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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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Obtain F(L,K) given F(L*,K*), K* and L*

The question is, given the optimal production function to maximize profit F(L*,K*)=p/w , and the optimal demands of capital and labour K*=0, L*=(p^2)/(2w^2), retrieve the original production function ...
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Is “$ f(k,l) \: is \: decreasing\:return\:to\:scale \Leftrightarrow f_{ll}f_{kk}-f_{kl}^2>0$” always true?

For the productions $f(k,l) $ that are continuously differentiable, is the proposition that "$ f(k,l) \: is \: decreasing\:return\:to\:scale \Leftrightarrow f_{ll}f_{kk}-f_{kl}^2>0$" always true, ...
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How to prove if production set Y satisfies free-disposal and CR of scale, then Y is convex, when there are only 2 commodities

Free-disposal and CR of scale can not imply convex production set. But it is true for single-input and single-output model. Therefore, I am wondering how to prove if production set Y satisfies free-...
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Demand Elasticity, Factor Substitution: Independent?

Given $ Y=f(K,L;\sigma) $, the effect on labor from a change in the price of capital can be gauged through a substitution effect and a scale effect: \begin{align*} \frac{\partial L}{\partial r} & ...
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1answer
65 views

Wages, capital: Substitution and Output Effects

Consider a CES production function $Y=f(K,L)$ with elasticity of factor substitution $\sigma>0$. The substitution effect of higher real wages naturally implies a shift along the isoquant to more $...
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38 views

interest rate costs and capital accumulation

Traditionally, we view the real interest rate as being the “cost” of capital accumulation -- and wages being the cost of labor. Thus, an increase in $r$ moves the firm along a smooth production ...
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55 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 ...
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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 ...
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1answer
32 views

Quasiconcavity and homogeneity

How to prove that if $f$ is strictly quasi-concave and homogeneous of degree 1, then $f$ is concave? It was left as an exercise by Silberberg & Suen (2001), p.140. I simply could not elaborate ...
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2answers
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2 firms production decision for one agent

I'm trying to solve the optimal production $\{x,y\}$ for a risk neutral agent with weight $w$ in firm $X$ and weight $1-w$ in firm $Y$. Each firm has marginal cost $c^X$ and $c^Y$ respectively. The ...
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1answer
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General Equilibrium with Linear Production

I don't think I understand how optimization problems with a linear function work as of now. If you have a production economy with two agents, two goods and Cobb-Douglas utility representation, and you ...
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Price Derivation in a Production Function

Here is a composite goods production function: And here is the price ratio of Ys and Yg, derived from their marginal products: Then the author normalized the price of final goods Y to 1 and somehow ...
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1answer
34 views

Nested CES Production Function

If I have four input factors (a, b, c, b) and I want to construct a nested CES production function such that (a, b) are substitutes, (c, d) are substitutes and [(a, b), (c, d)] are complements, I.e. a,...
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Cobb Douglas Production Function Substitutes or Complements?

I see that there is a question similar to mine but it is does not specifically answered. Here is my question: I understand that, in a Production Function, when elasticity of substitution is greater ...
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1answer
41 views

intuitive interpretation of the marginal return/productivity of capital being less than one

Let's assume you have a production function, f, and you want to know how the output changes with respect to capital, everything else constant (ceteris paribus), so you want to know the marginal ...
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1answer
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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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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 ...
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Question on returns of scale in isoquants

What does the dotted line in the diagram show and what are they called? I am also wondering why the steepness of the isoquants changes at Q=3 for the third diagram. Thanks
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Production function question

Some help with this would be appreciated.
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2answers
354 views

Perfect complement graph and isoquant

$f(x_1,x_2) = min \{x_1,x_2\} + x_2$ if that was the production, what would the isoquant be? Would it simply follow $x_1 = x_2$? I'm not entirely sure what it the graph would look like.
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Partial production function question

Hi everyone, I am learning about the partial production function. I don't understand why a tangent it drawn next to the curve; I get that it shows the MPL. Does this mean diminishing marginal product ...
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1answer
28 views

Estimating production functions with time series data

A while back I asked How do we estimate production functions? The answers given address cases when dealing with cross-sectional data, However most of the data I've been seeing is given by a time ...
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1answer
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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 ...
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Is it constant returns to scale if the output of a production function is purely a function of one variable?

