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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 so-called 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 capital-labour 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 capital-labour 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
- Monotonic function, Convex function, Types of functions