Questions tagged [cobb-douglas]

The Cobb-Douglas function is a commonly used functional form for a firm's production function or for consumers' utility, with a variety of convenient properties.

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Cobb-Douglas utility function

Why in Cobb-Douglas utility function, the exponents have to sum to one? Can they not be equal to 1 and why?
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Confusion on deriving Walrasian price changes using IFT

I am trying to derive price changes $$\frac{\partial x_1^*(p_1,p_2,w)}{\partial p_1}$$ for Cobb Douglas utility functions $$u(x_1,x_2)=x_1^\alpha x_2^{(1-\alpha)}$$ and I must use the implicit ...
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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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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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Final demand for CES nested in Cobb-Douglas production function

I am deriving the final demand for intermediate good $x_j$ for the following final producer's problem: \begin{equation} \max_{{ k_{t}, x_{j,t}}} K^{1-\alpha}X^\alpha - R_tK_t - \int_{0}^{N_t}p_{j,...
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Equilibrium in Edgeworth box with Cobb-Douglas utility function

Given an Edgeworth box environment, with two individuals and two goods $(x,y)$ with maximum quantities $Q_x$ and $Q_y$. Suppose that the utility functions for each individual are $$ U_1(x,y) = x^\...
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How to get from a CES production function with inverse elasticity on weights to the special cases Cobb-Douglas & Leontief

I am dealing with a CES production function, and I have attempted some of the "traditional" ways to derive the Cobb-Douglas (logs & l'Hôpital) but I am not sure how to deal with the ...
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Calculating cost-minimizing inputs with 3 inputs in production function [closed]

How can I determine the cost-minimizing input bundle with a standard Cobb-Douglas production function with three inputs. despite its simple process, the algebra becomes very hard as you go through the ...
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Trying to understanding how an equation is arrived at in Sach's et al "Geography and Economic Development"

Within the paper's discussion of formal models, I'm struggling to understand how the authors arrive at a certain equation. First, a model of the growth rate of economy is presented, which I do ...
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Cobb Douglas Production: Identification issues for technical change

It's well know that under a Cobb Douglas production function, capital and labor augmenting technical progress cannot be individually identified. Accordingly, people usually assume Hicks or Harrod ...
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Cobb-Douglas Production Function - Finding units of labour to maximise production

Given production function $f(L,K)=16L^\frac{1}{4}K^\frac{3}{4}$, where each unit of labour costs £50 and each unit of capital costs £100 and you have a budget of £500,000. Find the number of units of ...
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how to solve such UMP where utility function is quasi-linear with cobb-douglas function as the non-linear part [closed]

U =$X_1+X_2^aX_3^{1-a}$ $a ∈[0,1]$ $s.t. p·x≤w , x≥0 $ I have tried FOC for x1 x2 x3 and λ, but I cannot get two pairs of equalities separately in order to express two unknowns as a function of ...
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What is α in a Cobb-Douglas utility function?

Sorry if this is not the place to ask, I'm new here. I'm studying economy but I'm struggling to understand the Cobb-Douglas utility function. If we've one such that xt is consumption in period t, and ...
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Does global maximum of CRS Cobb-Douglas profit exist

In most macroeconomic papers it is taken as given that the aggregate prodution function is $Y=AK^{\alpha}L^{1-\alpha}$, and that the optimality conditions for inputs determine input demands: $$ \max_{...
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Cobb Douglas relation with uncompensated law of demand

Does a Cobb Douglas or homothetic function satisfy the uncompensated law of demand?
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Is it true that for Cobb-Douglas preferences, the demand function is always iso-elastic?

As we know that $Q*P=const.$ for Cobb-Douglas preferences, we can thus conclude that $\frac{dQ/Q}{dP/P}$ is always $-1$: $$ QP=const. \implies 0=d(PQ)=Q\ dP+P\ dQ \implies \frac{dQ}{Q}=-\frac{dP}{P} $$...
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Is an income tax always more favourable for consumers compared to ad valorem/quantity tax?

