# 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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### How can I obtain Leontief and Cobb-Douglas production function from CES function?

In most Microeconomics textbooks it is mentioned that the Constant Elasticity of Substitution (CES) production function, $$Q=\gamma[a K^{-\rho} +(1-a) L^{-\rho} ]^{-\frac{1}{\rho}}$$ (where the ...
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### Solow Model: Steady State v Balanced Growth Path

Okay, so I'm having real problems distinguishing between the Steady State concept and the balanced growth path in this model: $$Y = K^\beta (AL)^{1-\beta}$$ I have been asked to derive the steady ...
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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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### CES: Production function: Elasticity of substitution $\sigma = 1/(1 + \rho)$

I have to prove that $\sigma = 1/(1 + \rho)$ for the CES production function: \begin{align} q = (l^\rho + k^\rho)^\frac{1}{\rho} \end{align} I found out that I need to solve the following equation: \...
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### CES Production Function with $\rho>1$

In using CES production functions of the form $f(x_1,x_2)=(x_1^\rho+x_2^\rho)^{1/\rho}$, we always assume that $\rho\leq1$. Why do we make that assumption? I understand that if $\rho>1$, the ...
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### How to derive firm's cost function from production function?

I recently learned how to solve the following type of problem using the method of Lagrangian multipliers: Given a consumer with utility function $u(x,y)$, wealth $w$, prices $p =(p_x,p_y)$, budget ...
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### How does inflation impact the welfare of the economy?

When the government causes inflation through printing money, the individuals who saved their money in the bank are poorer. Is there a way to determine how different inflation rates impact the welfare ...
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### List of production functions that satisfy the Inada conditions

It is known that in the class of CES production functions, only the Cobb-Douglas production function satisfies the Inada conditions. Which other functions satisfy the Inada conditions?
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### Concave production function implies convex cost function

Let's assume we have an increasing production function $f:\mathbb{R^+} \to \mathbb{R^+}$ Now, assume this production function is concave and that the price of input z is fixed (this is a single-input ...
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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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### CES production function estimation

Introduction There are different ways of estimating the parameters of a production function. For example, single-equation and system equation techniques are both possible. Another difference among ...
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### Is a production function bilinear?

I believe the following is the multiplicative property of bilinearity: $$Y=F(K,AL)$$ $$c_1 F(K,AL) = F(c_1 K, AL)$$ $$c_2 F(K,AL) = F(K, c_2 AL)$$ But when we have multiplied through the ...
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### Lessons From Successfully small island economies

What economic and development lessons/strategies can developing Caribbean countries learn from successfully small island nations like Singapore?
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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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### 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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### Why labour, capital, and output levels cannot be pinned down in perfect competition?

Consider a firm producing with the following technology: \begin{equation} Y = AL^{\alpha}K^{\beta} \end{equation} Assuming that factors are paid their marginal contribution to output, it can be ...
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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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### 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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### 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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### Can capital still be paid its marginal product in the absence of a homogeneous capital stock?

Thomas Piketty's best-selling book on inequality, "Capital in the Twenty-First Century" has attracted a lot of criticism on the right for its data analysis. Less well-known, however, is the criticism ...
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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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### What exactly is L in a Cobb-Douglas production function?

Cobb-Douglas is $Y = AK^{1-\alpha}L^{\alpha}$. What exactly is $L$ in Cobb-Douglas? Is $L$ the number of workers available in a single year? Working hours?
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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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### Financial investment in the composition of GDP

In the production function Y = C + I + G + NX Does foreign investment in domestic assets (i.e. foreign buying of domestic bonds) - and vice versa - come under the Net Exports variable? Which ...
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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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### Leontief function marginal product of labor/capital

Find marginal product of labor of a Leontief production function. For example, $$f(L, K) = min\{\frac{L}{a}, \frac{K}{b}\}$$ MY ATTEMPT Now as I understand it the marginal product of labor, $MP_L$ ...
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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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### Cobb-Douglas nested in CES model

Assume, using the following equation $Y=[((A_1L)^\alpha K^\beta)^\sigma+ (A_2X^\gamma)^\sigma]^{1/\sigma}$, we back out (obtain) the evolution of $A_1$ & $A_2$. Can we interpret $A_1$ as labour-...
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### Solow Model, Growth rate of K/L and Y/L in steady state

I have been given the following setup: $$Y=K^\theta (AL)^{1-\theta }$$ Where Y = Output, K = Capital, L = Labour and A = Productivity. $$\frac{\dot{L}}{L} = n$$ $$\frac{\dot{A}}{A} = g$$ The ...
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### Convenient S-shaped production function (i.e. with IRS and DRS) to derive a discontinuous demand for labor

Let say that a firm produces a commodity using only one input (i.e. Labor if we suppose to be in the very short run). Then we have a general production function of the following form $y=f(L)$, for \$L≥...
A standard formulation of the Cobb-Douglas production function (e.g. here) is: $$Y=AK^{\alpha}L^{\beta}$$ Have there been estimates, at aggregate level for any large economy, of the absolute value of ...