digitalization on economic growth in China and Germany". My idea is to take a technology such as Industry 4.0, IoT or ...
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42 views

How to find the cost function for perfect complements [closed]

Imagine I got a production function like it : $$ \min\{x_1, x_2\} $$ How can I find the cost function?
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1answer
50 views

Unbalanced panel data in prodest package in R

I have a question regarding the usage of unbalanced panel data for TFP estimation by using the prodest package. The dataset could be found here: https://drive.google.com/file/d/...
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1answer
64 views

Production function and elasticity

Let $y=x_1^\alpha x_2^\beta$ where $\beta=1-\alpha$ be a Cobb-Douglas production function. Find the elasticity of the optimal demand functions (for minimizing production cost) for both goods wrt. $w_2/...
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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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Inverting the production function - Okun's Law: Fit at 50? (BALL; LEIGH; LOUNGANI, 2012)

Consider the following equations: $$ \tag{1} E_t - E_t^* = \gamma \cdot (Y_t - Y_t^*) + \eta_t, \gamma > 0, $$ $$ \tag{2} U_t - U_t^* = \delta \cdot (E_t - E_t^*) + \mu_t, \delta < 0 $$ where $...
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Marginal cost given (Cobb-Douglas) production

My function is $y=x_1^\alpha x_2^\beta$ with $\beta={1-\alpha}$. I found: the minimization problem for demand to be $x_1^{*}(w_1,w_2,y)=\left ( \frac{w_2}{w_1}\frac{\alpha}{\beta} \right )^{\frac{\...
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How to explain the flattening of the SRAC curve?

I discovered that there is a way that Short-run average cost curve could become 'flatter' instead of shifting. Yet I cannot find an explanation of why and how it can become flatter. For example, in ...
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1answer
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Finding the conditional input demand function

Find the conditional input demand function and cost function for the given production function $$f(a,b,c,d)=\min\{ a,2b\} + \max\{3c,4d\} $$ In The solution, The production function is defined as $f(x,...
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1answer
56 views

A question about Nash Equilibrium

I have some trouble with Nash Equilibrium. The specific question as follows. Suppose that there are $2N$ people in the village, of which $N$ residents live in the first district, and each person ...
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1answer
92 views

If production function is concave, then demonstrate that profit function will also be concave

Show that concavity of firm's production function implies concavity of its profit function. (Hint: For a concave function, first order conditions gives the vector that maximizes the function) ...
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1answer
68 views

Elasticity Cobb-Douglas productionfunction

I am given the production function $y=x_1^\alpha x_2^{1-\alpha}$, where $0< \alpha <1$ I found the demand functions for minimum production cost to be $ x_1^{*}(w_1,w_2,y)=\left ( \frac{w_2}{w_1}\...
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Calculate supply function based on production or cost function

Q1: A company has the following production function: $$f(x_1,x_2) = 2x_1 + x_2$$. The factor prices are $w_1=4$ and $w_2=3$. Calculate the company's supply function. Q2: A company's cost function is $$...
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Economic interpretation of Returns to Scale

There are three possible types of returns to scale: increasing returns to scale, constant returns to scale, and diminishing (or decreasing) returns to scale. If output increases by the same ...
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Estimating $TFP$ using Cobb-Douglas production function

Suppose we want to estimate total factor productivity (TFP) under time series framework. Let assume that the production function is given in the Cobb-Douglas form, i.e. $$Y_t=A_tK_t^\alpha L_t^\beta,$$...
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Prodest package in R for TFP estimation, gives different results

I have the following issue: each time I run the estimation of the TFP by using prodest package in R 4.0.3, I obtain different coefficients before the variables as well as omega variable is different: ...
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Why is my elasticity of substitution wrong?

I am calculating elasticity of substitution for the following production function: $$F(K,L) = A(aK^{-\gamma}+bL^{-\gamma})^{-\mu/\gamma}$$ where $A, a, b, \mu, \gamma$ are constants. $A, a, b, > 0$,...
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194 views

Increasing returns, implications?

If a firm has increasing returns to scale (i.e., doubling inputs more than doubles output) would that firm logically end up being the sole firm in its sector in the long run? If not, what is the ...
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Does labor negatively affect output (from empirical perspective)?

I am estimating Cobb-Douglas production function for US using time series framework: $$Y_t=A_t \times L_t^\alpha K_t^\beta $$ where $Y$ is output, $A$ is $TFP$, $L$ is labor and $K$ is capital. After ...
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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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1answer
66 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
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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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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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1answer
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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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1answer
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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
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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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1answer
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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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1answer
28 views

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
99 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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124 views

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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