One general version of the theorem consists of two parts.[2][3] The first states that, under the normal assumptions of the Solow and Neoclassical models, if (after some time T) capital, investment, consumption, and output are increasing at constant exponential rates, these rates must be equivalent. Building on this result, the second part asserts that, within such a balanced growth path, the production function, (where is technology, is capital, and is labor), can be rewritten such that technological change affects output solely as a scalar on labor (i.e. ) a property known as labor-augmenting or Harrod-neutral technological change.
Uzawa's theorem demonstrates a significant limitation of the commonly used Neoclassical and Solow models. Imposing the assumption of balanced growth within such models requires that technological change be labor-augmenting. By contraposition, any production function for which it is not possible to represent the effect of technology as a scalar on labor cannot produce a balanced growth path.[2]
Throughout this page, a dot over a variable will denote its derivative with respect to time (i.e. ). Also, the growth rate of a variable will be denoted .
Uzawa's theorem
(The following version is found in Acemoglu (2009) and adapted from Schlicht (2006))
Model with aggregate production function , where and represents technology at time t (where is an arbitrary subset of for some natural number ). Assume that exhibits constant returns to scale in and . The growth in capital at time t is given by
where is the depreciation rate and is consumption at time t.
Suppose that population grows at a constant rate, , and that there exists some time such that for all , , , and . Then
1. ; and
2. There exists a function that is homogeneous of degree 1 in its two arguments such that, for any , the aggregate production function can be represented as , where and .
Since and are constants, is a constant. Therefore, the growth rate of is zero. By Lemma 1, it implies that
Similarly, . Therefore, .
Next we show that for any , the production function can be represented as one with labor-augmenting technology.
The production function at time is
The constant return to scale property of production ( is homogeneous of degree one in and ) implies that for any , multiplying both sides of the previous equation by yields
^Uzawa, Hirofumi (Summer 1961). "Neutral Inventions and the Stability of Growth Equilibrium". The Review of Economic Studies. 28 (2): 117–124. doi:10.2307/2295709. JSTOR2295709.