Examples and exercises on isoquants and the marginal rate of technical substitition isoquants for a fixed proportions production function consider the fixed proportions production function f z 1, z 2 minz 1,z 2. Matehmatically, the cobb douglas production function can be represented as. In economics, a production function is an equation that describes the relationship between input and output, or what goes into making a certain product, and a cobbdouglas production function is a specific standard equation that is applied to describe how much output two or more inputs into a production process make, with capital and labor being the typical inputs described. Deriving shortrun cost functions from a cobbdouglas. An assessment of ces and cobbsdouglas production functions. The original formulation of the cobb douglas production function is. In a cobb douglas production function, the law of diminishing returns does not apply to a variable input if. The cobb douglas production function is still today the most ubiquitous form in theoretical and empirical analyses of growth and productivity. Exponential functions of this sort are commonly used by economists to describe production functions. Constant elasticity of substitution production function and 4. Cobb douglas suppose that the production function is q fz1,z2 z1az2b a,b 0 returns to scale. Therefore, before we explain the production function with two variable factors and returns to scale, we shall explain the concept of.
Isoquants contour lines that connect points with same in k,l space. In other words, log y is a linear function of log x1 and log x2. If a production function f 2is a monotonic transformation of another production function f 1then they represent different technologies. Applications of cobb douglas production function in construction timecost analysis. For the period 19952005, the authors do not find significant difference between the calculation of the supply side of the czech economy by the cobb douglas production function and a more. With a single output and input, a cobbdouglas production function has the. Pdf the paper treats various aspects concerning the cobbdouglas.
On the one hand were highlighted conditions for the existence of the cobb douglas function. The estimation of the parameters of aggregate production functions is central to much of todays work on growth, technological change, productivity, and labor. Hence, the cobbdouglas production function has an elasticity of substitution. For the analysis of production function with two variable factors we make use of the concept called isoquants or iso product curves which are similar to indifference curves of the theory of demand. Isoquants are a basic tool for analyzing the choice of inputs by competitive firms. Q f l, k it is also called as production with two variable factor inputs, labour l and capital k in particular. In economics, the cobbdouglas functional form of production functions is widely used to represent the relationship of an output to inputs. Under what circumstances will the production function exhibit a decreasing b constant c increasing returns to scale. This set is shown in the following figure, together with the isoquant for the output 2. The production function is the central part of production theory and as such there is a theoretical.
Zhigang shen cobb douglas production function was widely used in economics and productivity studies across many sectors. This is a particular example of a multiple inputs example 3 production function with diminishing returns example 2. The cobb douglas production function 173 the resulting equation is referred to as linear in the parameters or linear in the coefficients. As seen earlier, the sum of exponents of cobb douglas production function is equal to one. In 1928, cobb and douglas introduced a famous twofactor production function, nowadays called cobb douglas production function, in order to describe the distribution of the national income by help of production functions. Ces production function isoquants comparison of cobb douglas and ces cobb from agricultur ec 741 at egerton university. Notes on labor demand under a cobbdouglas technology r. Examples and exercises on isoquants and the marginal rate. This production function is of the cobbdouglas form. Suppose the production of digital cameras is characterized by the production function q lk, where q represents the number of digital cameras produced. Without doubt the most celebrated production function is the cobb douglas function. As with the linear production function, a and b represent the output elasticities of k and l. Now consider the fixed proportions production function f z 1, z 2 minz 1 2,z 2, which models a technology in which 2 units of input 1 and 1 units of input 2 are required to produce every unit of output.
The range of the function consists of each output level y that results from each level of input x being used. It was proposed by knut wicksell 18511926, and tested against statistical evidence by charles cobb and paul douglas in 19001928. Hence, the cobb douglas production function has an elasticity of substitution. They are called cobbdouglas functions, though the termis. The figure given below captures how the production function looks like in case of increasingdecreasing and constant returns to scale. Does production function 1 have decreasing, constant, or increasing returns to scale. The transformed function is the equation for a simple two variable regression line in which all observations in the data set used. One example of such function is the famous cobbdouglas production function. As we know, a production function explains the functional relationship between inputs or factors of production and the final physical output. The paper treats various aspects concerning the cobbdouglas production function. Ces production function isoquants comparison of cobb. Isoquants an isoquant shows the combinations of z1 and z2 that can produce a given level of output q0. Isocosts, isoquants, isocline lines, and scale lines for. We let the production technology in our model be represented by a cobb douglas production function y.