I'm studying the optimal choice of consumers with regards to taxation. I read that for consumers, income tax is generally (for Cobb-Douglas preferences) preferred compared to ad valorem tax: If the ...
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Finding restrictions on parameters for a demand function

I have a question that asks: Let $x_1$ be the quantity of a good 1, $p_1$ the price of good 1, $p_2$ the price of good 2, and $M$ is income. Let $𝑥_1(𝑝_1, 𝑝_2, 𝑀; 𝐴) = 𝐴𝑝_1^𝛼𝑝_2^𝛽𝑀^𝛾$ ...
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Values of A,K,N,a in Cobb-Douglas function expressing GDP

In many basic macroeconomics textbooks a Cobb-Douglas production function with constant returns to scale is used to express the output of the economy as a function of labor and capital: $Y=AK^aN^{1-a}$...
Giampiero's user avatar
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CES First order Condition with two labour types

I am struggling to derive a first order in this model with Cobb-Douglas production function and CES labour aggregator with two types labour (here male and female, but could be equally low and high ...
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What's special about Cobb–Douglas utility relative to the rest of the CES family?

My question concerns CES utility functions, which have the form $$u(x) = \left(\sum_{j=1}^n a_j x_j^{\rho} \right)^{1/\rho}$$ for utility parameters $a_j$, an elasticity parameter $\rho$, and some ...
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Cobb–Douglas utility maximized by spending a "fixed fraction of income on each good"?

Consider a Cobb–Douglas utility function having the form $$u(x) = \prod_{j=1}^n x_j^{a_j} $$ where $x$ is an allocation vector and $a_j$ are utility parameters with $\sum a_j = 1$. My question has to ...
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References on Cobb-Douglas Function

I am wondering why everybody uses Cobb-Douglas production functions? Nowadays they are so standard that they are just written down without further discussion. Does anybody know more about this and can ...
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Labor Supply- how to do comparative statics?

Consider an economy with a competitive industry where the representative firm's production function takes the form of a Cobb Douglas production function $Y=z K^{\theta} L^{1-\theta}$. $z$ is an index ...
Karan Kumar's user avatar
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Neoclassical Two-Sector Model of Endogenous Growth: Getting the consumption growth rate

I'm struggling to derive the growth of consumption from a two-sector model with the traditional Cobb-Douglas function. The model I am speaking about incorporates the fractions used by physical and ...
John M. Riveros's user avatar
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CobbDouglas: Constant marginal costs and constant returns to scale

A company has a production function: $$y=x_1^{\alpha}x_2^{1-\alpha}$$ where $0<\alpha<1$. Factor input 1 costs $w_1> 0$ and factor input 2 costs $w_2> 0$. The company wants to minimize its ...
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Production function and elasticity

Let $y=x_1^\alpha x_2^\beta$ where $\beta=1-\alpha$ be a Cobb-Douglas production function. Find the elasticity of the optimal demand functions (for minimizing production cost) for both goods wrt. $w_2/...
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Isn't the parameter A denoting technological progress from Cobb-Douglas already included in alpha and beta?

Does it not follow that if capital is better performing than labor it would be used more frequently? What exactly does parameter A actually mean, then?
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Marginal cost given (Cobb-Douglas) production

My function is $y=x_1^\alpha x_2^\beta$ with $\beta={1-\alpha}$. I found: the minimization problem for demand to be $x_1^{*}(w_1,w_2,y)=\left ( \frac{w_2}{w_1}\frac{\alpha}{\beta} \right )^{\frac{\...
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Elasticity Cobb-Douglas production function

I am given the production function $y=x_1^\alpha x_2^{1-\alpha}$, where $0< \alpha <1$ I found the demand functions for minimum production cost to be $ x_1^{*}(w_1,w_2,y)=\left ( \frac{w_2}{w_1}\...
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Can someone help me prove that the CES function is also a Cobb-Douglass function [duplicate]

I would like some assistance with a problem that I have showing a CES function is also a Cobb-Douglass utility function. Question: we have a CES function: $Y=A[\alpha K^{((1-\sigma)/\sigma))}+(1-\...
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Utility functions and positive monotone transformations

We let $g(z)$ be a strictly monotonous function so: $$\frac{dg(z)}{dz}>0$$ Consumer 1 has preferences given by the utility function $u(x_1,x_2)=ln(x_1)+2ln(x_2)$, while consumer 2 has preferences ...
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Compare taxes Cobb-Douglass and more

Let a utility function for a consumer be defined as $u(x_{1},x_{2})=x_{1}^{1/2} x_{2}^{1/2}$. With the budget $x_{1}p_{1}+x_{2}p_{2}=m$. With values $p_1=p_2=1, m=32$. The state now adds a tax of unit ...
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Estimating $TFP$ using Cobb-Douglas production function

Suppose we want to estimate total factor productivity (TFP) under time series framework. Let assume that the production function is given in the Cobb-Douglas form, i.e. $$Y_t=A_tK_t^\alpha L_t^\beta,$$...
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How was CES utility function derived?

