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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Log differentiation of aggregate production function [closed]

Suppose we have an aggregate production function $Y = A F(K, L)$. I'm following some slides which then states that we if log differentiate we get the following: $\frac{\dot{Y}}{Y} = \frac{\dot{A}}{A} +...
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The Household Production Function

I was wondering whether someone can explain the household production function. Specifically, the variant presented in Patanayak et al 2005. The paper uses the HPF to determine household willingness to ...
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CES v. Leontief Aggregator in Production

Consider a production process with two distinct capital types such that there is a capital aggregator. You could view $k_v$ as a more versatile capital (e.g. can be converted into many different ...
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Production technology of $y=2x$

I have a given product $y$ that is produced by the input $x$ in the following relation: $2x=y$. In our example, we are given the unit price of $x$ is $16$. Find the unit cost of $y$. The answer is $8$....
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Why is the price of capital ‘r’ ? (From Cost function)

according to the Cost formula in microeconomics class, Total Fixed Cost is represented as “rK” (K as in unchanging, fixed K) now my economics teacher tells me this ‘r’ is the interest rate at which ...
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Production technology and efficiency

I have been practicing some questions on production function and efficiency and I came across the following questions. For question 1 what I did is, the efficiency of firm A in q1 = production of ...
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Stochastic frontier analysis in a unit out put production function. Taking logs is causing issues?

I wish to perform stochastic frontier analysis to calculate inefficiency of firms, but for a unit output isoquant ( imp) now y'=1, k'=k/y and l'=l/k. Now, these values lie between 0 and 1 (including 0)...
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43 views

Can a CES parameter be negative?

I estimated the CES function (https://en.wikipedia.org/wiki/Constant_elasticity_of_substitution) using national accounts data for France. I got (using notation in wikipedia): a = 1.19 r = -0.48 so ...
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Continuity of Prices for Constant Elasticity of Substitution Production Functions

Imagine I have a CES production function $$ Y_{\sigma} = Z [ \sum_{i=1}^N \alpha_{i} X_i^{\frac{\sigma}{1-\sigma}}]^{\frac{1-\sigma}{\sigma}}$$ I know that as $\sigma \to 1$, the corresponding ...
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Production function involving profit maximisation

​Hi, I don't get how the answer of d is deduced in this question because I don't think I made any mistakes in my calculation and have also used all the information given. After knowing L is 800, I ...
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MRTS question involving production function [closed]

My work out shows constant MRTS and also increasing returns to scale. I thought the answer was C as I only found increasing marginal products of labour and capital. I really don't see how the answer ...
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1answer
80 views

CES production function profit and supply function

I need to derive the profit function for the following CES function: $$ f(z) = (\sqrt{z_{1}^{\rho} + z_{2}^{\rho}})^{1/ \rho}$$ where $\rho \leq 1$. This is the answer that I am supposed to be getting:...
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Decreasing and increasing returns to scale question

Hi, I have deduced that this function exhibit increasing returns to scale but I am not sure how to verify part d. My answer doesn't show that there is decreasing returns to scale but I can't be sure d ...
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42 views

Keynesian-cross analysis [closed]

I have a question from my textbook which is: Using the Keynesian-cross analysis, assume that the consumption function is given by C = 100 + 0.6(Y – T). If planned investment is 100 and T is 100, then ...
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Relationship between Elasticity of substitution of sectoral outputs and elasticity of substitution of inputs

There are two sectors Y1 and Y2. Composite output is given by CES form - Each sector employs Capital and Labor in combination through Cobb-Douglas Production Technology. The paper mentions that ...
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If the relative share of inputs is constant in a production process, does that mean that the inputs are used in fixed proportion?

One of the assumptions in Euler's adding up theorem is: Fourth, the relative shares of the factors are constant and independent of the level of the product. Does this mean that the factors are ...
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Can technical change have a negative growth rate?

