28
votes
Accepted
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 ...
- 31.8k
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}{\...
- 673
13
votes
Accepted
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.
- 15.9k
11
votes
Accepted
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 ...
- 14.3k
9
votes
Accepted
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 ...
- 3,202
8
votes
Accepted
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 ...
- 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 ...
- 81
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 ...
- 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 ...
- 6,652
6
votes
Accepted
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 ...
- 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
Accepted
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 ...
- 26k
5
votes
Accepted
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 ...
- 26k
5
votes
Accepted
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 ...
- 14.3k
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
$$rK ...
- 31.8k
5
votes
Accepted
How was the Cobb Douglas function derived?
What is the proof of this formula?
There is actually no proof for what the production function should be. There are infinite many possible production functions and to discover which one is the most ...
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5
votes
Accepted
Cobb–Douglas utility maximized by spending a "fixed fraction of income on each good"?
A "fixed fraction" doesn't mean an "equal fraction", or at least that's not the intended meaning.
It can be easily verified that the solution to
\begin{equation}
\max_{x_1,x_2}\;...
- 14.3k
5
votes
Accepted
Cobb-Douglas Production Function - Finding units of labour to maximise production
If your aim is to maximize the production then your approach is correct.
But if the aim is to find the optimal number of units of labor, then you should solve it as a profit maximization problem with ...
- 342
5
votes
Cobb Douglas Production: Identification issues for technical change
Maybe the following is useful:
Sato (1970), The Estimation of Biased Technical Progress and the Production Function, International Economic Review, 11, 179-208
JSTOR link
- 8,737
5
votes
Accepted
Calculating cost-minimizing inputs with 3 inputs in production function
The production function
$$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} = \...
- 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 ...
- 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 $\sum_{i=1}^na_i =1$),
$$U = \left[\sum^n_{i=...
- 31.8k
4
votes
Are Cobb Douglas goods complements or substitutes?
Try to think what you mean when you ask whether they're complements or substitutes.
You could mean: "Does my marginal utility in $x$ increase when I get more $y$? That would correspond to the cross ...
- 10.4k
4
votes
In the C.E.S. utility function do the parameters need to add up to unity to obtain the Cobb-Douglas utility function?
Yes.
Write
$$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 ...
- 31.8k
4
votes
Accepted
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 ...
- 2,330
4
votes
Accepted
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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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 ...
- 31.8k
4
votes
Accepted
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{...
- 2,021
4
votes
Accepted
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 $p_1x_1$...
- 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.
- 4,517
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