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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 ...

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