Using the following nested CES function I have backed out augmentation indices $A_L$ and $A_E$. Where $A_L$ is a labour-augmenting technical change index and $A_E$ is an energy-augmenting technical ...
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What is the difference between imperfect substitutes and complements in a production function?

According to the following definition: ...
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When is the PPF convex to the origin?

Given a 2X2 model (2 goods, 2 inputs), if the factor intensities (capital/labour ratio) of the two goods along the Pareto set are unequal, then we get a concave PPF. Can we get a convex PPF in some ...
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Question about an economy with 3 components household, firm and government with functions given

I've spent considerable amount of time on this question, in vain. It is from one of the competitive exams for admission to a grad econ program. Help would be tremendously appreciated. Thanks. An ...
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Solve for the steady state with CRS Cobb-Douglas, problem with the system of equations

There is one agent with utility function given by: \begin{equation} U(c,l) = \frac{c^{1-\sigma}}{1-\sigma}-\frac{l^{1+\gamma}}{1+\gamma}\tag{1} \end{equation} With budget constraint: \begin{...
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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})...
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In a perfectly competitive industry, why is apple considered best example of commodity? [closed]

Out of these options, shampoo apples ice cream hotels Why is apple considered to be the best example of commodity in a perfectly competitive industry? And Why not others?
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What is the economic meaning of distribution parameter in a CES-production function?

This is the production function (two input factors: $x_1$ and $x_2$) $$q=A[δx_1^ρ+(1-δ)x_2^ρ]^{\frac{1}{ρ}} $$ If distribution factor $δ$ is set to increase, what are the economic impacts on these ...
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What's the relationship between Output elasticity and Returns to scale?

https://en.wikipedia.org/wiki/Output_elasticity "If the production function contains only one input, then the output elasticity is also an indicator of the degree of returns to scale. If the ...
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Economic interpretation of a CES production function

I am following a paper where a production function of this type is used. $$Y=\left [\beta K^{- \rho}+\alpha \eta \left (\frac{K}{L} \right )^{-c(1+\rho)}L^{- \rho} \right ]^{-1/\rho}$$ It is a ...
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Derive the cost function for a Homothetic production function

I'm having trouble understanding the steps in showing that a Homothetic production function's cost function must be expressible in the form $C(w, q) = a(w)b(q)$. Since the production function is ...
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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 ...
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General Equilibrium Involving Production

I need a little conceptual clarification. For a standard $N*K*M$ general equilibrium model, would an allocation, say, $y^k$ be Pareto Optimal if it does not solve $max(py^k)$? I understand that the ...
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Is the CES a special case of the translog production function?

I was discussing today with a classmate about the relationship between prodution functions, and we tried to prove that the CES is a special case of the translog production function, but we failed. We ...
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Diminishing MRTS range

So I was solving a question which says find the range in which MRTS is diminishing $f(k,l) = 600k^2l^2-k^3l^3$ is the production function I got the MRTS = $ -(1200kl^2-3k^2l^3) \over 1200k^2l-3k^3l^2$ ...
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Marginal Product and Average product

So I've seen in my textbooks and many places online that when marginal product is more than average product average product is increasing, when average product is falling average product is greater ...
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Whats the rationale for using taylor series in economics?

I've been reading about the translog production function and know its really just a log-log production function approximated using a first order maclauren series. Why not just leave the function as ...
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Homogenous production function

Could one define a production function which is homogenous as having constant elasticity of substitution. Just want clarification . Thanks
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Why does a homothetic function have constant ratio of marginal products along rays?

A homothetic ordering is defined as $x \succeq y \Rightarrow \lambda x \succeq \lambda y \qquad \forall \lambda >0$ where $x,y \in \mathbb{R}^n$ Then, any differentiable function representing ...
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Derive the cost function and supply function from production function

I didn't study economics, but am quite interested in the topic. I came to the question whether I could derive the supply curve / marginal cost function from the production function and I actually ...
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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, I have ...
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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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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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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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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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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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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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288 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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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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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 ...