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. $\dot{X}(t)\equiv \frac{dX(t)}{dt}$). 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 $\tilde{F}:{ℝ}_{+}^2\times 𝒜\to {ℝ}_{+}$ and $\tilde{A}(t)\in 𝒜$ represents technology at time t (where $𝒜$ is an arbitrary subset of ${ℝ}^N$ 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 $T<\mathrm{\infty }$ such that for all $t\ge T$, $\dot{Y}(t)/Y(t)=g_Y>0$, $\dot{K}(t)/K(t)=g_K>0$, and $\dot{C}(t)/C(t)=g_C>0$. Then

1. $g_Y=g_K=g_C$; and

2. There exists a function $F:{ℝ}_{+}^2\to {ℝ}_{+}$ that is homogeneous of degree 1 in its two arguments such that, for any $t\ge T$, the aggregate production function can be represented as $Y(t)=F(K(t),A(t)L(t))$, where $A(t)\in {ℝ}_{+}$ 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 $t\ge T$ . Therefore, the previous equation implies

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

for all $t\ge T$. 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 $t\ge T$, 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 $t\ge T$, 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 $t\ge T$, 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 $t\ge T$.

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

$\frac{\dot{A}(t)}{A(t)}=\frac{\dot{Y}(t)}{Y(t)}-\frac{\dot{L}(t)}{L(t)}=g_Y-n$. $◼$

See also

• Dynamic efficiency
• Growth accounting