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Part 1 / Section 1: Labor and Capital as Aggregate Inputs: Substitutability Section 1.1

The neoclassical production function

We start from the most general description of the production technology, embodied in the production function,

\[\begin{equation} Y=zF(K,L), \tag{1.1} \end{equation}\]

where \(Y\) is output, \(K\) is aggregate capital, \(L\) is aggregate labor, and \(z>0\) is total factor productivity. The function \(F\) describes how physical inputs become output; \(z\) measures how effective the entire production process is. Later on we will introduce factor-augmenting productivity that affects a given input directly.

Formally, let \(F:\R_+^2\rightarrow\R_+\) be twice continuously differentiable and satisfy

  1. Productive inputs: Holding the other input fixed, more capital or labor raises output.

    \[ F_K(K,L)>0, \qquad F_L(K,L)>0. \]

  2. Marginal products diminish: Adding more machines or workers increases output, but ever less so.

    \[ F_{KK}(K,L)<0, \qquad F_{LL}(K,L)<0. \]

  3. Constant returns to scale: Doubling all inputs doubles output.

    \[ F(\lambda K,\lambda L)=\lambda F(K,L), \qquad \lambda>0. \]

  4. The production function is jointly concave. We need this assumption to ensure the firm's problem is well posed mathematically.

Constant returns to scale and Euler's theorem

The fact that the production function has constant returns to scale has a very useful consequence that follows from Euler's Theorem for Homogeneous Functions: The production function can be expressed in terms of the marginal product of capital and labor.

\[\begin{equation} \text{Euler's Theorem:}\quad F(K,L)\;=\;F_K(K,L)K\,+\,F_L(K,L)L. \tag{1.2} \end{equation}\]

The reason this identity is so useful is that when markets are competitive we can express output in terms of the cost of inputs. In a competitive market, factors receive the value of their marginal products, so \(\,r\,=\,pzF_K\,\) and \(\,w\,=\,pzF_L\). So, if output sells at price \(\,p\,\) we can write

\[\begin{equation} rK\,+\,wL \;=\; pzF_K(K,L)K\,+\,pzF_L(K,L)L \;=\; pz\left(F_K(K,L)K\,+\,F_L(K,L)L\right) \;=\; pY \tag{1.3} \end{equation}\]

Thus constant returns imply that all revenue is paid to factors according to their marginal products in competitive markets. In other words, profits are zero when the production function has constant returns to scale. So, the real question is not about profits, but about how labor and capital are remunerated. Going forward we will focus on how the firm chooses inputs, how this choice is affected by input prices, and how inputs are remunerated.