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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Labour saving technological change VS Capital saving technological change
I have been learning about technological change in relation with production functions(cobb Douglas) and isoquants and I'm very confused as the explanations can be somewhat vague and not clear enough.
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Cambridge Capital Controversy: Why is Reswitching and Reverse Capital Deepening a Problem Exactly?
This question requires some background so Im going to be a bit more formal than usual for a question I post here.
Introduction
The Cambridge Capital Controversy is a hard thing to understand in the ...
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Return to scale of a production function, $q = L^\lambda + K^\gamma$, is determining it possible in that general form?
Given the production function $q = L^\lambda + K^\gamma$, how do we determine the return to scale for different value of $\lambda$ and $\gamma$?
I know we have to determine the homogeneous degree of ...
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How to determine if a production function in a functional form has diminishing marginal rate of technical substitution?
For production function $q(L,K) = L^\lambda + K^\gamma$ The MRST is defined as
$\frac{\lambda L^{\lambda-1}}{\gamma K^{\gamma-1}}$. Is it correct and sufficient to say that in order for MRTS to be ...
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Understanding Second-Order Approximations in Translog Production Functions
Consider a model with the following production technology:
\begin{equation}
Q_i=F(\Omega_i,K_i,S_i,N_i) = \Omega_i\Big(\nu N_i^\sigma+(1-\nu)(\tau K_i^\rho+(1-\tau)S_i^\rho)^{\frac\sigma\rho}\Big)^{\...
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Is increasing Average Product(AP) always implying increasing Marginal Product(MP) in microeconomics?
I'm studying microeconomics and came across a statement that I'm not sure is correct:
"If average product ($AP = F(X)/X$) always increases from $X=0$, then marginal product ($MP = F'(X)$) also ...
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Calculating Production Functions using Input Prices
I am trying to estimate production functions and want to correct the endogeneity problem from direct estimation.
I found from these notes that it is possible to estimate production functions using ...
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Costs and Increasing returns to scale
If a firm has increasing returns to scale, does that mean that "costs" will always decrease as production increases. If so, does that mean the firm will end up being a monopolist?
And what ...
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any contingent labor and capital demand functions shortcuts for cobb-douglas functions?
for cobb-douglas utility functions,
aI/px and (a-1)I/py gives the marshallian demand for x and y.
Are there any parallels for cobb-douglas production functions,
q = x^a * y^(1-a) ?
Thank you.
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Is CES production representing the average of inputs?
I know that the Constant elasticity of substitution production function is given by:
$$Q=\gamma\left[\delta L^{-\alpha}+\left(1-\delta\right)K^{-\alpha}\right]^{\frac{-1}{\alpha}}$$
where $\gamma$ is ...
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Origin of terms Harrod neutral, Solow neutral and Hicks neutral
I just looked for the same question on the exchange but I didn't find anything, I even tried to look for the answer on Google but it seems it's not present. Could someone give me the origin of these ...
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question about production optimization
the question is,
if Q = AK^a(HL)^b
and the parameters are:
(A =100) (K = 10000$) (H = 1) (L = 100 person) (a = 0.5) (b = 0.5)
P = 5 per unit, R = interest rate of 3 percent per year , W = 3 per ...
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Derivation from Solow-type neoclassical growth equation?
In Chapter 6 of the 12th edition of "Economic Development" by Michael P. Todaro & Stephen C. Smith, an equation is introduced to illustrate the consequences of rapid population growth, ...
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Introducing productive sector into an exchange economy where only one agent is endowed with input
I'm trying to find a competitive equilibrium for an economy with consumers and some outside productive sector.
Consider an economy with two consumption goods $x_1, x_2$ and two individuals $A,B$ .
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Toy model equilibrium calculation: How a small system's firm determines wage and price?
I am trying to model a small system, with limited population and a single firm. Personally I have no economic background but only taken a simple microeconomics course and I am quite interested in ...
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Could you give an example of production function such that involves sunk costs?
I am looking for an example of a single output-single input production function such that involves sunk costs. I have in my mind that a drug - firm that is motivated to make a new drug, the drug has ...
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From Cobb-Douglas Production Function to Profit Function
A firm's output is given by the Cobb-Douglas production function $$Y_t=X_tK_t^{\alpha_K} L_t^{\alpha_L}$$
where $\alpha_K\approx\frac{1}{3}$ is the capital share and $\alpha_L$ the labor share.
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Slope of isoquants
Consider a production function $f(L,K)=\sqrt{KL}$.
The |MRTS|=$K/L$, and $\frac{d|MRTS|}{dl}=\frac{-K}{L^2}$
However, if I use the expression given in Nicholson and Snyder (Microeconomic Theory, ...
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Derive the input requirement set from production set
This question relates to the book Varian Microeconomic Analysis 3rd edition exercise 1.1. Much like this question but my emphasize is different.
Q: True or False? If V(y) is a convex set, then the ...
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Solow model with three input factors
My problem is that I want to construct a Solow model with three input factors; labour, capital and energy. But when trying to divid the equation by labour to get the per capita variables, it doesn’t ...
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A problem with "Returns to Scale"
Suppose that $Y\subset R^3$ is a production set satisfying the free disposal condition: if $y\in Y$ and $y'\leq y$ then $y'\in Y.$ Suppose the technology of production uses good 1 and good 2 as inputs ...
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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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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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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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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$$
where $L$ is labor and wage $w = 20$ find supply function of a price-taking firm under perfect competition. Fixed costs equal $10$.
Following ...
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Are homothetic additively separable preferences always equivalent to CES?
Are homothetic additively separable preferences always a monotonic transformation of CES preferences?
In technical language, the question is the following:
Let $n>1$, and let $f:\mathbb{R}^n_{\ge 0}...
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Solving Lagrangian FOCs: a few difficulties
I have an optimization problem from microeconomics that yields me the following first-order conditions based on a Lagrangian:
$ p_1 = \lambda \qquad(1)$
$ p_2 - \lambda (x_2^2+x_3^2)^{-1/3}x_2=0 \...
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Total Factor Productivity and Leontief Production Function
Just a bit of a conceptual question regarding the Leontief production function and the concept of total factor productivity (TFP).
In particular, I wondered if these two concepts are at odds?
The ...
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CES production function: How to show that $\sigma < 1$ implies essentialness?
Consider the CES production function:
$$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 ...
