# Tag Info

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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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### 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 ...

### 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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### 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 ...

### 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 ...

### 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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### 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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### 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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### 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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### 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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### 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 ...

### 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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### 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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### 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 $p_1x_1$...

### 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.

### 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-...

### 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 $K$.

### 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 ...

### 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 ...
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 ...