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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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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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Interpretation of Interesting Utility Function

Solving introductory microeconomics problems I have come across the following type of utility function: $$ f(K,L) = (\alpha K^{\frac{\sigma - 1}{\sigma}} + (1 - \alpha) L^{\frac{\sigma - 1}{\sigma}})^{...
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Regression on derived consumer preference

I have a data set with some demographics of consumers who bought a product that can be used to imply their preference (beta) using Cobb-Douglas (see comments of original question). I’d like to check ...
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Consumer preference and price in the Cobb-Douglas function

I believe I’m using the most basic version of Cobb-Douglas: $U(x,y)=x^\beta * y ^{(1-\beta)}$. The question I have is: in this example would a consumer’s preference ($\beta$) change if the price of ...
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Heckscher-Ohlin with different technologies

Consider two countries: Home and Foreign that produce two goods, cars and wheat. The production technologies are such that: $q_{c} = K_{c}^{0.5} L_{c}^{0.5}$ and $q_{w} = 0.5 K_{w}^{0.5}L_{w}^{0.5}$ ...
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MRS in Walrasian equilibrium price expression?

I have the utility function $U^A=\alpha ln(x^A)+\beta ln(y^A)$ for consumer $A$ and $U^B=\gamma ln(x^B)+\phi ln(y^B)$ for consumer $B$. They are endowed with $\omega^h_x$ and $\omega^h_y$ of $x$ and $...
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Constant elasticity of substitution utility function to Cobb-Douglas [duplicate]

How to prove that constant elasticity of substitution utility function $u=(\frac{1}{\mid{M}\mid} \int_{i\in M}q(i)^{\frac{\sigma-1}{\sigma}}di)^{\frac{\sigma-1}{\sigma}}$ is the same as Cobb-Douglas ...
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Marginal product of capital net of depreciation

I am trying to understand how marginal product of capital net of depreciation is the following: Given that the production function is quite standard I understand the first term of the marginal ...
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557 views

Profit maximization with Cobb-Douglas function

I'm trying to maximize a firm's profit given the production function $F(L,K)=L^\alpha K^\beta$ (where $L$ is labor and $K$ is capital) and that $\alpha + \beta \neq 1$. So, I know that this ...
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Notation of a Cobb-Douglas function printed in 1989

I am trying to understand a paper written back in 1989 about long run population growth. It seems like the PDF is a scanned image of the paper. The notation for the function is on page 11 of the pdf (...
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Growth Accounting with variable factor shares

Under the assumption of competitive markets, the usual growth accounting equation using Cobb-Douglass function is as follows: $\Delta Y/Y = \Delta A/A+w_l\Delta L/L+w_k\Delta K/K$ $Y$ is output $K$...
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Are Cobb-Douglas preferences homothetic?

Our lecture defined a preference to be homothetic, if the following is true: $$(x_1, x_2) \thicksim (y_1, y_2) \Leftrightarrow (kx_1, kx_2) \thicksim (ky_1, ky_2)$$ Cobb-Douglas preferences can be ...
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Find Engel Curve with a Cobb-Douglas

I have $U(x,y)=xy$, $p_1=4$ and $p_2=1$. Income is unknown. Where do I start?
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Unitary model with cobb-douglas function

We have these following from the first order conditions: ${U_{LA} \over U_C} = w_A \to MRS_{C,LA}=w_A$ ${U_{LB} \over U_C} = w_B \to MRS_{C, LB}=w_B$ ${U_{LA} \over U_{LB}} = {w_A \over w_B} \to ...
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Using the Slutsky equation

Suppose we have utility: $$U(x,y)=x^{0.5}y^{0.5}$$ Then Marshallian Demand for good $x$ is: $$x(p_{x},p_{y},I)=\frac{0.5I}{p_{x}}$$ And Hicksian Demand for good $x$ is: $$x^{c}(p_{x},p_{y},U)=p_{...
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The factor elasticities from a Cobb-Douglas function in Romer's macroeconomy book

Good night, I'm reading the Romer's macroeconomy book in the page 42, "A complication" section title. The begin of third paragraph say: This is not a general property of production functions, ...
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Calculating growth rate of capital when not in steady state

Given a Cobb-Douglas production function, the annual population growth rate, savings rate, alpha, annual depreciation rate, and annual technological progress rate, how would one calculate the growth ...
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Indirect changes in Marshallian Demand

Suppose we have a Cobb-Douglas utility function: $$U(x,y)=x^\alpha y^\beta$$ and a budget constraint: $$p_{x}x+p_{y}y=I$$ where $\alpha+\beta=1$. It can be shown that the Marshallian demand for $x$ ...
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Is the Cobb-Douglas Utility Function Locally Non-Satiated at (0,0)?

My understanding of local non-satiation is that increasing your allocation of one good by a marginal amount increases utility. Suppose your utility takes the following form: $$U(x,y)=x^\alpha y^\beta$$...
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Given a standard Cobb-Douglas production function, how to estimate the output elasticity of labour and capital by country?

