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

The proofs I will present are based on techniques relevant to the fact that the CES production function has the form of a generalized weighted mean. This was used in the original paper where the CES ...
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15 votes

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

The regular method of obtaining Cobb-Douglas and Leotief is L'Hôpital's rule. Another methods should be used too. Setting $ \gamma=1$ will be return $Q=[a K^{-\rho} +(1-a) L^{-\rho} ]^{-\frac{1}{\...
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  • 673
13 votes
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Marshallian Demand for Cobb-Douglas

Since $a + b=1$ the equations are exactly the same. Substituting in for $a+b$ with $1$ in the third and fourth equations gives the first and second equations.
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  • 15.9k
11 votes
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Cobb-Douglas and Logarithm Utility Functions

Utility functions are invariant with respect to positive monotonic transformations (PMT). Take $U(x,y)=x^\alpha y^{1-\alpha}$, and let $V(x,y)=\log(U(x,y))$ be a PMT of $U$. Thus $V$ and $U$ both ...
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  • 14.3k
9 votes
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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?

It is not true that this function is equivalent to the Cobb-Douglas utility function when $c \sim 0$ for any values of $(a,b)$; you have to assume $a+b=1$ for that, i.e. $b=1-a$. To see why it is ...
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  • 3,202
8 votes
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Is the Cobb-Douglas Utility Function Locally Non-Satiated at (0,0)?

No. Cobb-Douglas utility is monotonic and monotonicity implies L.N.S. The issue here is that you're only considering edge cases. You've correctly reasoned that edge points are not more desirable that ...
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  • 2,871
8 votes

CobbDouglas: Constant marginal costs and constant returns to scale

Since the exponents add to one the production function has constant returns to scale, which means that, given factor prices, total cost is linear, which means that it's derivative (= marginal cost) is ...
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7 votes

Marshallian Demand for Cobb-Douglas

This is how you get from your first equation to your second. your utility function is $u(x_1, x_2)=x_1^a x_2^b$ since $a+b=1$ I'll change it slightly to a and (1-a) In order to optimise these two ...
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  • 3,721
7 votes

How was CES utility function derived?

To understand the CES utility functions, which I guess is your question, a good starting point is the Wikipedia page on constant elasticity of substitution. In particular, The CES aggregator is also ...
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  • 6,652
6 votes
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Constant Elasticity of Substitution: Special Cases

We know that if $u$ represents $\succeq$ on $X$, then for any strictly increasing function $f: \mathbb{R} \rightarrow \mathbb{R}$, then $v(x) = f(u(x))$ represents $\succeq$ on $X$ ($X$ in this case ...
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  • 6,409
6 votes

How was CES utility function derived?

The C.E.S functional has been introduced in Economics in the context of production theory, by Arrow, K. J., Chenery, H. B., Minhas, B. S., & Solow, R. M. (1961). Capital-labor substitution and ...
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5 votes
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Notation of a Cobb-Douglas function printed in 1989

Seems to be the second one, so $$ Q_t = A_t*(K_t^\alpha N^\beta_t T_t^\rho). $$ Two clues: This is the usual specification. On the top of page 14 it is written that $$ w_t = \beta Q_t/N_t. $$ Given ...
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  • 26k
5 votes
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Are Cobb-Douglas preferences homothetic?

Note that the wikipedia article is very specific: [...] defined a preference to be homothetic, if they CAN be represented by A utility function [...] You chose a specific utility function to ...
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  • 26k
5 votes
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Interpretation of Interesting Utility Function

This is the CES production function, where CES stands for constant elasticity of substitution. The parameter $\sigma$ captures the (constant) elasticity of substitution and $\alpha$ is the share ...
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5 votes

Is Cobb-Douglas the only output function corresponding to a competitive economy?

Let $a+b<1,\;\; a,b>0$. Pay the factors of production their marginal product: $$rK = \frac {aQ}{K} K= aQ,\;\;\ wL=\frac {bQ}{L} L=bQ$$ So total payments to factors of production will be $$rK ...
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5 votes
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How was the Cobb Douglas function derived?

