From the SelectedWorks of Jürgen Antony 2010 On the Dynamic Implications of the Cobb- Douglas Production Function Jürgen Antony, CPB Netherlands Bureau for Economic Policy Analysis Available at: https://works.bepress.com/antony/7/
On the Dynamic Implications of the Cobb-Douglas Production Function Jürgen Antony CPB Netherlands Bureau for Economic Policy Analysis February 27, 2010 Abstract This note is concerned with an economy employing capital and labor as production factors. The capital stock is accumulated by foregone consumption. It shows that under perfect competition, the Cobb Douglas production function fulfills the necessary and sufficient conditions of maximizing at the same time the present values of consumption expenditure out of labor income and out of capital income. Keywords: Cobb-Douglas production function, dynamic optimization, consumption and wealth JEL Classification Number: E21, E23, C61, CPB Netherlands Bureau for Economic Policy Analysis, Van Stolkweg 14, 2585 JR The Hague, The Netherlands, Phone: +31(0)70-3383451, Fax: +31(0)70-3383350, e-mail: j.antony@cpb.nl. 1
1 Introduction This note demonstrates the dynamic implications of the Cobb Douglas production function for an economy using labor and capital as input factors in production. In particular it is shown that the Cobb Douglas production function, combined with perfect competition, fulfills the necessary and sufficient conditions for a twofold maximization problem. This problem can be seen as maximizing at the same time the present values of consumption expenditure out of labor income and out of capital income when capital is accumulated by foregone consumption. This contribution is closest related to the analysis in de La Grandville (2008, chap. 15) who demonstrates in his proof of Adam Smith s invisible hand conjecture that an interest rate given by the marginal product of capital maximizes the present value of consumption expenditure in the total economy. The present contribution goes a step further in treating consumption out of labor and capital income separately. The necessary and sufficient conditions for maximizing these two present values are developed, and second, it is shown that perfect competition combined with a Cobb-Douglas technology fulfills them. A second relation to the existing literature might be seen in the implied rational for the Cobb Douglas production technology if it were desired to maximize the present value of consumption expenditures. This would give a underlying framework justifying the use of a Cobb-Douglas technology. There are some other, although distinct, attempts in the theoretical literature to justify the use of a Cobb Douglas production technology. Houthakker (1955-1956) provides an aggregation result showing that a Cobb-Douglas production function can be obtained by aggregating over Leontief production functions in capital and labor. Jones (2003, 2005) gives an argument related to the distributional properties of labor and capital augmenting technologies providing a long run production function that is Cobb Douglas. However, these approaches are quite distinct from the present one, having in common only a possible economic rational for the Cobb-Douglas production technology. The next section states the basic assumptions underlying the economy and formulates the intertemporal maximization problems. Section three provides the necessary 2
and sufficient conditions for an optimum and shows that the Cobb-Douglas production function fulfills them. Finally, the last section concludes. 2 Assumptions It is assumed that the economy produces output using labor and capital with a constant returns to scale technology to produce output 1 Y t = F (L t, K t ), where L t is labor input at time t and K t is the economy s capital stock. Since F ( ) is homogenous of degree one Y t = L t F (1, K t /L t ) = L t f(k t ) with k t = Kt L t the capital stock per worker and f(k t ) per capita production. Without loss of generality L t is normalized to one. Output can be used for consumption or can be foregone to accumulate capital according to k t = f(k t ) c t, where k t = kt t and c t is total consumption expenditure. The economy is assumed to be further characterized by perfect competition. The price for one unit of output is equal to its marginal cost which is normalized to 1. The wage and the rental price of capital equal the marginal product of labor and capital, i.e. r t = f (k t ), w t = f(k t ) f (k t )k t, where f (k) = f(k) k economy wide savings rate s t =. In the following it will be useful to use the definitions of the k t f(k, the capital share π t) t = f (k t)k t f(k t) 1 Output denotes production net of any capital depreciation. and the labor 3
share 1 π t. Consumption out of labor income is defined as c l t = (1 s t )w t = (1 π t ) [f(k t ) k ] t and consumption out of capital income as c k t = (1 s t )r t k t = [ π t f(k t ) k t ]. The two stated intertemporal maximization problems read therefore as V V l,0 = max k = max k k,0 = max k = max k 0 0 0 0 c l te R t 0 i(τ)dτ dt (1 π t ) c k t e R t 0 i(τ)dτ dt [f(k t ) k t ] e R t 0 i(τ)dτ dt, (1) π t [ f(k t ) k t ] e R t 0 i(τ)dτ dt, (2) where k denotes the whole trajectory of capital between time 0 and infinity and i(τ) is the rate of interest at time τ. It is important to note that we are searching for a solution with respect to the path of the capital stock and not with respect to a specific production function. Indeed, as the problem is formulated above, we are searching for a solution given a particular production function. It will be important to keep this in mind when interpreting the results below. Further, the interest rate is not per definition equal to the (net) marginal product of capital. However, the results below will show that the solution of the optimization problems will imply such an equality. 3 Necessary and Sufficient Conditions for an Optimum Denote the integrands of (1) and (2) as G l (k, k, t) and G k (k, k, t) respectively. The necessary conditions for a solution of (1) and (2) are then G l k d G l dt k G k k d G k dt k = 0, = 0, 4
