Finance: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, [math]\displaystyle{ Y = \tilde{F}(\tilde{A},K,L) }[/math] (where [math]\displaystyle{ A }[/math] is technology, [math]\displaystyle{ K }[/math] is capital, and [math]\displaystyle{ L }[/math] is labor), can be rewritten such that technological change affects output solely as a scalar on labor (i.e. [math]\displaystyle{ Y = F(K,AL) }[/math]) 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. [math]\displaystyle{ \dot{X}(t)\equiv {dX(t) \over dt} }[/math]). Also, the growth rate of a variable [math]\displaystyle{ X(t) }[/math] will be denoted [math]\displaystyle{ g_X\equiv \frac{\dot{X}(t)}{X(t)} }[/math].

Uzawa's theorem

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

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

[math]\displaystyle{ \dot{K}(t)=Y(t)-C(t)-\delta K(t) }[/math]

where [math]\displaystyle{ \delta }[/math] is the depreciation rate and [math]\displaystyle{ C(t) }[/math] is consumption at time t.

Suppose that population grows at a constant rate, [math]\displaystyle{ L(t)=\exp(nt)L(0) }[/math], and that there exists some time [math]\displaystyle{ T \lt \infty }[/math] such that for all [math]\displaystyle{ t\geq T }[/math], [math]\displaystyle{ \dot{Y}(t)/Y(t)=g_Y\gt 0 }[/math], [math]\displaystyle{ \dot{K}(t)/K(t)=g_K\gt 0 }[/math], and [math]\displaystyle{ \dot{C}(t)/C(t)=g_C\gt 0 }[/math]. Then

1. [math]\displaystyle{ g_Y = g_K = g_C }[/math]; and

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

Sketch of proof

Lemma 1

For any constant [math]\displaystyle{ \alpha }[/math], [math]\displaystyle{ g_{X^\alpha Y}=\alpha g_X+g_Y }[/math].

Proof: Observe that for any [math]\displaystyle{ Z(t) }[/math], [math]\displaystyle{ g_{Z}= \frac{\dot{Z}(t)}{Z(t)}= \frac{d\ln Z(t)}{dt} }[/math]. Therefore, [math]\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 }[/math].

Proof of theorem

We first show that the growth rate of investment [math]\displaystyle{ I(t)=Y(t)-C(t) }[/math] must equal the growth rate of capital [math]\displaystyle{ K(t) }[/math] (i.e. [math]\displaystyle{ g_I=g_K }[/math])

The resource constraint at time [math]\displaystyle{ t }[/math] implies

[math]\displaystyle{ \dot{K}(t)=I(t)-\delta K(t) }[/math]

By definition of [math]\displaystyle{ g_K }[/math], [math]\displaystyle{ \dot{K}(t)=g_K K(t) }[/math] for all [math]\displaystyle{ t\geq T }[/math] . Therefore, the previous equation implies

[math]\displaystyle{ g_K+\delta=\frac{I(t)}{K(t)} }[/math]

for all [math]\displaystyle{ t\geq T }[/math]. The left-hand side is a constant, while the right-hand side grows at [math]\displaystyle{ g_I-g_K }[/math] (by Lemma 1). Therefore, [math]\displaystyle{ 0=g_I-g_K }[/math] and thus

[math]\displaystyle{ g_I=g_K }[/math].

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

[math]\displaystyle{ Y(t)=C(t)+I(t) }[/math]

Differentiating with respect to time yields

[math]\displaystyle{ \dot{Y}(t)=\dot{C}(t)+\dot{I}(t) }[/math]

Dividing both sides by [math]\displaystyle{ Y(t) }[/math] yields

[math]\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)} }[/math]

[math]\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 }[/math]

Since [math]\displaystyle{ g_Y, g_C }[/math] and [math]\displaystyle{ g_I }[/math] are constants, [math]\displaystyle{ \frac{C(t)}{Y(t)} }[/math] is a constant. Therefore, the growth rate of [math]\displaystyle{ \frac{C(t)}{Y(t)} }[/math] is zero. By Lemma 1, it implies that

[math]\displaystyle{ g_c-g_Y=0 }[/math]

Similarly, [math]\displaystyle{ g_Y= g_I }[/math]. Therefore, [math]\displaystyle{ g_Y = g_C = g_K }[/math].

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

The production function at time [math]\displaystyle{ T }[/math] is

[math]\displaystyle{ Y(T)=\tilde{F}(\tilde{A}(T), K(T), L(T)) }[/math]

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

[math]\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)}) }[/math]

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

[math]\displaystyle{ Y(t)=\tilde{F}(\tilde{A}(T), K(t), L(T)\frac{Y(t)}{Y(T)}) }[/math]

For any [math]\displaystyle{ t\geq T }[/math], define

[math]\displaystyle{ A(t)\equiv\frac{Y(t)}{L(t)}\frac{L(T)}{Y(T)} }[/math]

and

[math]\displaystyle{ F(K(t),A(t)L(t))\equiv\tilde{F}(\tilde{A}(T), K(t), L(t)A(t)) }[/math]

Combining the two equations yields

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

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

Moreover, by Lemma 1, the growth rate of [math]\displaystyle{ A(t) }[/math] is given by

[math]\displaystyle{ \frac{\dot{A}(t)}{A(t)}=\frac{\dot{Y}(t)}{Y(t)}-\frac{\dot{L}(t)}{L(t)}=g_Y-n }[/math]. [math]\displaystyle{ \blacksquare }[/math]