For $Y=F(K,L)= 2L$ If I multiply them by an constant $z$: $Y= F(zK,zL0)$, i'll get $2(zL) = z(2L)$. Inputs increase proportionally therefore constant returns to scale. This doesnt seem right ...
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1answer
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How does hedging using futures work?

I can't understand how it makes sense. From what I've read, corn producers (for example) sell their corn at current price but deliver the corn later, in order to protect themselves from decline in ...
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Estimating Endogenous vs Exogenous growth models

Exogenous economic growth models (mostly those that take technological progress, or change in TFP as exogenous and random) are rather easy to estimate econometrically. But how about endogenous ...
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Linearization of VES production funtions

I know that the linearization of a CES (constant elasticity of substitution) funtion is a bit complicated. There is even an R package dedicated just for that - the econometric estimation and ...
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1answer
51 views

Choosing Data for CES Production Function

So I am currently trying to write a paper estimating the Constant Elasticity of Substitution Production Function of the USA. I am using the simple version with two inputs capital and labour. Since the ...
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How change in technology affects the price of labour and capital?

If the technology improves how does the price of labour and capital change. We take the neo-classical assumption that demand for each factor depends on it's marginal productivity, and demand and ...
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1answer
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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. ...
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1answer
396 views

How was the CES production function derived?

The Constant Elasticity of Substitution production function is defined as: (Taken from Wikipedia) $$Q=F \boldsymbol{\cdot}\left(a\boldsymbol{\cdot}K^r+(1-a)\boldsymbol{\cdot}L^r \right)^{1\over{r}}$$...
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1answer
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Aggregate production function, factor shares and cointegration

When estimating an aggregate production function you fit your data to a selected functional form of the production function, derive the parameters and inference from there. My question is, is there ...
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Anything written on the process behind “multiplicative” production function?

Intuitively, one might make a naive first guess that the production function of the economy, or of a firm, should be a "leontief" production function: "for example, you need both a factory and a ...
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How to determine sensible values for economies of scale?

I'm analysing a contest-game in which players exert effort to increase their chance of winning. Each player has a has a cost of marginally increasing their probability of winning the contest given by $...
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1answer
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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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1answer
53 views

Consider the following production function $Q=min \left(\frac{L}{2a}, \frac{K}{4b}\right)$. Let $w$ and $r$ be the wage and rental rate respectively [closed]

The cost function associated with this production function is $A) 2awQ\\ B) 4brQ\\ C) (wa + 2br)Q\\ D) None\; of\; the\; above$ What I have tried is: We have the cost function $wL+rK=C$. Since, here, ...
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modelling Inconsistent production functions

last year I asked How do we estimate production functions?. That answer provided was insightful from an econometric perspective and has helped me in applying such an understanding to the workplace. ...
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How to find the “cost function” given the production function *as well as* the cost per unit produced and the fixed costs?

I'm working on the following homework problem, transcribed verbatim: A firm has a production function defined as $y = 8L^{1/4}K^{3/4}$. The firm faces costs of \$20 wage, \$60 rental rate of ...
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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,...
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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. ...
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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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1answer
78 views

Interpretation: Elasticitity of Substitution [closed]

I have this production function: $$P(x_1,x_2)=x_1+x_1*x_2$$ I am trying to find the elasticity of substitution, and I found this: $$\sigma = -\frac{d \ln (\frac{x_2}{x_1})}{d \ln(\frac{x_1}{1+x_2})}...
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1answer
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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 ...
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1answer
156 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 ($I_{b,t}$),...
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2answers
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What is the use of knowing elasticity of average product?

Firstly they have given output elasticity of a factor, use of which is clear to me. It says to what degree my total output will vary due to a change in the quantity of a factor. But what else do we ...
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2answers
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Proof that for increasing AC MC(Q)>AC(Q) at any Q>0? [closed]

I know the "common sense" proof, but how can we prove it algebraically?
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Example production function with increasing returns to scale but diminishing marginal product [duplicate]

I know that diminishing marginal returns even to all factors of production doesn't imply decreasing returns to scale. But could you please give me just an example of such production function?
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1answer
64 views

How to Graph Short-Run Average Cost?

I've learnt to roughly draw graphs of various functions like isoquants of Cobb Douglas function, i.e., $k=√q/L$. Here first derivative is negative so it's downward sloping and second derivative is ...
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1answer
105 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 (...
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Sources of Growth and co-integration: production function approach

I am experimenting with time series data to gauge the importance of factors of production i.e. labour force, capital stock, energy, land, etc. in output growth. One venue I am looking into is the ...