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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22
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2answers
28k views

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 ...
11
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2answers
11k views

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 ...
11
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4answers
2k views

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. ...
10
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3answers
1k views

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 ...
8
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1answer
12k views

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: \...
7
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2answers
35k views

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 ...
6
votes
3answers
2k views

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 ...
6
votes
3answers
126 views

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?
6
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2answers
6k views

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 ...
6
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2answers
950 views

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 ...
6
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2answers
1k views

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 ...
5
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2answers
66 views

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 ...
5
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2answers
307 views

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{\...
5
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2answers
326 views

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^{...
5
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1answer
250 views

Lessons From Successfully small island economies

What economic and development lessons/strategies can developing Caribbean countries learn from successfully small island nations like Singapore?
5
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1answer
85 views

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 ...
5
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1answer
534 views

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 ...
4
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3answers
773 views

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 ...
4
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2answers
5k views

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{...
4
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1answer
220 views

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 (...
4
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2answers
323 views

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 ...
4
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2answers
115 views

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 ($K$) ...
4
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3answers
944 views

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 ...
4
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1answer
140 views

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 ...
4
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1answer
175 views

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 ...
4
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1answer
220 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), ...
4
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2answers
2k views

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 (...
4
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1answer
125 views

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 ...
4
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1answer
54 views

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 ...
4
votes
1answer
158 views

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. ...
4
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1answer
766 views

Could someone please explain the proof of Hotelling's lemma?

According to https://en.wikipedia.org/wiki/Hotelling%27s_lemma, the maximum of the firm's profit at some output is given by the minimum of the difference between the profit and the revenue. However,...
4
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1answer
330 views

Complementarity in CES Production Function

I'm reading Fisher (1997, Journal of Monetary Economics). From the intermediate goods produced ($Y_t$), the final goods firm allocates into consumption ($C_t$), business capital investment ($I_{b,t}$),...
4
votes
1answer
235 views

Can capital still be paid its marginal product in the absence of a homogeneous capital stock?

Thomas Piketty's best-selling book on inequality, "Capital in the Twenty-First Century" has attracted a lot of criticism on the right for its data analysis. Less well-known, however, is the criticism ...
4
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0answers
15 views

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)^...
4
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0answers
50 views

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 $$ \forall i: \lim_{x_i \to \infty} h(\mathbf{x}) = \infty. $$ ...
4
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0answers
60 views

How to econometrically identify perfect complements in production?

The production $$f(x_i,...,x_n)=\min\{x_i,...,x_n\}$$ is pretty straight forward and usually with smaller size data sets and can usually be picked up on rather quickly in an intuitive sense. ...
4
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1answer
244 views

Elasticity of substitution in Jehle and Reny Advanced Micro (3rd ed) exercise 3.8

Letting $f_i(\mathbf{x})=\partial f(\mathbf{x})/\partial x_i$, ($\mathbf{x}$ is a vector, a commodity bundle, and $x_i$ is a scalar, commodity $i$ in the bundle) show that, $\sigma_{ij}(\mathbf{x})\...
3
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3answers
380 views

Cobb-Douglas production function, given $w$ get $r$ regardless of input levels. Why?

There is a market economy with technology given by: $$Y = K^\alpha L^{1-\alpha} \tag{1}$$ Firms behave competitively and input prices are: $$r = \alpha K^{\alpha-1}L^{1-\alpha} = \alpha(\frac{L}{K})...
3
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1answer
1k views

Euler's Theorem

Can anyone give me connection and intuition behind each of the following euler's equation- Euler's equation in production function represents that total factor payment equals degree of homogeneity ...
3
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1answer
291 views

What exactly is L in a Cobb-Douglas production function?

Cobb-Douglas is $Y = AK^{1-\alpha}L^{\alpha}$. What exactly is $L$ in Cobb-Douglas? Is $L$ the number of workers available in a single year? Working hours?
3
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1answer
2k views

How to show the production function is concave in K and L but not strictly so?

Suppose we have a production function with constant returns to scale. Let us denote it by $F(A,K,L)$ where $A$ is the technology, $K$ the capital and $L$ Labor. Further assume the first partial ...
3
votes
1answer
538 views

Financial investment in the composition of GDP

In the production function Y = C + I + G + NX Does foreign investment in domestic assets (i.e. foreign buying of domestic bonds) - and vice versa - come under the Net Exports variable? Which ...
3
votes
2answers
96 views

Is there an economic interpretation of the production transformation function?

The production set has a simple meaning: It is the set of all production vectors that are feasible to a firm. The production function also has a simple meaning: It gives the output quantity for a ...
3
votes
1answer
3k views

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, $MP_L$ ...
3
votes
1answer
3k views

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 ...
3
votes
1answer
633 views

Cobb-Douglas nested in CES model

Assume, using the following equation $Y=[((A_1L)^\alpha K^\beta)^\sigma+ (A_2X^\gamma)^\sigma]^{1/\sigma}$, we back out (obtain) the evolution of $A_1$ & $A_2$. Can we interpret $A_1$ as labour-...
3
votes
1answer
1k views

Solow Model, Growth rate of K/L and Y/L in steady state

I have been given the following setup: $$ Y=K^\theta (AL)^{1-\theta }$$ Where Y = Output, K = Capital, L = Labour and A = Productivity. $$ \frac{\dot{L}}{L} = n $$ $$ \frac{\dot{A}}{A} = g $$ The ...
3
votes
1answer
86 views

Impact of Natural Disaster on Production Function

Say a natural disaster strikes. People were killed but the losses were small relative to the size of the work force. However, many buildings and infrastructures have been severely damaged. I'm ...
3
votes
2answers
287 views

Absolute Value of Total Factor Productivity in an Aggregate Cobb-Douglas Production Function

A standard formulation of the Cobb-Douglas production function (e.g. here) is: $$Y=AK^{\alpha}L^{\beta}$$ Have there been estimates, at aggregate level for any large economy, of the absolute value of ...
3
votes
1answer
100 views

Homothetic Production Technologies

Can someone suggest a good resource on homothetic technologies and what properties they imply about cost function, profit function, input demands, output supply etc? Also is it possible to have ...