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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production functions and theory of firm [closed]

Consider a consumer whose preference can be described by the following utility function: ​ a) Derive the optimal demand for x1 and x2 as a function of prices and income. b) Assume the price of good 1 ...
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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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In a production function, the technology can vary across the firms and times. But the “power” must be same for the same industry?

Let $F_t=A_tK_t^\alpha L_t^{1-\alpha}$ be the production function with two parameters. In regression, we know the firm level $F_t$, $K_t$, and $L_t$. We want to estimate $A_t$ and $\alpha$. I've ...
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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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Too much of a good thing - examples of production (or utility) functions that are initially increase in a factor and then decrease

I'm attempting to build a spatial model that estimates labour distributions after climate change impacts agricultural production. The key point to note is that some rainfall is good, and some ...
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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^{...
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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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How to derive cubic cost function from a problem of constrained optimization?

The cubic total cost function usually take the form $TC(q)=a+bq+cq^{2}+dq^{3} \qquad a,b,d>0, c<0$ and $c^{2}<4bd$ I know that from a constraint maximization problem $min\quad wL+vK$ ...
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Production for multiplant firm?

A multi-plant firm will never produce at a plant with an increasing marginal cost when they own another plant with a decreasing marginal cost. Is this true? My reasoning behind this is that firms ...
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When will the reserves be exhausted?

A rough estimate of the oil and gas reserves in some country at the beginning of 2010 was 15 billion tons. production that year was approximately 250 million tons. when will the reserves be exhausted ...
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How to prove that profit function is convex in price (with smaller price)?

According to this site, if output price increases from $p*$ to $p'$ and factor prices remain constant, then a new production bundle chosen must yield at least the same amount of profits as the old ...
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How to prove that a concave production imply that the input requirement sets are convex?

According to page 7 of this slide, "A convex production set Y implies that the associated input requirement set V(y) is convex". How can one go about proving it?
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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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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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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 ...
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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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Total cost function and (dis)economies of scale

I'm really confused about how I'd tackle this. I have an equation, $TC(Q) = 100Q + 20Q^2 + 3Q^3$. And I'm trying to find where the economies of scale, diseconomies of scale and constant return 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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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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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 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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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 (...
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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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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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Quick Question on Production Possibilities Frontier Curve

Can someone please tell me how or why the curve shifts outward. In the textbook, I was given that: "But if we cut production of mobile phones to 3 million this year, we can produce 2 mobile phone ...
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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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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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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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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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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 ...
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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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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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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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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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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'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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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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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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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 ...