Uzawa's theorem
Uzawa's theorem, also known as the steady state growth theorem, is a theorem in economic growth theory concerning the form that technological change can take in the Solow–Swan and Ramsey–Cass–Koopmans growth models. It was first proved by Japanese economist Hirofumi Uzawa. One general version of the theorem consists of two parts. 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, formula_1 (where formula_2 is technology, formula_3 is capital, and formula_4 is labor), can be rewritten such that technological change affects output solely as a scalar on labor (i.e. formula_5) 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. Statement. Throughout this page, a dot over a variable will denote its derivative with respect to time (i.e. formula_6). Also, the growth rate of a variable formula_7 will be denoted formula_8. Uzawa's theorem Model with aggregate production function formula_9, where formula_10 and formula_11 represents technology at time t (where formula_12 is an arbitrary subset of formula_13 for some natural number formula_14). Assume that formula_15 exhibits constant returns to scale in formula_3 and formula_4. The growth in capital at time t is given by formula_18 where formula_19 is the  depreciation rate and formula_20 is consumption at time t. Suppose that population grows at a constant rate, formula_21, and that there exists some time formula_22 such that for all formula_23, formula_24, formula_25, and formula_26. Then 1. formula_27; and 2. For any formula_28 , there exists a function formula_29 that is homogeneous of degree 1 in its two arguments, such that the aggregate production function can be represented as "formula_30", where formula_31 and formula_32. Sketch of proof. Lemma 1. For any constant formula_33, formula_34. Proof: Observe that for any formula_35, formula_36. Therefore, formula_37. Proof of theorem. We first show that the growth  rate of investment formula_38 must equal the growth rate of capital formula_39 (i.e. formula_40) The resource constraint at time formula_41 implies By definition of formula_43, formula_44 for all formula_23 . Therefore, the previous equation implies for all formula_23. The left-hand side is a constant, while the right-hand side grows at formula_48 (by Lemma 1). Therefore, formula_49 and thus From national income accounting for a closed economy, final goods in the economy must either be consumed or invested, thus for all formula_41 Differentiating with respect to time yields Dividing both sides by formula_54 yields Since formula_57 and formula_58 are constants , formula_59 is a constant. Therefore, the growth rate of formula_59 is zero. By Lemma 1, it implies that Similarly, formula_62. Therefore, formula_63. Next we show that for any formula_23, the production function can be represented as one with labor-augmenting technology. The production function at time formula_65 is The constant return to scale property of production (formula_15 is homogeneous of degree one in formula_3 and formula_4) implies that for any formula_23, multiplying both sides of the previous equation by formula_71 yields Note that formula_73 because formula_74(refer to solution to differential equations for proof of this step). Thus, the above equation  can be rewritten as For any formula_23, define and Combining the two equations yields By construction, formula_81 is also homogeneous of degree one in its two arguments. Moreover, by Lemma 1, the growth rate of formula_82 is given by