# Cobb douglas production function calculator

We focus on providing fast, comprehensive, convenient collections of online Calculators. Cobb-Douglas Production Function Calculator helps calculating the quantity of products, the marginal product of Labor and the marginal product of capital, given Cobb-Douglas Production Function. In economics, a production function represents the relationship between the output and the combination of factors, or inputs, used to obtain it.

The Cobb-Douglas production function is a particular form of the production function. It is widely used because it has many attractive characteristics.

Free Online Calculators. Icalc - easy to calculate. We Create Awesome Calculators. Cobb-Douglas Production Function. Hundreds of Free online Calculators. To see the various calculators, press the relevant calculator's title. Accounting Calculators. How to Calculate Retained Earnings? Conversion Calculators. Reverse a Text. Economics Calculators. How to Calculate AVC? Electrical Calculators. Watts to kilovolt-amps kVA Calculator.

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Becareful about interpreting the term we are treating it as a function of continuous Lsome define MPL in terms of discrete increases in Lit is a derivative, which as we have discussed is the slope of the tangent line to the production function line with fixed K and L along the. Which means if you increase labor by a infinitestimally small amount when existing andthe slope of output increase will be. The actual output increase is that infinitestimally small increase in labor multiplied by.

It is perhaps difficult to conceptualize what it means to multiply something infinitely small by another number. To make the idea more conconcret, we will think using to approximate the increase in output given a small increase in labor. Continuing with the two numbers we can calculate without a calculator:. Suppose we are interested in the increase in output when labor increases from towhat is the new output? What is the increase in output?

You can think of this as increasing the number of workers by 3 percentage points. Exact Solution : We can directly calculate this, very hard by hand, but using matlab:. Remember as we have seen, the slope of the tangent line at is similar to the slope of the line between andfrom the definition of derivative, for h small, the following should be true:.

Just move the h from the right to the left, the increase in output is approximately :. Furthermore, the level of output is approximately :. In our case above, we can now approximate output levels using the two numbers we calculated by hand, with and :. Now withthat is something we can use very easily, back to 1st grade math. We calculated previously that ifthe exact new level of output is :. What is our approximated increase that we can calculate by hand? It is. What we have just done is called First Order Taylor Polynomial Approximationwhich can be written more generally as:.

This is just another way to write down the differential formula described at the beginning. When solving economics problems, we often end up with functions that takes too much time to evaluate. To save time, we often approximate functions by the first order taylor approximation. We do this when we are solving for points around a point where we have already evaluated a point where perhaps it is easier to evaluate the function. We just demonstrated this idea using the MPL example here, where we used something we can approximate using 1st grade algebra something that we would need a calculator matlab to compute accurately for us.

Analyze the functional form of MPL, what accurate is the 1st order taylor approximation or differential approximation for the same h increase in L if existing L is high vs if it is low?The Cobb-Douglas production function calculator helps you calculate the total production of a product according to Cobb-Douglas production function.

Briefly, a production function shows the relationship between the output of goods and the combination of factors used to obtain it. The Cobb-Douglas production function is a special form of the production function. It uses the relationship between capital and labor to calculate the number of goods produced. To learn more about the characteristics of the Cobb-Douglas production function, read the article below where you can find more about the production function definition and production function equation.

We've also presented the Cobb-Douglas production function formula, scroll down and check it out! Development of this production function started in the s when Paul Douglas calculated estimates for production factors for labor workers and capital here in a broad sense: money, buildings, machines.

He wanted to show how they relate to each other and he wished to express this relation as a mathematical function. Charles Cobb suggested using an existing production function equation proposed by Kurt Wicksell as a base, which Douglas and Cobb improved and expanded upon. The results they got very closely reflected American macroeconomic data at the time. Although accurate, economists criticized the results for using sparse data.

Even when conducting small scale research you need a proper sample size to make your results statistically significant. It is even more important when you want to try to estimate industry-wide macroeconomic theories.

Over the years, the theory was improved and expanded using US census data, and proved accurate for other countries as well. Paul Douglas formally presented the results in The Cobb-Douglas production function is known for being the first time a proper aggregate production function was estimated and developed to be accurately used to analyze whole branches of industry. It was a cornerstone for macroeconomics, and has been widely used, adopted and improved since its inception. The importance of Cobb-Douglas production function to macroeconomics can be compared to the importance of the Pythagorean theorem to math.

The Cobb-Douglas production function formula for a single good with two factors of production is expressed as following:. Output elasticity is the responsiveness of total production quantities to changes in quantities of a production factor. It is a percentage change in total production resulting from a percentage change in a factor. The more capital or labor we use, the more of a good we are going to get, but it is not a one to one conversion.