which are the two Euler equations corresponding to the two problems above. These conditions give after computing the derivatives [ f (k t )k t (1 π t )i(t) ] e R t 0 i(τ)dτ = 0, [ f (k t )k t + f (k t ) π t i(t) ] e R t 0 i(τ)dτ = 0, where f (k t ) = f (k t) k t. These conditions can be rewritten as i(t) = f (k t )k t 1 π t, (3) i(t) = f (k t )k t + f (k t ) π t. (4) For these to be also sufficient conditions for a maximum, G l (k, k, t) and G k (k, k, t) have to be globally concave with respect to k and k (see e.g. Chiang 1992, chap. 4). This is e.g. the case when 2 G l k 2 0, 2 G l k 2 analogously for G k (see e.g. de La Grandville 2008). = 0 and 2 G l k 2 Turning now to the Cobb-Douglas case, production is given by 2 G l k 2 2 G l = 0 and k k f(k t ) = Ak α t, where A is a positive constant and 0 < α < 1. f (k t ) = αak α 1 t this in (3) and (4) gives It can easily be verified that, f (k t ) = α(1 α)ak α 2, π t = α and 1 π t = 1 α. Inserting t i(t) = αak α 1 t, (3 ) i(t) = αak α 1 t. (4 ) In addition, the concavity conditions above are all satisfied 2. Thus the necessary and sufficient conditions for a maximum of the present value of consumption out of 2 G k k 2 t 2 For G l we have 2 G l k 2 t = α 2 (1 α)k te R t 0 i(τ)dτ < 0, 2 G k k 2 = α(1 α) 2 k te R t 0 i(τ)dτ < 0, 2 G l k 2 = 0 and 2 G k = 0. k k 5 = 0 and 2 G l k k = 0. For Gk we have
labor and capital income are always fulfilled if production is Cobb-Douglas and the interest rate is given by the marginal product of capital. 4 Conclusion We just have proven that an economy with a Cobb-Douglas technology and a perfectly competitive environment always fulfills the necessary and sufficient conditions for maximizing the present value of consumption expenditures out of labor and capital income. However, this does not imply that an economy has to chose the Cobb-Douglas technology in order to achieve this. This is because, the necessary conditions (3) and (4) do not say anything about how to choose f(k t ). Instead, they provide conditions on how to chose the trajectory of k t in order to solve the two Euler equations. In particular these equations say that the trajectory should be chosen in such a way that the elasticity of substitution between capital and labor f (k t ) [f(k t ) f (k t )k t ] / [f (k t )k t f(k t )] is unity for all t (to see this one has just reinsert (3 ) in (3) or (4 ) in (4). It just happens that this is always the case with the Cobb-Douglas function. Additionally, in the Cobb-Douglas case it is easy to show that the concavity requirements are globally fulfilled. Given an arbitrary production function f(k t ), both objectives might be maximized if for just one value for k t for every t a unit elasticity is attainable and provided that the concavity requirements would be fulfilled by this function. Natural candidates might be certain variable elasticity of substitution production functions which need not to have nice formal representations. An interesting observation is also that following these arguments, the equation defining a constant elasticity of substitution of unity can be seen as an Euler equation related to the above two maximization problems. For future research it might be interesting to interpret the equation defining an arbitrary constant elasticity of substitution σ as an Euler equation as well. The main task would be to find the problem which is solved by 6
σf (k t )k t f(k t ) + f (k t ) [ f(k t ) f (k t )k t ] = 0. This might give further important insights in the meaning of the elasticity of substitution and maybe why we observe empirically specific values for it. Recent empirical evidence (e.g. Duffy and Papageorgiou 2000) point to an elasticity of substitution between capital and labor that is increasing in the stock of capital (per worker). In the light of the arguments given in the present paper, this points to a direction of development towards a maximization of consumption present values. Also this is of importance when interpreting this empirical finding. 7
Literature Chiang, A. C., 1992, Elements of Dynamic Optimization, McGraw-Hill. de La Grandville, O., 2008, Economic Growth - A Unified Approach, Cambridge University Press. Duffy, J. and C. Papageorgiou, 2000, A Cross-Country Empirical Investigation of the Aggregate Production Function Specification, Journal of Economic Growth 5, 87-120. Houthakker, H. S., 1955-1956, The Pareto Distribution and the Cobb-Douglas Production Function in Activity Analysis, Review of Economic Studies 23, 27-31. Jones, C. I., 2003, Growth, Capital Shares, and a New Perspective on Production Functions, University of California, Berkeley. Jones, C. I., 2005, The Shape of Production Functions and the Direction of Technical Change, Quarterly Journal of Economics 120, 517-549. 8