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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Marginal and Average costs for constant returns to scale production function being constant
Suppose that we are dealing with a production function $q = f(k,l)$, of inputs capital and labor. If this function exhibits constant returns to scale then I know that both the marginal cost and ...
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The consideration of export in calculating national income
Is export/imports a part of factor income i.e returns from the act of production? If not, then here is my counter argument.
Export is the equivalent value which is added to the goods when it is ...
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What could a negative output elasticity of an input imply?
Output elasticity of an input means (consider the non-calculus formulation) the percent change in output for a percent change in input (it is customary to substitute “change” with “increase”).
Let’s ...
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Modelling the optimal mix of labour
I am trying to model the profit maximization decision of a firm that uses two types of labor, workers A and workers B. I started by drawing the marginal product and marginal cost curves (lines, for ...
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Where does the optimal point of production occur in the graph of short-run cost curves?
When MC, ATC, AVC, and AFC are in the same diagram, where does the optimal point of production occur? I know production is optimal when MR=MC but the question doesn't give you the price of the good. ...
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Help with Solow model
I need help with the following question, I would really appreciate any help. For the general case of any production function, the differential equation for k(superscript dot) looked as follows: k(...
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relation between elasticity of substituion and mrts
elasticity of substituion has been defined as the as the percentage change in capital labour ratio given change in mrts.
but mrts itself tries to explain how a change in labour results in a change in ...
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how is output elasticity different from marginal product of a factor input?
marginal product has been defined as the addition to total product given the employment of one more unit of a factor input.
output elasticity has been defined as the percentage change in output given ...
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Connection between roundabout production and decreasing returns to scale?
I've heard the statement that roundabout production as in Basu 1995 leads to decreasing returns to scale in production.
What is the connection between the two?
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Leontief function nested in a cobb-douglas function for a computable general equilibrium
I am currently trying to build a CGE model, and I'm stuck with the specification of the agriculture sector. I'm trying to understand how to do nested production functions and also how to solve them. I ...
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Why is the Hicksian form of the CES demand used in CGE model forms rather than the Marshallian
I am curious to know why the Hicksian form of the CES is used in CGE models rather than the Marshallian form. I have a few hypotheses, but I am not sure which one is correct. If any?
Hypothesis 1: In ...
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Stata command for Dynamic Panel Production Function estimation
Consider a production function to be estimated,
$$(*) y_{it} = \beta_0 +\beta_k k_{it} +\beta_l l_{it} + a_i +\omega_{it} +\varepsilon_{it}$$
where $\omega_{it}=\rho\omega_{i,t-1}+\xi_{it}$. The ...
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How do I derive optimal tax on pollution causing intermediate products?
I am reading "The Environment and Directed Technical Change" by Acemoglu et al. (2012).
I cannot understand how the optimal tax in \eqref{eqA10} is derived.
$$\tau_t = \frac{\omega_{t+1} \xi}...
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Technology Parameter In Converted Minimisation Problem
Question:
I want to understand what's going on with respect to the technology parameter $A$ when i convert this minimisation problem into a maximisation problem. The issue is only revealed when i use ...
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Global returns to scale
I have a production function of the form $f(x_1,x_2) = x_1^a x_2^b$ and I am trying to figure out what the global returns to scale would be given that $a,b \in (0,1)$.
This production function is ...
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How to derive the input demand functions from a perfect substitutes production function
I am struggling to derive the input demand functions from a production function with inputs that are perfect substitutes.
The production function is as follows:
$f(x_1,x_2) = (x_1+x_2)^\frac{1}{2}$
I ...
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Hessian Matrix Test - When does it fail?
When does the hessian matrix test fail. I understand we are testing the definiteness of the Matrix, and i also understand that because it's a symmetric $n•n$ matrix, we have a principal minor ...
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Arguments for Concavity or Quasi-concavity
I'm faced with questions that want me to show that a utility or production function is either concave, or if not then quasi-concave so that we can apply the KKT conditions.