In practice they have to be smaller than 1 because a perfect production process does not exist - inefficiencies in labor and capital occur. Output elasticities can be found using historic production data for an industry. Now that you know a little more about Cobb-Douglas production function, its history and the main components, it is time to move on to the Cobb-Douglas production function characteristics:. It means that doubling the amount of both capital and labor would result in double the output.

With the United States industry data available, this is what Paul Douglas observed when he was first establishing the function. Let's assume that A is 2, our labor is 10 and capital Our production in this case would be:.

Multiplying If you are having trouble calculating labor and capital raised by an alpha and beta check out our handy exponent calculator. The proportional change in factors will result in a smaller proportional change in output. Likewise, the proportional change in factors will lead to a higher proportional change in output. In this example, you will see how our Cobb-Douglas production function calculator uses the data you provide to calculate the total production.

Let's say you want to calculate the total production of goods in a particular industry; for example, you are producing glass balls.While discussing the production theory of the firm, economists C. Cobb and P. Douglas used a special form of production function, which is known as the Cobb-Douglas Production Function. Cobb-Douglas C-D production function is of the form. Actually, the parameter A is the efficiency parameter. It serves as an indicator of the state of technology.

## The Solow Growth Model

The higher the value of A, the higher would be the level of output that can be produced by any particular combination of the inputs. They have to do with the relative factor shares in the product. Here it is assumed that the firm uses two inputs, labour L and capital K and produces only one product Q. The C-D production functions possess a number of important properties which have made it widely useful in the analysis of economic theories.

We shall now discuss them. C-D production function 8. For here we obtain. We obtain from 8. Also 8. In that case, 8. On the other hand, if the C-D function is homogeneous of degree one as given by 8. Let us now establish this property. Therefore, if the firm changes the quantities of L and K keeping their ratio unchanged, all these average and marginal products would remain constant. In other words, they can change only when firm changes L and K in different proportions.

That is, if the firm increases the use of one of the inputs, that of the other remaining unchanged, then the AP and the MP of the former input would decrease. Let us establish this property. It is clear from 8. We have already obtained, of course, in 8. We have obtained above that in the case of C-D production function 8. Similarly, from equations 8. We may establish this property in the following way.

Hence, the property iv is established. The C-D production function is. Hence, we have obtained that for the C-D function 8. Again 8.

This implies that an IQ would be convex to the origin. This property may also be established if we use the general version of the C-D function as given by 8. Therefore, for the C-D homogeneous function of degree one 8.For example, variable X and variable Y are related to each other in such a manner that a change in one variable brings a change in the other. The relationship between X and Y can be shown with the help of a formula, which is shown as follows:. In the aforementioned formula, the value of Y can be determined with the help of the given value of X.

Similarly, production function is the mathematical representation of relationship between physical inputs and physical outputs of an organization. In other words, production function represents the maximum output that an organization can attain with the given combinations of factors of production land, labor, capital, and enterprise in a particular time period with the.

It acts as a collection of different production possibilities of an organization. In the words of Prof. If it is presented mathematically, it is called Production Function. It is related with a given state of technological change. The relationship between input and output is represented in the form of table, graph, or equation. The input-output relationship is presented in a quantitative form.

However, the production function has reduced to capital and labor, so that it can be easily understood. Other factors are excluded from the production function due to various reasons. Land and building are excluded because they are constant for aggregate production function. However, in case of individual production function, they are included in capital factor Raw materials are excluded because they represent a constant relationship with the output at all phases of production.

For example steel, tires, steering, and engines used for manufacturing cars explains a constant relationship with the number of cars. The algebraic or equation form of production function is most commonly used to analyze production. Let us understand the algebraic form of production function with the help of an example. Suppose a diamond mining organization has used two inputs capital and labor in the production of diamonds. This production function implies that quantity of diamond production depends on labor engaged in producing diamond and capital required to carry out production.

The production of diamonds would increase with the increase in labor and capital.

On the basis of time period required to increase production, an organization decides whether it should increase labor or capital or both. An organization takes into account either long- run production or short-run production for increasing the level of production. In short-run, the supply of capital is inelastic except for individual organization in perfect competition. This implies that capital is constant. In such a case, the organization only increases labor to increase the level of production.

On the other hand, in the long- run, the organization can increase labor and capital both for increasing the level of production.

Therefore, on the basis of time period, production function can be classified in two types, namely, short-run production function and long-run production function. On the other hand, the long-run production function can be algebraically represented as follows:.

