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Unformatted text preview: Chapter 3. A Production Function of Central Importance In 1961, Kenneth Arrow, Hollis Chenery, Bagicha Minhas and Robert Solow made a major discovery. Not only did it modify forever the way economists looked at production functions, but it also had a considerable in&uence on another ¡eld of research, namely utility theory. We will explain here how the socalled constant elasticity of substitution (C.E.S.) production function was found. We will start with the concept of the elasticity of sub stitution, and then turn to the derivation of the CES function itself and its main properties. 1 Motivation Throughout history, the rental price of capital has decreased while the wage rate has increased. At the same time the ratio capitallabour has increased. For any given, ¡xed, level of production there is also a positive relationship between the wage/rental ratio and the capital/labour ratio. The elasticity of substitution is the elasticity of the latter relationship. We then need to recall ¡rst the concept of factor demand because the capitallabour ratio is precisely the ratio of the demand functions of capital and labour. Suppose that to the use of capital during a given time span cor responds a price that we will call the rental rate, denoted q ; the measurement units of this rental rate is $ per unit of capital, per unit of time. So in units we have: q = ¢dollars£ ¢unit of capital£ : ¢unit of time£ & The use of labor costs the wage rate w , expressed in $ per unit of labour, per unit of time. Also in units we have: w = ¢dollars£ ¢unit of labour£ : ¢unit of time£ & The cost of using both factors in quantities K and L is thus C = qK + wL . 1 Suppose now that the &rm, or society, wants to use those rare resources in the most e¢ cient way. An e¢ cient way may have di/erent meanings. For instance, it could be minimizing the cost of production subject to a given production level; or it could mean the ¡dual problem£of maximizing output subject to a cost constraint. Let us see what the implications of the &rst of these cases are. Minimizing the cost of production with respect to a production level Y leads to writing a Lagrangian L = qK + wL + & [ Y & F ( K; L )] where F ( K; L ) is a concave function in ( K; L ) . Setting to zero the gradient of L , we have: @ L @K = q & & @F @K ( K; L ) = 0 (1) @ L @L = w & & @F @L ( K; L ) = 0 (2) Y = F ( K; L ) (3) The above system of 3 equations in the three unknowns K; L and & can generally be solved in K; L and & to yield the factor demand functions K and L , which depend on the parameters Y ; w and q 1 . So we can derive the functions K = K ( Y ; w; q ) L = L ( Y ; w; q ) : An example may be helpful. Suppose that Y = K & L 1 & & . With @F @K = ¡K & & 1 L 1 & & and @F @L = (1 & ¡ ) K & L & & , equations (1), (2) and (3) are: q & &¡K & & 1 L 1 & & = 0 (4) w & & (1 & ¡ ) K & L & & = 0 (5) Y = K & L 1 & & (6) Eliminating & between (4) and (5), and using (6), we get the demand func tions 1 Note that the set of equations (1)(3) plus the convexity of the Lagrangian ensures...
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 Fall '08
 OLIVIERDELAGRANDVILLE
 Monotonic function, Convex function, Types of functions

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