For example the production ...
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Envelope Theorem and Factor Mix Intuition
Brief Summary of question:
I'm frustrated with the intuition of the envelope theorem that when input costs change, our envelope theorem tells us we do not need to re-optimise demand levels.
Context:
I ...
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CRS, Homothetic Functions, and constant MRTS
Questions
When our Isoquant map exhibits constant MRTS along a ray from the origin making. Why do we make specific reference to.
Constant returns to scale
Homothetic Functions
I'm asking because it ...
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Conceptualising the effect of changes to the substitution parameter in a CES production function
I'm trying to have a conceptual understanding of what happens to the CES production function when the substitution parameter $\rho = \frac{\epsilon - 1}{\epsilon}$ changes, where $\epsilon$ is the ...
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Constant returns and (weak/strict) concavity
Suppose I have a constant returns production function $Q = f(X,Y,Z)$, where $X$, $Y$, and $Z$ are the inputs. Because of constant returns, the Hessian matrix of second-order partial derivatives (f_ij) ...
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Mixed Partial Derivatives in Profit Function
$\pi(x,z) = p(a\ln(x) + b\ln(z)) - w_xx - w_zz$
Question 1:
Using the first order conditions, we get:
$x = \frac{pa}{w_x}$
$z = \frac{pb}{w_z}$
What do we call these Input demand functions as a ...
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Solow Model - speed of convergence
This is a question also for those with a good expertise in micro. For micro guys who wanna go streight to the question, just jump to equation $(1)$
I'm studying the Solow growth model.
Let's write the ...
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Homothetic Functions and Monotonic Transformations
Using the following definition of a homotheic function (taken from my Mathematical Economics course pack).
A function $f: \mathbb{R^{n+}} \to \mathbb{R}$ is homothetic if it has the form:
$f(x,y) = q(...
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Homotheic Function Definitions
There are a number of different definitions of Homothetic functions i have come across. I have used each of them to prove that a function $f(x, y) = x^a y^b$ with $a+b > 0$ is homothetic. But i ...
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Derive cost function from production function
proportions production function as follows:
where the price of input is 1 and z2 is supposed to be a fixed factor of production. I've been having trouble finding the cost function because if z2 isn't ...
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Optimal production plan in monopol
Suppose a monopolist faces the following demand curve: $𝑃(𝑄) = 120 − 3𝑄$, where $𝑄 = 𝑞_1 + 𝑞_2$. The monopolist has two factories. Factory 1 and factory 2 have the following marginal costs:
$$
𝑀...
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Solve long run production function of a firm using technical rate of substitution
I don't understand the solution to a question which deals with the long run production function of a firm.
The question is:
Suppose a firm has a production function $f(x_1, x_1) = x_1^{0.5}x_2^{0.5}$, ...
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Maximization of CD production function
I was reading the paper "Optimal Investment Under Uncertainty" (Abel, 1982). At one point the author addresses the following problem:
$$\max_{L_{t}}\left\{ p_{t}L_{t}^{\alpha}K_{t}^{1-\alpha}...
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Elasticity of substitution between capital and effective labour
While going through the derivation of elasticity of substitution between capital and effective labour in economic materials for a Slow growth model, I found the following step there:
$\frac{\partial ...
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Supply function of a price-taking firm with a quadratic production function
For a firm with the production function
$$Q = 40L-L^2$$
where $L$ is labor and wage $w = 20$ find supply function of a price-taking firm under perfect competition. Fixed costs equal $10$.
Following ...
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How to rationalize the following behaviors using preference relations?
The producer wishes to produce at least $y^*$ units. Once he has
achieved that goal, he maximizes profit.
The producer maximizes profit, but already employs $a^*_1$ workers
and will incur a cost c (...
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Are homothetic additively separable preferences always equivalent to CES?
