Uzawa's theorem

Uzawa's theorem, also known as the steady-state growth theorem, is a theorem in economic growth that identifies the necessary functional form of technological change for achieving a balanced growth path in the Solow–Swan and Ramsey–Cass–Koopmans growth models. It was proved by Japanese economist Hirofumi Uzawa in 1961.^[1]

A general version of the theorem consists of two parts.^[2][3] The first states that, under the normal assumptions of the Solow-Swan and Ramsey models, if capital, investment, consumption, and output are increasing at constant exponential rates, these rates must be equivalent. The second part asserts that, within such a balanced growth path, the production function, ${\displaystyle Y={\tilde {F}}({\tilde {A}},K,L)}$ (where ${\displaystyle A}$ is technology, ${\displaystyle K}$ is capital, and ${\displaystyle L}$ is labor), can be rewritten such that technological change affects output solely as a scalar on labor (i.e. ${\displaystyle Y=F(K,AL)}$) a property known as labor-augmenting or Harrod-neutral technological change.

Uzawa's theorem demonstrates a limitation of the Solow-Swan and Ramsey models. Imposing the assumption of balanced growth within such models requires that technological change be labor-augmenting. Conversely, a production function that cannot represent the effect of technology as a scalar augmentation of labor cannot produce a balanced growth path.^[2]

Statement

Throughout this page, a dot over a variable will denote its derivative concerning time (i.e. ${\displaystyle {\dot {X}}(t)\equiv {dX(t) \over dt}}$ ). Also, the growth rate of a variable ${\displaystyle X(t)}$  will be denoted ${\displaystyle g_{X}\equiv {\frac {{\dot {X}}(t)}{X(t)}}}$ .

Uzawa's theorem

The following version is found in Acemoglu (2009) and adapted from Schlicht (2006):

Model with aggregate production function ${\displaystyle Y(t)={\tilde {F}}({\tilde {A}}(t),K(t),L(t))}$ , where ${\displaystyle {\tilde {F}}:\mathbb {R} _{+}^{2}\times {\mathcal {A}}\to \mathbb {R} _{+}}$  and ${\displaystyle {\tilde {A}}(t)\in {\mathcal {A}}}$  represents technology at time t (where ${\displaystyle {\mathcal {A}}}$  is an arbitrary subset of ${\displaystyle \mathbb {R} ^{N}}$  for some natural number ${\displaystyle N}$ ). Assume that ${\displaystyle {\tilde {F}}}$  exhibits constant returns to scale in ${\displaystyle K}$  and ${\displaystyle L}$ . The growth in capital at time t is given by

${\displaystyle {\dot {K}}(t)=Y(t)-C(t)-\delta K(t)}$ 

where ${\displaystyle \delta }$  is the depreciation rate and ${\displaystyle C(t)}$  is consumption at time t.

Suppose that population grows at a constant rate, ${\displaystyle L(t)=\exp(nt)L(0)}$ , and that there exists some time ${\displaystyle T<\infty }$  such that for all ${\displaystyle t\geq T}$ , ${\displaystyle {\dot {Y}}(t)/Y(t)=g_{Y}>0}$ , ${\displaystyle {\dot {K}}(t)/K(t)=g_{K}>0}$ , and ${\displaystyle {\dot {C}}(t)/C(t)=g_{C}>0}$ . Then

1. ${\displaystyle g_{Y}=g_{K}=g_{C}}$ ; and

2. There exists a function ${\displaystyle F:\mathbb {R} _{+}^{2}\to \mathbb {R} _{+}}$  that is homogeneous of degree 1 in its two arguments such that, for any ${\displaystyle t\geq T}$ , the aggregate production function can be represented as ${\displaystyle Y(t)=F(K(t),A(t)L(t))}$ , where ${\displaystyle A(t)\in \mathbb {R} _{+}}$  and ${\displaystyle g\equiv {\dot {A}}(t)/A(t)=g_{Y}-n}$ .

Sketch of proof

Lemma 1

For any constant ${\displaystyle \alpha }$ , ${\displaystyle g_{X^{\alpha }Y}=\alpha g_{X}+g_{Y}}$ .

Proof: Observe that for any ${\displaystyle Z(t)}$ , ${\displaystyle g_{Z}={\frac {{\dot {Z}}(t)}{Z(t)}}={\frac {d\ln Z(t)}{dt}}}$ . Therefore, ${\displaystyle g_{X^{\alpha }Y}={\frac {d}{dt}}\ln[(X(t))^{\alpha }Y(t)]=\alpha {\frac {d\ln X(t)}{dt}}+{\frac {d\ln Y(t)}{dt}}=\alpha g_{X}+g_{Y}}$ .