Given a standard Cobb-Douglas production function: $$Y_t=(A_t L_t)^{\alpha} K_t^{1-\alpha}$$ Moreover, the production function has constant returns to scale: $$\alpha + (1-\alpha)=1$$ How to estimate ...
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In the C.E.S. utility function do the parameters need to add up to unity to obtain the Cobb-Douglas utility function?

Consider the C.E.S. utility function $$U(x, y) = (ax^{-c} + by^{-c})^{-\frac{1}{c}} $$ Is it true that we must have $a+b=1$ in order to obtain a Cobb-Douglas utility function as $c\rightarrow 0$?
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How does the limit of $U(x, y) = (ax^{-c} + by^{-c})^{-\frac{1}{c}}$ as c approaches 0 yield the Cobb-Douglas utlity function? [duplicate]

\begin{equation*} U(x, y) = (ax^{-c} + by^{-c})^{-\frac{1}{c}} \end{equation*} I ask this mainly because after logging both sides of the Utility equation (the first step to proving the assertion, I ...
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Marginal Product of Capital in the Solow Model

In the classic form of the Solow Model: $$ Y=K^\alpha (AL)^{1-\alpha } $$ Describe circumstances in which the marginal product of capital could rise over time, at least for a temporary period. I've ...
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Constant Elasticity of Substitution: Special Cases

Take an $n$-commodity constant elasticity of substitution utility function, $$U = \left[\sum^n_{i=1} \alpha_i x^\rho_i \right]^\frac{1}{\rho}$$ How do we show the following: Show that as $\rho \...
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Optimizing Cobb-Douglas like functions

Economics isn't my home field, but I'm looking for references for a paper I'm working on and I'm hoping one of you can help. Are there many good references for optimizing a Cobb-Douglas like utility ...
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Optimizing Cobbs-Douglas given output

I am trying to find L$^*$ and K$^*$ (Labour and Capital respectively) given the following: Q = $K^{1/4}$ $L^{3/4}$ where Q = 120, w=$24, and r=$128 (Not sure if you need w and r) I know that if I ...
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Given Cobb-Douglas production functions for 2 factories (same owner), how will the owner produce $y$?

So my question is this: A company owns two factories, A and B, each with the following production functions: $f_A(x_1,x_2)=x_1^{\alpha}x_2^{1-\alpha}$ $f_B(x_1,x_2)=x_1^{\beta}x_2^{1-\beta}$ Now ...
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Fast way to write out the utility optimization problem for a Cobb-Douglas function?

In my last problem set, I had to solve both the Utility Maximization Problem (UMP) and Expenditure Minimization Problem (EMP) for a Cobb Douglas utility function. Recall, Cobb Douglas is defined as $$...
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Are Cobb Douglas goods complements or substitutes?

Given $$U(x,y)= x^\alpha y^{1-\alpha}$$ $\alpha \in (0,1)$, are Cobb Douglas goods (here $x$ and $y$) complements, substitutes, or neither? Why? An explanation with mixed partial derivatives would ...
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Interpreting how graphs of Cobb-Douglas utility functions. How does MRS vary as we vary $\alpha$?

Suppose I have the following Cobb-Douglas function $$U(x,y) = x^\alpha y^{1-\alpha} = 1$$ where $\alpha \in [0,1]$. $$MRS = -\frac{U_x}{U_y} = - \frac{\alpha}{1-\alpha} \frac{y}{x} $$ $$\frac{\...
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Editing formula for finding Marshallian Demand with Cobb-Douglas utility function

Suppose a utility function $u=x_1^ax_2^b$ with $a+b=1$. The following formula finds the values for $x$: $x_1 = \frac{am}{p_1}\\ x_2 = \frac{bm}{p_2}$ But what if the utility function looks like $u=...
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Marshallian Demand for Cobb-Douglas

When trying maximize the utility having a cobb-douglas utility function $u=x_1^ax_2^b$, with $a+b = 1$, I found the following formulas (Wikipedia: Marshallian Demand): $x_1 = \frac{am}{p_1}\\ x_2 = \...
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Derive a cost production function give prod f, only K

In this question, only K is included and L is excluded how would I go about deriving it? Total cost= Fixed costs + Average costs. Since the variable input costs r per unit, the variable costs is r ...
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Marginal product versus marginal productivity in a Cobb-Douglas production function [closed]

CFA Kaplan Schweser claim that Cobb-Douglas function exhibits constant marginal product of capital but diminishing marginal productivity of capital. I think this statement is not right. My view ...
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Cobb-Douglas and Logarithm Utility Functions

Suppose I have a consumer with a utility function $U(x,y) = x^\alpha y ^{1-\alpha} $ where $a \in (0,1)$. Suppose this consumer has wealth $w$ and the prices for $x$ and $y$ are $p_x$ and $p_y$ ...
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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 ...