Are homothetic additively separable preferences always a monotonic transformation of CES preferences?
In technical language, the question is the following:
Let $n>1$, and let $f:\mathbb{R}^n_{\ge 0}...
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Solving Lagrangian FOCs: a few difficulties
I have an optimization problem from microeconomics that yields me the following first-order conditions based on a Lagrangian:
$ p_1 = \lambda \qquad(1)$
$ p_2 - \lambda (x_2^2+x_3^2)^{-1/3}x_2=0 \...
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Total Factor Productivity and Leontief Production Function
Just a bit of a conceptual question regarding the Leontief production function and the concept of total factor productivity (TFP).
In particular, I wondered if these two concepts are at odds?
The ...
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CES production function: How to show that $\sigma < 1$ implies essentialness?
Consider the CES production function:
$$Y = f(K, L) = (a \cdot K^\rho + (1 - a) \cdot L^\rho )^{1/\rho}$$
The elasticity of substitution is $\sigma = 1/(1 - \rho)$. I remember that, if the elasticity ...
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Naive Question About PPFs
I am a maths major, and am taking an introduction to microeconomics course this semester, and am confused by how we deduce the shape of PPF's.
For example, I was given the following problem:
Larry, ...
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Production function with intermediate goods producer
I was reading the theoretical discussions of the following paper
Aghion, P., Boustan, L., Hoxby, C., & Vandenbussche, J. (2009). The causal impact of education on economic growth: evidence from US....
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Convexity of production sets and input requirement sets
The following question is from Microeconomic Analysis by Hal R Varian.
True or false? If V(y) is a convex set, then the associated production set Y must be convex.
The solution available says;
False. ...
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Explain the definition of a primal shifter versus an input shifter parameters in the standard CES function
I have run into a CES function that seems to be very closer to standard but with a small disaggregation of the share parameter into two parameters (primal share) and (input shift). I am hoping someone ...
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Effects of retirement and unemployment on PPF
Which of the following will not shift a country's production possibility frontier (PPF) ? An increase in the age at which people retire or a fall in unemployment ?
To me, it is the increase in the age ...
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How are returns to scale of a non homogeneous production function defined?
Most of the production functions encountered in Intermediate Microeconomics are homogeneous (Cobb-Douglas, perfect substitutes, perfect complements).
So their returns to scale are easy to get, ...
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Is elasticity of substitution defined for non-homogeneous production functions?
The elasticity of substitution between two inputs $x_1$ and $x_2$ is typically given as
$$\frac{d \ln \left( \frac{x_2}{x_1} \right)}{d \ln(\mathrm{MRTS}_{21})}.$$
As these notes show, if the ...
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Is there a standard term for the elasticity of an isoquant?
Isoquants - the level sets of a production function $f$ - are very useful in microeconomics. For example, if we hold all but two inputs fixed, then the isoquant is a plane curve that quantifies the ...
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Isn't producer surplus less important due to the Marginal Utility of money not being factored in?
The demand curve is a good model to find of utility because it takes marginal utility of the good into account. However, the supply curve is based on marginal cost of production and marginal revenue ...
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Total factor productivity (TFP) estimation in R via estprod package
good morning
I am trying to calculate the total factor productivity (TFP) for companies in the manufacturing industry through the Levinsohn-Petrin model. To do so, I use the prodest package in R. ...
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TFP and the residual with prodest() package in r
I am estimating a firm level production function in R and I am using the prodest() package, which is the equivalent of the prodest package in Stata, written by the ...
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Elasticity of substitution for 3 and more goods (interpretation)
Elasticity of substitution for 3 and more goods (interpretation)
Hello everyone,
I have a problem regarding the understanding of how would the elasticity of substitution work in the case of function ...
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How is production managed with respect to the long run vs the short run?
Assuming perfect competition, I think that firms are price takers in the labor/capital markets as well (in the short and long run), correct?
And I know that the Long-run total cost curve is derived by ...
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