# 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.
A general version of the theorem consists of two parts. 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, $Y={\tilde {F}}({\tilde {A}},K,L)$ (where $A$ is technology, $K$ is capital, and $L$ is labor), can be rewritten such that technological change affects output solely as a scalar on labor (i.e. $Y=F(K,AL)$) 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.

## Statement

Throughout this page, a dot over a variable will denote its derivative concerning time (i.e. ${\dot {X}}(t)\equiv {dX(t) \over dt}$). Also, the growth rate of a variable $X(t)$ will be denoted $g_{X}\equiv {\frac {{\dot {X}}(t)}{X(t)}}$.
Uzawa's theorem
The following version is found in Acemoglu (2009) and adapted from Schlicht (2006):
Model with aggregate production function $Y(t)={\tilde {F}}({\tilde {A}}(t),K(t),L(t))$, where ${\tilde {F}}:\mathbb {R} _{+}^{2}\times {\mathcal {A}}\to \mathbb {R} _{+}$ and ${\tilde {A}}(t)\in {\mathcal {A}}$ represents technology at time t (where ${\mathcal {A}}$ is an arbitrary subset of $\mathbb {R} ^{N}$ for some natural number $N$). Assume that ${\tilde {F}}$ exhibits constant returns to scale in $K$ and $L$. The growth in capital at time t is given by
${\dot {K}}(t)=Y(t)-C(t)-\delta K(t)$
where $\delta$ is the depreciation rate and $C(t)$ is consumption at time t.
Suppose that population grows at a constant rate, $L(t)=\exp(nt)L(0)$, and that there exists some time $T<\infty$ such that for all $t\geq T$, ${\dot {Y}}(t)/Y(t)=g_{Y}>0$, ${\dot {K}}(t)/K(t)=g_{K}>0$, and ${\dot {C}}(t)/C(t)=g_{C}>0$. Then
1. $g_{Y}=g_{K}=g_{C}$; and
2. There exists a function $F:\mathbb {R} _{+}^{2}\to \mathbb {R} _{+}$ that is homogeneous of degree 1 in its two arguments such that, for any $t\geq T$, the aggregate production function can be represented as $Y(t)=F(K(t),A(t)L(t))$, where $A(t)\in \mathbb {R} _{+}$ and $g\equiv {\dot {A}}(t)/A(t)=g_{Y}-n$.

## Sketch of proof
### Lemma 1

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

### Proof of theorem

We first show that the growth rate of investment $I(t)=Y(t)-C(t)$ must equal the growth rate of capital $K(t)$ (i.e. $g_{I}=g_{K}$)
The resource constraint at time $t$ implies
    ${\dot {K}}(t)=I(t)-\delta K(t)$
By definition of $g_{K}$, ${\dot {K}}(t)=g_{K}K(t)$ for all $t\geq T$ . Therefore, the previous equation implies
    $g_{K}+\delta ={\frac {I(t)}{K(t)}}$
for all $t\geq T$. The left-hand side is a constant, while the right-hand side grows at $g_{I}-g_{K}$ (by Lemma 1). Therefore, $0=g_{I}-g_{K}$ and thus
    $g_{I}=g_{K}$.
From national income accounting for a closed economy, final goods in the economy must either be consumed or invested, thus for all $t$
    $Y(t)=C(t)+I(t)$
Differentiating with respect to time yields
    ${\dot {Y}}(t)={\dot {C}}(t)+{\dot {I}}(t)$
Dividing both sides by $Y(t)$ yields
    ${\frac {{\dot {Y}}(t)}{Y(t)}}={\frac {{\dot {C}}(t)}{Y(t)}}+{\frac {{\dot {I}}(t)}{Y(t)}}={\frac {{\dot {C}}(t)}{C(t)}}{\frac {C(t)}{Y(t)}}+{\frac {{\dot {I}}(t)}{I(t)}}{\frac {I(t)}{Y(t)}}$
    $\Rightarrow g_{Y}=g_{C}{\frac {C(t)}{Y(t)}}+g_{I}{\frac {I(t)}{Y(t)}}=g_{C}{\frac {C(t)}{Y(t)}}+g_{I}(1-{\frac {C(t)}{Y(t)}})=(g_{C}-g_{I}){\frac {C(t)}{Y(t)}}+g_{I}$
Since $g_{Y},g_{C}$ and $g_{I}$ are constants, ${\frac {C(t)}{Y(t)}}$ is a constant. Therefore, the growth rate of ${\frac {C(t)}{Y(t)}}$ is zero. By Lemma 1, it implies that
    $g_{c}-g_{Y}=0$
Similarly, $g_{Y}=g_{I}$. Therefore, $g_{Y}=g_{C}=g_{K}$.
Next we show that for any $t\geq T$, the production function can be represented as one with labor-augmenting technology.
The production function at time $T$ is
    $Y(T)={\tilde {F}}({\tilde {A}}(T),K(T),L(T))$
The constant return to scale property of production (${\tilde {F}}$ is homogeneous of degree one in $K$ and $L$) implies that for any $t\geq T$, multiplying both sides of the previous equation by ${\frac {Y(t)}{Y(T)}}$ yields
    $Y(T){\frac {Y(t)}{Y(T)}}={\tilde {F}}({\tilde {A}}(T),K(T){\frac {Y(t)}{Y(T)}},L(T){\frac {Y(t)}{Y(T)}})$
Note that ${\frac {Y(t)}{Y(T)}}={\frac {K(t)}{K(T)}}$ because $g_{Y}=g_{K}$(refer to solution to differential equations for proof of this step). Thus, the above equation can be rewritten as
    $Y(t)={\tilde {F}}({\tilde {A}}(T),K(t),L(T){\frac {Y(t)}{Y(T)}})$
For any $t\geq T$, define
    $A(t)\equiv {\frac {Y(t)}{L(t)}}{\frac {L(T)}{Y(T)}}$
and
    $F(K(t),A(t)L(t))\equiv {\tilde {F}}({\tilde {A}}(T),K(t),L(t)A(t))$
Combining the two equations yields
    $F(K(t),A(t)L(t))={\tilde {F}}({\tilde {A}}(T),K(t),L(T){\frac {Y(t)}{Y(T)}})=Y(t)$ for any $t\geq T$.
By construction, $F(K,AL)$ is also homogeneous of degree one in its two arguments.
Moreover, by Lemma 1, the growth rate of $A(t)$ is given by
    ${\frac {{\dot {A}}(t)}{A(t)}}={\frac {{\dot {Y}}(t)}{Y(t)}}-{\frac {{\dot {L}}(t)}{L(t)}}=g_{Y}-n$. $\blacksquare$