Notes on labor demand under a cobbdouglas technology. There are several classes of utility functions that are frequently used to generate demand functions. Another common form for utility is the constant elasticity of substitution ces utility function. Another monotonic transformation of cobb douglas is shown in of her income on good 2. Also, it would imply that you somehow measured the utility that is, selecting a unit or a measure for the utility, sometimes this can be in monetary units or more complex ideas deriving from subjective utility measures. Similarly, it is not necessarily the case that a macro cobb douglas applies at the disaggregated level. Notice how neither fraction depends on income m or the prices of the two goods, p 1 and p 2. Without wanting to labor the point too much, suppose in this example. X r is called a utility function representing the preference relation i. The cobb douglas utility function is a special case where this income. I take a cobb douglas utility function in general functional form and solve for the utility maximizing consumption bundle, along with the demand functions. Curves the collection of all indifference curves for a given preference relationcurves for a given preference relation is an indifference map.
If dd 12 1, that is if the cobb douglas function exhibit constant returns to scale, then 26 and 28 can be simplified as 12 2. Utility functions and dimensional analysis now we look at some utility functions. We can also solve this by transforming the utility function to a cobbdouglas. In turn, a utility function tells us the utility associated with each good x 2 x, and is denoted by ux 2 utility function ux represents an agents preferences if ux uy if and only if x utility function. Is y in the first problem the same as m in the second. A macroeconomic production function is a mathematical expression that describes a systematic relationship between inputs and output in an economy, and the cobb douglas and constant elasticity of substitution ces are two functions that have been used extensively. The function f is a convex function on u if and only if d2fx is positive semide.
Cobbdouglas production function differentiation example. Solving the utility maximization problem with ces and cobb. The cobbdouglas function is widely used to represent production functions and also utility func tions in economics. Students could increase their skills by using a wider range of utility functions. Dimensional analysis of production and utility functions in. It is part of a larger category called constant elasticity of substitution ces utility functions. By deriving the first order conditions for the emp and substituting from the constraints u h 1 p, u, h 2 p, u u, we obtain the hicksian demand functions. Consumer utilitymaximizing outcome given a cobb douglas utility function.
The cobb douglas function is used to represent consumers preferences or the production function of a firm. The engel curve then describes how the demand changes with income 5. The problem is maximizing utility ux 1,x 2 x 1 25x 2 35 subject to nonnegativity constraints x 1. The utility function that produced the demand function x. P x p y 23 properties of hicksian demand functions. In order for the cobb douglas function to meet this condition, coefficients dd12,0t have to satisfy the condition dd 12 d1. Concave and quasiconcave functions concave and convex functions theorem 21. Finally, in order to derive the cobb douglas and leontief functions we need lhopitals rule. The cobb douglas form was developed and tested against. Assume the cobb douglas utility function to be 1 uaxx12. Cobb douglas utility cobb douglas utility functions n r2, for two goods x,y are given by the general formula u xay1 a the function can be shown to be.
Another monotonic transformation of cobb douglas is shown in notes 7, at the top of p. Based on the structure of preferences and the nature of the optimizationproblem, they will have certain properties which we will discuss shortly. As an example, consider the cobb douglas utility function, ux yn i1 x i i x 1 1 x 2 2 x 3 3 7 with i 0 and p n i1 i 1. The cobb douglas function is often used as a utility function. Apparently a 0 0 issue arises when one evaluates uas 1. We consider a consumer with cobb douglas preferences. Dimensional analysis of production and utility functions. In a world where there are two commodities, the cobb douglas utility function for an individual is expressed.
The cobbdouglas production function 1 introduction in general, a productionfunctionis a speci. This is a special case of the cobb douglas utility function, which has the form. He is aware that in econ 101a we only use two goods, but he is too attached to all of them to let go of one. The lagrangian for the utility maximization problem is 12 ,x x x p x p x moo. For example, if the utility function is u xy then mrs y x this is a special case of the cobb douglas utility function, which has the form. Measuring consumer preferences and estimating demand. For example, if bundle a and bundle b are distinct bundles and the consumer. In turn, a utility function tells us the utility associated with each good x 2 x, and is denoted by ux 2 utility function ux represents an agents preferences if ux uy if and only if x utility function ux. Some aspects of the translog production function estimation. Many identical households choose goods x and y to maximize utility subject to income i. Note that this function is similar to a cobb douglas function, if you take expux q l l1 x l l l. Cobbdouglas production function handout jae wook jung. The cobb douglas utility function is a special case where this \income e ect exactly cancels out the substitution e ect, so the consumption of one good is independent of the price of the other goods.
Pdf the substitution and the revenue effects for a cobb. L l with 0, continuous, strictly monotone, and strictly. These functions play an important role in the economic forecasts and policy. P x net substitutes if 0,complementsif understanding. Q b0 x y b1 b2 the cobb douglas functional form of production functions is widely used to represent the relationship of an output and two inputs. With cobb douglas, quantity demanded of each good is a function. First, we look at cobb douglas utility function, as in prescott 1986.
