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Uzawa's theorem

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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.[1]

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]

Statement

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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 .

Sketch of proof

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Lemma 1

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For any constant , .

Proof: Observe that for any , . Therefore, .

Proof of theorem

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We first show that the growth rate of investment must equal the growth rate of capital (i.e. )

The resource constraint at time implies

By definition of , for all . Therefore, the previous equation implies

for all . The left-hand side is a constant, while the right-hand side grows at (by Lemma 1). Therefore, and thus

.

From national income accounting for a closed economy, final goods in the economy must either be consumed or invested, thus for all

Differentiating with respect to time yields

Dividing both sides by yields

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

Note that because (refer to solution to differential equations for proof of this step). Thus, the above equation can be rewritten as

For any , define

and

Combining the two equations yields

for any .

By construction, is also homogeneous of degree one in its two arguments.

Moreover, by Lemma 1, the growth rate of is given by

.

References

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  1. ^ 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. JSTOR 2295709.
  2. ^ a b Jones, Charles I.; Scrimgeour, Dean (2008). "A New Proof of Uzawa's Steady-State Growth Theorem". Review of Economics and Statistics. 90 (1): 180–182. doi:10.1162/rest.90.1.180. S2CID 57568437.
  3. ^ Acemoglu, Daron (2009). Introduction to Modern Economic Growth. Princeton, New Jersey: Princeton University Press. pp. 60-61. ISBN 978-0-691-13292-1.