Let us convert the equation of production function into a table of production function with the help of Cobb-Douglas production function. This production function can be used to determine value of Q when the combination of K and L are different.

These values can be represented in the form of a table that is known as tabular form of production function, which is shown in Table In Table-2, it can be seen that there are four combinations of K and L, which are yielding the same value of Q, On joining these four combinations, a curve is drawn known as isoquant. Article Shared by Nitisha. Iso-cost Lines Explained With Diagram.Egwald's popular web pages are provided without cost to users.

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Now I will show how cost functions look when they are obtained from a production function. Recall that a production function produces levels of output for combinations of inputs. A profit maximizing firm will try to use a combination of inputs that will minimize its cost of producing a given level of output. If you have not already done so, look at how the parameters of a Cobb-Douglas production function can be estimated: Estimating a Cobb-Douglas production function.

Production functions need to have certain properties, to ensure that we can solve the least-cost problem: Check any of the many textbooks.

### Total Factor Productivity

If for given values of L,K, and M, the Hessian of the production function f is negative definitethen its isoquants at that point are concave to the origin.

With decreasing returns to scale, a proportional increase in all inputs will increase output by less than the proportional constant. But, these combinations will be more costly at the given factor prices.

If for each feasible amount of product, we compute the cost of producing the product using the cost minimizing combination of inputs, we obtain the cost function, from which the average cost and marginal cost functions can be obtained.

From the graphs, we see that both average cost and marginal cost are increasing, and marginal cost is greater than average cost. Both of these results are the consequence of our Cobb-Douglas production function having decreasing returns to scale. Also, we see that these cost functions don't look like the U-shaped cost functions I used in the oligopoly model. It is always a good idea to look at some numbers, to get an understanding for the beast at hand.

With increasing returns to scale, a proportional increase in all inputs will increase output by more than the proportional constant.

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Our Cobb-Douglas production function might now have the form:. With the same factor prices as before, we compute the cost of producing the product using the cost minimizing combination of inputs, obtaining the cost function, and the average cost and marginal cost functions.

Now we see that both average cost and marginal cost are decreasing, with marginal cost below average cost, a consequence of increasing returns to scale. If we splice together the two cases above, we do get something like the U-shaped average and marginal costs that I used in the oligopoly model. We usually assume that capital is fixed in the short run.

Suppose our firm is to operate efficiently using the cost minimizing combination of inputs producing product in the 25 - 35 unit range using the decreasing returns Cobb-Douglas production function. Because marginal cost is virtually a linear function of q, total cost with capital fixed is virtually a quadratic function of q since its derivative, marginal cost, is linear.

By fixing the amount of input for one factor, we obtain a 2-dimensional isoquant curve. The red rays emanating from the origin in the diagrams intersect each isoquant at the same angle. Consequently, any isoquant is a radial projection of each other isoquant. In particular, any isoquant is a radial projection of the unit isoquant, i. Production functions with this property are called homothetic production functions. The unit cost function c wL, wK, wM looks, interestingly, like its parent â€” the Cobb-Douglas production function.

The Cobb-Douglas production function is called homotheticbecause the Cobb-Douglas cost function can be separated factored into a function of output, q, times a function of input prices, wL, wK, and wM. Check that the Hessian for the function c is negative semi definite. The Hessian, H, of a function, f is negative definite, if the principal minors of H alternate in sign, starting with negative.Total factor productivity TFP is a measure of productivity calculated by dividing economy-wide total production by the weighted average of inputs i.

It represents growth in real output which is in excess of the growth in inputs such as labor and capital. Productivity is a measure of the relationship between outputs total product and inputs i.

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It equals output divided by input. There are two measures of productivity: a labor productivity, which equals total output divided by units of labor and b total factor productivity, which equals total output divided by weighted average of the inputs.

### Free Online Calculators

The most widely used production function is the Cobb-Douglas function which is as follows:. If we rearrange the Cobb-Douglas function, we get the following formula for total factor productivity:. TFP represents the increase in total production which is in excess of the increase that results from increase in inputs.

It results from intangible factors such as technological change, education, research and development, synergies, etc. It is more useful to look at productivity increase over a period instead of the absolute value of total factor productivity.

The following growth accounting equation gives us the relationship between growth in total product, growth in labor and capital and growth in TFP:. We need to isolate the increase in total product that is not explained by the increase in inputs i. You are welcome to learn a range of topics from accounting, economics, finance and more. We hope you like the work that has been done, and if you have any suggestions, your feedback is highly valuable.

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