24
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 function was introduced, Arrow, K. J., Chenery, H. B., Minhas, B. S., & Solow, R. M. (1961). Capital-labor substitution and economic efficiency. The Review ...
13
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 $Q=[a K^{-\rho} +(1-a) L^{-\rho} ]^{-\frac{1}{\rho}}$ and
$$Q^{-\rho}=[a K^{-\rho} +(1-a) L^{-\rho} ]$$
By The total derivative via differentials we will have
$$-\rho Q^{-\rho-1}dQ=- a\rho K^{-\rho-1}dK -(...
13
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.
10
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 represent the same preference, and thus demand functions for $x$ and $y$ are the same.
8
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 the origin. However, LNS simply claims that there exists a more desirable bundle within the open epsilon ball of your allocation under consideration (and this ...
8
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 true, fix $(x,y)$ and consider the following Taylor expansion of $U(x,y)$ when $c$ gets close to $0$. We have
\begin{align*}
(ax^{-c}+by^{-c})^{-\frac{1}{c}} &...
8
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 contant. If you change the exponent 1-alpha to beta where alpha+beta < 1, there will be decreasing returns to scale (but still homotheticity) and you will ...
7
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 sometimes called the Armington aggregator, which was discussed by Armington (1969).
Then, the CES utility function was popularized by Dixit and Stiglitz (1977) ...
6
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 is $\mathbb{R^n}$)
Consider $v(x, \rho) = \ln(u(x, \rho)) - \frac{\ln\left[\sum^n_{i=1}\alpha_i \right]}{\rho}$, which is strictly increasing.
$$v(x, \rho) = \...
6
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 choices, you need to maximise utility, wrt your choice variables.
subject to $p_1x_1 + p_2x_2 = w$
using Walras Law. Basically, in order to optimise utility, all ...
6
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 economic efficiency. The review of Economics and Statistics, 225-250.
There you can find a discussion of how it was derived.
A more pedagogic and detailed ...
5
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 a competitive labor market you have
$$
w_t = \frac{\partial Q_t}{\partial N_t}.
$$
If you assume the production function above then you indeed have
$$
\frac{\...
5
This is the CES production function, where CES stands for constant elasticity of substitution.
The parameter $\sigma$ captures the (constant) elasticity of substitution and $\alpha$ is the share parameter.
The Cobb-Douglas production function can be obtained as a special case of the CES class by taking $\sigma\to1$. For proof, I'd refer you to this post.
5
Let $a+b<1,\;\; a,b>0$. Pay the factors of production their marginal product:
$$rK = \frac {aQ}{K} K= aQ,\;\;\ wL=\frac {bQ}{L} L=bQ$$
So total payments to factors of production will be
$$rK +wL = aQ + bQ = (a+b)Q < Q$$
And the question is: who gets the rest of the output that has been produced?
5
What is the proof of this formula?
There is actually no proof for what the production function should be. There are infinite many possible production functions and to discover which one is the most appropriate we need to make some empirical observations. In different cases different production functions are appropriate. Cobb-Douglas is popular production ...
4
If $\alpha$ rises, the utility puts more weights to the $x$. Then you must give up more $x$ for one $y$ (for the same utility). Your MRS, for a given $y$, increases in absolute value. Graphically:
If you set $y$ and $x$, the slope is lower for higher $\alpha$ (be careful to only change one thing at a time).
Concerning your "loop" problem, look at the ...
4
These are standard mathematical results for generalized means. For example,for the $\rho \rightarrow 0$ result, write (setting without loss of generality $\sum_{i=1}^na_i =1$),
$$U = \left[\sum^n_{i=1} \alpha_i x^\rho_i \right]^\frac{1}{\rho} = \exp\left\{\frac 1\rho\ln \left(\sum^n_{i=1} \alpha_i x^\rho_i\right)\right\}$$
Apply L'Hopital's rule on
$$\...
4
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 derivative $\frac{\partial U^2}{\partial x \partial y}$.
You could (and this is the convention) mean the response to a change in prices. Denote with stars the ...
4
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
$$\frac {\ln \left(ax^{-c} + by^{-c}\right)}{-c} \tag{2}$$
will be an indeterminate form $0/0$ and so we can apply L'Hopital's rule on it to get
$$\frac {1}{-c}\...
4
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 to vary for different price levels we'd let utility depend on prices, which it does not. Of course the actual utility that can be obtained depends on prices, but ...
4
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 even in non CRS competitive economy can exist.
3
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 price of said good. This is an interesting feature of CD-utility, that when the price of good $y$ changes the demand for $x$ doesn't change. This means that $x$ ...
3
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$.
3
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 represent your Cobb-Douglas preferences. However there are infinitely many others. All monotonic transformations of your utility function represent the same ...
3
Start with the more general CES function $U=[a*x^b+(1-a)*y^b]^{1/b}$. Compute the elasticity of substitution of this function. Then compute the functional form of U for $b=0$,$b=1$ and $b=-\infty$. You will find the two extreme (perfect complements / substitute) cases for CES, and the common case C-D for b=0.
3
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 at many places, like section 4 here. But you can surely do it yourself!
3
Let $y=Y/L$ and $k=K/L$ be the per-worker levels of output and capital. Observe that $y=Ak^\alpha$.
Steady state is given by: $$k^*=sy^*+(1-\delta)k^*,$$ or $$k^*=sA(k^*)^\alpha+(1-\delta)k^*.$$
Doing the algebra: $$k^*=\left(\frac{sA}{\delta}\right)^{\frac{1}{1-\alpha}}.$$
And: $$y^*=A\left(\frac{sA}{\delta}\right)^{\frac{\alpha}{1-\alpha}}=A^{\frac{1}{1-...
3
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 the demand function for good 1) we get
$$M=p_1x_1 + \sum_{j=2}^n p_j \frac{a_j p_1}{a_1 p_j} x_1$$
$$M=p_1x_1 + \sum_{j=2}^n \frac{a_j p_1}{a_1} x_1$$
$$M=...
3
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 function (over $n$ goods, $x_1,\dots,x_n$) takes the form
\begin{equation}
u(x_1,\dots,x_n)=\bigl[\alpha_1x_1^\rho+\cdots+\alpha_nx_n^\rho\bigr]^{1/\rho},
\end{...
3
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 account two properties that had been theoretically discussed in the past:
That production exhibits constant returns to scale, meaning that doubling all inputs will ...
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