Cobb-Douglas functions in economics

In summary, the conversation discusses Cobb-Douglas functions in economics and a question about how to solve for Y algebraically. The solution involves taking the root of both sides and using LaTeX for formatting.
  • #1
kurvmax
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[SOLVED] Cobb-Douglas functions in economics

Hi. I'm an economics major. This problem isn't actually required, but I'm trying to learn more about Cobb-Douglas functions.

Homework Statement


http://rds.yahoo.com/_ylt=A0oGkj3DT...http://www.uvm.edu/~wgibson/cobb-douglas.pdf" [pdf] is a description of them. On that first page they go from [tex]U_{0} = X^{B}*Y^{1-B}[/tex]. Then they solve for Y and get [tex]Y = U^{1/1-B}_{0}*X^{-B/1-B}[/tex]. The question is simple: how did they do that algebraically?


Homework Equations





The Attempt at a Solution




I tried to sort things around to get their solution. I got [tex]Y^{1-B} = U_{0}/X^{B}[/tex] Assuming that is right, I then took the natural log of both sides and divided both sides by [tex]1-B[/tex]. But here is where I'm stuck. Obviously the -B exponent on X in the solution makes it a fraction, but I'm just too rusty/untalented at this stuff to figure out how dividing by (1-B) can find a way into the exponents of the variables.

Incidentally, how do I group things with fractions in Latex?

Nevermind, I'm retarded. Of course if I take the root (1-B) of both sides I get that answer...duh...still curious about the Latex stuff though...
 
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  • #3





Hello, thank you for your question about Cobb-Douglas functions in economics. As an economist, I can explain the algebraic steps to solving for Y in the Cobb-Douglas function described in the paper you provided.

Firstly, the Cobb-Douglas function is a mathematical model used in economics to represent the relationship between two or more inputs (usually capital and labor) and outputs (usually goods or services). It is expressed as Y = A*X^a * Y^b, where A is a constant, X is the amount of capital, Y is the amount of labor, and a and b are the respective coefficients of capital and labor in the production function.

In the paper you provided, the authors start with the utility function U_0 = X^B * Y^(1-B), which represents the satisfaction a consumer gets from consuming a combination of X and Y goods. They then solve for Y by rearranging the equation to Y = U_0^(1/(1-B)) * X^(-B/(1-B)). This can be done algebraically by taking the natural log of both sides of the equation, as you mentioned, and then using the properties of logarithms to solve for Y.

To group fractions in Latex, you can use parentheses to indicate the numerator and denominator. For example, \frac{a}{b} will display as a/b.

I hope this explanation helps you understand the Cobb-Douglas function better. If you have any further questions, please don't hesitate to ask. Best of luck with your studies!
 

1. What is a Cobb-Douglas function in economics?

A Cobb-Douglas function is a mathematical equation used in economics to model the relationship between two or more inputs and the output of a production process. It is named after economists Charles Cobb and Paul Douglas, who first used it in their research in the 1920s.

2. How is a Cobb-Douglas function expressed?

A Cobb-Douglas function is typically expressed in the form of Y = A * X^a * Z^b, where Y is the output, X and Z are the inputs, A is a constant, and a and b are the output elasticities of the respective inputs.

3. What are the main assumptions of a Cobb-Douglas function?

The main assumptions of a Cobb-Douglas function are constant returns to scale, perfect competition, and that the input factors are substitutable but not interchangeable. This means that the proportion of inputs used in production remains constant, the market is efficient, and the inputs are not perfect substitutes for each other.

4. What are the advantages of using a Cobb-Douglas function in economic analysis?

One advantage of using a Cobb-Douglas function is its simplicity, which makes it easy to use and understand. It also allows for the measurement of the impact of each input on the output and can be used to determine the optimal combination of inputs for maximum output.

5. What are the limitations of a Cobb-Douglas function?

One limitation of a Cobb-Douglas function is that it does not account for technological advancements or changes in the production process. It also assumes a linear relationship between inputs and outputs, which may not always hold true in real-world situations. Additionally, it may not be suitable for analyzing complex production processes with multiple inputs and outputs.

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