10
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 ...
- 3,232
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 ...
- 2,911
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
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 ...
- 6,578
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 ...
- 6,885
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 ...
- 33.4k
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 ...
- 28.7k
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 ...
- 28.7k
5
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
$$\frac ...
- 33.4k
5
votes
Constant Elasticity of Substitution: Special Cases
These are standard mathematical results for generalized means. For example,for the $\rho \rightarrow 0$ result, write (setting without loss of generality $\sum_{i=1}^na_i =1$),
$$U = \left[\sum^n_{i=...
- 33.4k
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 ...
- 54.4k
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
$$rK ...
- 33.4k
5
votes
Accepted
Interpretation of Interesting Utility Function
This is the CES production function, where CES stands for constant elasticity of substitution.
The parameter $\sigma$ captures the (constant) elasticity of substitution and $\alpha$ is the share ...
- 15.3k
5
votes
Accepted
Cobb–Douglas utility maximized by spending a "fixed fraction of income on each good"?
A "fixed fraction" doesn't mean an "equal fraction", or at least that's not the intended meaning.
It can be easily verified that the solution to
\begin{equation}
\max_{x_1,x_2}\;...
- 15.3k
5
votes
Accepted
CES First order Condition with two labour types
Simply multiply and divide one $\left(a_{F} \boldsymbol{F}\right)^{\frac{(\sigma-1)}{\sigma}}$ in the bracket and then take one outside the bracket. And by the way your FOC is incorrect in that $\frac{...
- 2,339
5
votes
Accepted
Cobb-Douglas Production Function - Finding units of labour to maximise production
If your aim is to maximize the production then your approach is correct.
But if the aim is to find the optimal number of units of labor, then you should solve it as a profit maximization problem with ...
- 356
5
votes
Cobb Douglas Production: Identification issues for technical change
Maybe the following is useful:
Sato (1970), The Estimation of Biased Technical Progress and the Production Function, International Economic Review, 11, 179-208
JSTOR link
- 10.3k
5
votes
Accepted
Calculating cost-minimizing inputs with 3 inputs in production function
The production function
$$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} = \...
- 3,286
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 ...
- 2,375
4
votes
Accepted
Deriving a demand curve from a Cobb-Douglas utility
If you take the general class of CES utility functions, of which Cobb-Douglas is a special case, you do indeed get a demand function that depends on other prices. Specifically, the CES utility ...
- 15.3k
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 ...
- 33.4k
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 ...
- 1,344
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 $p_1x_1$...
- 6,550
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.
- 6,550
4
votes
Does global maximum of CRS Cobb-Douglas profit exist
It is generally true that a profit-maximizing firm with a constant-returns to scale technology can produce a positive output only if the profit is zero. Output prices are pinned down by the zero-...
- 11.7k
3
votes
Marginal Product of Capital in the Solow Model
According to your calculations MPK is not increasing in $K$. The Solow model assumes $0< \alpha < 1$, thus $\alpha - 1 < 0$ and $K^{\alpha - 1}$ is decreasing in $K$.
- 28.7k
3
votes
Profit maximization with Cobb-Douglas function
Should I solve for $L^∗$ by separating $K^∗$ from the equation and plugging into $pMP_{L}$
Yep, that's about it.
Wouldn't this yield a very complicated solution?
Somewhat. The math is available ...
- 41
3
votes
Indirect changes in Marshallian Demand
You correctly derived the Marshallian demand function for Cobb-Douglas utility, you notice that the optimal level of consumption of $x$ or $y$ is a function only of the individual's income and the ...
- 871
3
votes
Accepted
Growth Accounting with variable factor shares
Interesting question. In effect, while factor shares were thought to have remained fairly stable over a long time (the first of the Kaldor's facts), more recently they have varied, particularly in the ...
- 8,581
3
votes
Accepted
Demand derived from Cobb-Douglas utility, interpretation, check
From (1) and (2) you get
$$\frac{x_j}{x_i}=\frac{a_j p_i}{a_i p_j},$$
or equivalently,
$$x_j =\frac{a_j p_i}{a_i p_j} x_i.$$
Substituting this into equation 3 for $j=2,...,n$ and $i=1$ (solving for ...
- 595
Only top scored, non community-wiki answers of a minimum length are eligible