Is there any book/papers that I can refer to the proof (derivation) of the CES utility function? Or if anyone could help me with the derivation, I will be so much grateful to you.
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Are prices part of total factor productivity?

I am trying to understand how production is related to income/profit and where do prices enter. Suppose there is a single firm with a Cobb-Douglas production technology: $$Y=AK^{\alpha}L^{\beta}$$ ...
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How was the Cobb Douglas function derived?

In economics and econometrics, the Cobb–Douglas production function is a particular functional form of the production function, widely used to represent the technological relationship between the ...
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Capital in terms of labor

I have a question that asks to find $\frac{\partial K}{\partial L} $ from $Q=cL^aK^b$, when $Q$ and $c$ are constants. It lists 4 answer choices but I’m just not sure how to approach it. Implicit ...
lampshade's user avatar
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CES aggregator for intermediates with heterogeneous productivity

I'm following Atkeson, Burstein (2019 JPE), and cannot understand the aggregation result. There is measure $M(z)$ of firms with productivity $z$, with production function $$ y(z) = z k(z)^\alpha l(z)^...
JaySingsBlue's user avatar
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Deriving a demand curve from a Cobb-Douglas utility

Probably a daft question but I derived an equation for a demand curve from a general Cobb-Douglas utility function $$U(x,y)=\beta x^{\alpha}y^{1-\alpha}$$ given a budget constraint $$M=xP_x+yP_y$$ and ...
Jonathan Andrews's user avatar
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Optimization problem of a Cobb-Douglas function with 3 inputs

A perfectly competitive firm uses 3 inputs to manufacture a certain product according to the following Cobb-Douglas production function: $$ Q = A L_1^{\alpha_1} L_2^{\alpha_2} L_3^{\alpha_3} $$ ...
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K in the Cobb Douglas function

I'm using the Cobb Douglas production function for a Mathematics investigation into how optimisation works in Economics. The assumption is that the firm require's only one type of capital, that is, a ...
Saad's user avatar
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Cobb-Douglas function homotheticity

I've been given the Cobb-Douglas utility function: $\ u(q_1, q_2)=a\ln q_1+b\ln q_2=q_1^aq_2^b \ $ If I want to prove homothetic preferences, I use the following condition: $\ u(\lambda q_1, \...
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Determine if goods are substitutes or Complements based on demand function

So I have a consumer with a utility function of the Cobb-Douglas form $v(x_1,x_2)=x^{\frac{1}{2}}_1x^{\frac{1}{2}}_2$. From that I constructed the demand function for good 1 and good 2: $x_1=\frac{1}{...
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Derivation of demand function

Hello. I'm graduate student in Japan. This time, what I want to ask is how to solve the profit maximization problem using the image production function and derive the demand function. This ...
Kenta's user avatar
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Demand derived from Cobb-Douglas utility, interpretation, check

I derived demand, given a Cobb-Douglas utility function but I am not really sure if I did it correctly. I am especially struggling with the sum signs and the subscripts of $i$ & $j$. It would be ...
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How to Calculate the productivity multiplier?

Given a Cobb Douglas $Y_t = A (K_t^\alpha L_t^{1-\alpha}) $ $ K_{t+1} = sY_t + (1-\delta) K_t$ How do we get the multiplier on productivity to be equal to $ \frac{1}{1-\alpha}$? I understand ...
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Linking top-down and bottom-up models for analyzing electricity price-based demand response: Expenditure constraint is violated?

I have a question about the contents of this paper*, which links a building energy model and a utility-maximization component. In it, the author tests several electricity prices using a Cobb-Douglas ...
Harry Silver's user avatar
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1 answer
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Maximizing a Cobb-Douglas Function

Suppose that a competitive firm receives a price of $P$ for its output, and pays prices of w, r and v for its labor $(L)$, capital $(K)$ and natural resources $(R)$ inputs, respectively. The firm ...
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2 answers
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Is Cobb-Douglas the only output function corresponding to a competitive economy?

Apologies if this is a rather simple question, I appreciate any guidance. $$ Q(K,L) = AK^\alpha L^{\beta} $$ where A is a constant. Identify the conditions on $\alpha$ and $\beta$ for ...
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