Proof of theorem

We first show that the growth rate of investment ${\displaystyle I(t)=Y(t)-C(t)}$  must equal the growth rate of capital ${\displaystyle K(t)}$  (i.e. ${\displaystyle g_{I}=g_{K}}$ )

The resource constraint at time ${\displaystyle t}$  implies

${\displaystyle {\dot {K}}(t)=I(t)-\delta K(t)}$ 

By definition of ${\displaystyle g_{K}}$ , ${\displaystyle {\dot {K}}(t)=g_{K}K(t)}$  for all ${\displaystyle t\geq T}$  . Therefore, the previous equation implies

${\displaystyle g_{K}+\delta ={\frac {I(t)}{K(t)}}}$ 

for all ${\displaystyle t\geq T}$ . The left-hand side is a constant, while the right-hand side grows at ${\displaystyle g_{I}-g_{K}}$  (by Lemma 1). Therefore, ${\displaystyle 0=g_{I}-g_{K}}$  and thus

${\displaystyle g_{I}=g_{K}}$ .

From national income accounting for a closed economy, final goods in the economy must either be consumed or invested, thus for all ${\displaystyle t}$ 

${\displaystyle Y(t)=C(t)+I(t)}$ 

Differentiating with respect to time yields

${\displaystyle {\dot {Y}}(t)={\dot {C}}(t)+{\dot {I}}(t)}$ 

Dividing both sides by ${\displaystyle Y(t)}$  yields

${\displaystyle {\frac {{\dot {Y}}(t)}{Y(t)}}={\frac {{\dot {C}}(t)}{Y(t)}}+{\frac {{\dot {I}}(t)}{Y(t)}}={\frac {{\dot {C}}(t)}{C(t)}}{\frac {C(t)}{Y(t)}}+{\frac {{\dot {I}}(t)}{I(t)}}{\frac {I(t)}{Y(t)}}}$ 

${\displaystyle \Rightarrow g_{Y}=g_{C}{\frac {C(t)}{Y(t)}}+g_{I}{\frac {I(t)}{Y(t)}}=g_{C}{\frac {C(t)}{Y(t)}}+g_{I}(1-{\frac {C(t)}{Y(t)}})=(g_{C}-g_{I}){\frac {C(t)}{Y(t)}}+g_{I}}$ 

Since ${\displaystyle g_{Y},g_{C}}$  and ${\displaystyle g_{I}}$  are constants, ${\displaystyle {\frac {C(t)}{Y(t)}}}$  is a constant. Therefore, the growth rate of ${\displaystyle {\frac {C(t)}{Y(t)}}}$  is zero. By Lemma 1, it implies that

${\displaystyle g_{c}-g_{Y}=0}$ 

Similarly, ${\displaystyle g_{Y}=g_{I}}$ . Therefore, ${\displaystyle g_{Y}=g_{C}=g_{K}}$ .

Next we show that for any ${\displaystyle t\geq T}$ , the production function can be represented as one with labor-augmenting technology.

The production function at time ${\displaystyle T}$  is

${\displaystyle Y(T)={\tilde {F}}({\tilde {A}}(T),K(T),L(T))}$ 

The constant return to scale property of production (${\displaystyle {\tilde {F}}}$  is homogeneous of degree one in ${\displaystyle K}$  and ${\displaystyle L}$ ) implies that for any ${\displaystyle t\geq T}$ , multiplying both sides of the previous equation by ${\displaystyle {\frac {Y(t)}{Y(T)}}}$  yields

${\displaystyle Y(T){\frac {Y(t)}{Y(T)}}={\tilde {F}}({\tilde {A}}(T),K(T){\frac {Y(t)}{Y(T)}},L(T){\frac {Y(t)}{Y(T)}})}$ 

Note that ${\displaystyle {\frac {Y(t)}{Y(T)}}={\frac {K(t)}{K(T)}}}$  because ${\displaystyle g_{Y}=g_{K}}$ (refer to solution to differential equations for proof of this step). Thus, the above equation can be rewritten as

${\displaystyle Y(t)={\tilde {F}}({\tilde {A}}(T),K(t),L(T){\frac {Y(t)}{Y(T)}})}$ 

For any ${\displaystyle t\geq T}$ , define

${\displaystyle A(t)\equiv {\frac {Y(t)}{L(t)}}{\frac {L(T)}{Y(T)}}}$ 

and

${\displaystyle F(K(t),A(t)L(t))\equiv {\tilde {F}}({\tilde {A}}(T),K(t),L(t)A(t))}$ 

Combining the two equations yields

${\displaystyle F(K(t),A(t)L(t))={\tilde {F}}({\tilde {A}}(T),K(t),L(T){\frac {Y(t)}{Y(T)}})=Y(t)}$  for any ${\displaystyle t\geq T}$ .

By construction, ${\displaystyle F(K,AL)}$  is also homogeneous of degree one in its two arguments.

Moreover, by Lemma 1, the growth rate of ${\displaystyle A(t)}$  is given by

${\displaystyle {\frac {{\dot {A}}(t)}{A(t)}}={\frac {{\dot {Y}}(t)}{Y(t)}}-{\frac {{\dot {L}}(t)}{L(t)}}=g_{Y}-n}$ . ${\displaystyle \blacksquare }$ 

See also

• Dynamic efficiency
• Growth accounting

References

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. 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.