What is the proof of this formula? There is actually no proof for what the production function should be. There are infinite many possible production functions and to discover which one is the most ...
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5 votes
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Cobb–Douglas utility maximized by spending a "fixed fraction of income on each good"?

A "fixed fraction" doesn't mean an "equal fraction", or at least that's not the intended meaning. It can be easily verified that the solution to \begin{equation} \max_{x_1,x_2}\;...
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5 votes
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Cobb-Douglas Production Function - Finding units of labour to maximise production

If your aim is to maximize the production then your approach is correct. But if the aim is to find the optimal number of units of labor, then you should solve it as a profit maximization problem with ...
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  • 342
5 votes

Cobb Douglas Production: Identification issues for technical change

Maybe the following is useful: Sato (1970), The Estimation of Biased Technical Progress and the Production Function, International Economic Review, 11, 179-208 JSTOR link
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  • 8,737
5 votes
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Calculating cost-minimizing inputs with 3 inputs in production function

The production function $$F(x) = x_1^{\alpha_1}x_2^{\alpha_2}x_3^{\alpha_3},$$ has the derivative with respect to $x_j$ where for sake of example I choose $j=1$ $$\frac{\partial F(x)}{\partial x_1} = \...
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  • 3,247
4 votes

Interpreting how graphs of Cobb-Douglas utility functions. How does MRS vary as we vary $\alpha$?

If $\alpha$ rises, the utility puts more weights to the $x$. Then you must give up more $x$ for one $y$ (for the same utility). Your MRS, for a given $y$, increases in absolute value. Graphically: If ...
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  • 698
4 votes

Constant Elasticity of Substitution: Special Cases

These are standard mathematical results for generalized means. For example,for the $\rho \rightarrow 0$ result, write (setting without loss of generality $\sum_{i=1}^na_i =1$), $$U = \left[\sum^n_{i=...
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4 votes

Are Cobb Douglas goods complements or substitutes?

Try to think what you mean when you ask whether they're complements or substitutes. You could mean: "Does my marginal utility in $x$ increase when I get more $y$? That would correspond to the cross ...
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4 votes

In the C.E.S. utility function do the parameters need to add up to unity to obtain the Cobb-Douglas utility function?

Yes. Write $$U(x,y) = (ax^{-c} + by^{-c})^{-\frac{1}{c}} = \exp\left\{\frac {-1}{c}\ln \left(ax^{-c} + by^{-c}\right)\right\} \tag{1}$$ Now if $a+b=1$ then as $c\rightarrow 0$, expression $$\frac ...
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4 votes
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Consumer preference and price in the Cobb-Douglas function

No, preferences are stable. That is not to say that the quantity demanded or marginal utility obtained at the new price level is the same though. If we'd allow the exponent of the utility function ...
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  • 2,330
4 votes
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Is Cobb-Douglas the only output function corresponding to a competitive economy?

Any constant returns to scale function is compatible witha competitive economy. Cobb-Douglas is not the only one. Google CES production function. Also, product can be wasted or be an externality, so ...
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  • 1,314
4 votes

How was the Cobb Douglas function derived?

If one reads the original article by Cobb and Douglas (1928), https://www.aeaweb.org/aer/top20/18.1.139-165.pdf , one will find at the end of page 152 that the authors stress that they took into ...
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4 votes
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CES First order Condition with two labour types

Simply multiply and divide one $\left(a_{F} \boldsymbol{F}\right)^{\frac{(\sigma-1)}{\sigma}}$ in the bracket and then take one outside the bracket. And by the way your FOC is incorrect in that $\frac{...
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  • 2,021
4 votes
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Finding restrictions on parameters for a demand function

Demand is positive, so $A>0$. If $p_1$ goes to $\infty$, $x_1$ has to go to 0, since $p_1x_1$ is bounded by $M$. Thus $\alpha < 0$. If $p_2$ goes to 0, $x_1$ cannot go to $\infty$, since $p_1x_1$...
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  • 4,517
4 votes

Is it true that for Cobb-Douglas preferences, the demand function is always iso-elastic?

Yes. I have to include at least 30 characters in an answer, so let me repeat: Yes.
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