Note that with that transformation, we get the utility function. At the end, the paper presents a simple applied exam. In the second example, are a and b the exponents in the utility function x1 and x2. As illustrated in the twofactor example shown in figure a2. The mrs and the cobbdouglas simon fraser university. Properties of the cobb douglas production function. Review of microeconomics algebraic formulation of mrs in. Pdf we provide characterizations of preferences representable by a cobb douglas utility function. The readers are referred to 6 and 8 for the justification of the use of cobb douglas utility functions in reallife problems, to 1 for the use in economic growth theory, and to 15 for a. The translog production functions represent in fact a class of flexible functional forms for the production functions ch. One of the main advantages of the respective production function is that, unlike in case of cobb douglas production function, it does not. For example, if mux muy 1 and the price of x is twice that of y, she could give. Pdf from preferences to cobbdouglas utility researchgate. Msc in economics for development macroeconomics for development.
An assessment of ces and cobbsdouglas production functions. Estimating the utility function of the cobbdouglas type will require data of a set of goods x and y in this case and the utility. While there is a literature containing examples of utility functions where one good is an. Demand is a function of prices and income so is x 1 p 1,p 2, m x 2 p. We use utility functions to derive demand curve, by choosing the mix of goods that maximises utility subject to a budget constraint 4. Problems with solutions, intermediate microeconomics, lecture 12. We consider a consumer with cobbdouglas preferences. In economics and econometrics, the cobb douglas production function is a particular functional form of the production function, widely used to represent the technological relationship between the amounts of two or more inputs particularly physical capital and labor and the amount of output that can be produced by those inputs. A person likes one bundle twice as much as another if they are willing to pay twice as much for it utility function level set. The cobbdouglas function is widely used to represent production functions and also utility functions in economics. Review of microeconomics algebraic formulation of mrs.
In this case the marginal rate of substitution for the cobb douglas utility function is mrs. Given the basic form of the cobbdouglas production function, well find the partial derivatives with respect to capital, k, and labor, l. These functions are called marshallian demand equations. On the cobbdouglas production function division of the. In this section, the cobb douglas form is simply derived as an algebraic transformation of the identity. A cobbdouglas utility function in what follows, we assume that the utility function takes the form of a cobb douglas function. The cobbdouglas utility function is a special case where this income. Utility maximization with a cobbdouglas utility function. One of the most common is the cobb douglas utility function, which has the form ux, y x a y 1 a. What follows is a brief overview of the four types of utility functions you havewill. Recall from 103 that elasticity is the ratio of two variables. Then, f is a concave function on u if and only if the hessian d2fx is negative semide. An early microfoundation of the aggregate cobb douglas technology based on linear activities is derived in houthakker 1955.
In the 1920s the economist paul douglas was working on the problem of relating inputs. Measuring consumer preferences and estimating demand systems. Another monotonic transformation of cobb douglas is. The cd production functions possess a number of important properties which have made it widely useful in the analysis of economic theories. It is based on the statistical observation that \the division of national income between capital and. It is based on the statistical observation that \the division of national income between capital and labor have been roughly constant over time. If we assume that the consumer has cobb douglas utility function over the two goods. With very large and increasing per capita income, ultimately the utility function becomes indistinguishable from cobb douglas. One of the common assumption in economics is that utility function needs to be quasiconcave. Note that they depend on the prices of all good and income. Proving that a cobbdouglas function is concave if the sum of exponents is no bigger than 1. The utility function is cobb douglas cd in all the problems that follow. Some basic microeconomic concepts applied to a cobb douglas utility function. Setting up the lagrangian for this problem, we get the following rst order conditions i x i yn i1 x i i p i 0, i 1.
Msc in economics for development macroeconomics for. Consider the following two opinions, both of which can be found in the literature of consumer demand systems. Try to draw the indi erence curves, they will look similarly to those of cobb douglas but the asymptotes will now be in x 1 1. Appendix 2 modified cobbdouglas utility function in this paper. For a math example, imagine that a person has the cobb douglas utility function u x0. This concept can be applied at the level of individual. Aidiff i i lan indifference map is equivalent to a utility function. Note that 1 gives a keyimplication of cobb douglas utility on optimal consumption. Maximize utility subject to budget constraint and solve for endogenous variables as a function of the parameters. Cobb douglas preferences are easy to use and therefore commonly used. Since were doing macroeconomics we will be considering. The paper makes an analysis between the two effects, considering the general case of a cobbdouglas utility function and introduces three indicators which will characterize these shares. Nov 04, 20 during 19001947, charles cobb and paul douglas formulated and tested the cobb douglas production function through various statistical evidence. Utility maximization and choice university at albany, suny.
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