An isoquant curves that describe all the combinations of inputs that produce the same level of output. Monotone transformations contrary to utility functions, production functions are not an ordinal, but cardinal representation of the firms production set. L 1, where k measures aggregate capital and l aggregate labor in efficiency units. If we have s 1, then a 10% change in mrts will yield a 10% change in the input mix. However, these functions allow for variable elasticity of substitution. The function s quantitative modeling of resource inputs and. An isoquant is the implicit function relating l and k for a given q. This video derives all seven of the firms shortrun cost functions from a cobbdouglas production function. The cd production functions possess a number of important properties which have made it widely useful in the analysis of economic theories. If the production function is denoted by p pl,k, then the partial derivative. If the function f is quasiconcave the upper contour or isoquants are convex. Economists often work with homogeneous production functions.
To show how the isoquant curve is truly threedimensional, the demonstration also displays the production function as a 3d surface plot. Isoquants are linear with a slope equal to the negative of the constant mrts, which is here. In economics and econometrics, the cobbdouglas production function is a particular functional form of the production function, widely used to represent the. Using lagrange multipliers, this demonstration allows you to manipulate all aspects of both the cobb douglas and isocost functions while viewing how this affects the allocation of labor and capital for optimal production. A commonly discussed form of long run production function is the cobb douglas production function which is an example of linear homogenous production functions. The 1isoquant is the set of all pairs z 1, z 2 for which f z 1, z 2 1, or minz 1,z 2 1. Oaxaca university of arizona 1 cobb douglas production function q aegtl. This indicates that the capital and labor inputs are not perfect substitutes for each other. Cobb douglas production function is used in obtaining marginal rate of technical substitution the rate at which one input can be substituted for the other to produce same level of output between two inputs. To illlustrate production functions, isoquants, and total product curves. The longrun production function involving the usage of two factors say, capital and labour is represented by isoquants or equal product curves or production indifference curves. There can be input substitution, but it is not linear.
It owes part of its name to professor paul douglas who, from empirical observations, inferred its properties, and part to his colleague cobb, a mathematician, who suggested the mathematical form. Isoquant maps for production functions with different elasticities. Problem set 2 fe312 fall 2011 rahman page 3 of 5 2 suppose that the production function is cobb douglas. Economists call it the marginal production with respect to labor or the marginal productivity of labor. An assessment of ces and cobb douglas production functions 1 eric miller email. L is the rate at which production changes with respect to the amount of labor. Cobb douglas production functions is a functional relationship between input and out put. Production functions see chap 9 2 production function. The production function describes also the set of inputs vectors l,k that allow to produce a.
The production function tells us the maximum amount of production. Thus the elasticity of substitution of a constant returns to scale production function can be expressed as the elasticity of output per capita with respect to the marginal product of labor. This section will discuss the discovery of the production formula and how partial derivatives are used in the cobbdouglas model. Understanding the leontief production function lpf. This specification creates isoquants that are gently convex and easy to work with.
In the cobb douglas production function the elasticity of. The cobb douglas production function is widely used in economic models. F k, l so, this production function exhibits increasing returns to scale. All values of x greater than or equal to zero constitute the domain of this function. Twofactor cobb douglas production function is given by where denotes the labor input, is the capital input, is the total. They are called cobbdouglas functions, though the termis sometimes reserved for the special case in which the exponents sum to one. Variable elasticity substitution production function.
The isoquants in the cobb douglas production function are curved with the slope changing along them. Again, since production function 1 is a cobb douglas production function we can simply add the exponents together. Notes on labor demand under a cobbdouglas technology. Chapter 7 production functions turning inputs into outputs.