We introduce a simple model of production to incorporate substitution between capital and labor. A firm utilizes capital *K* and labor *L* to produce a single output, *Y*. We assume the firm’s cost-minimizing behavior. Thus, given output *Y*, a firm chooses *K* and *L* to minimize cost based on factor prices *r* (user cost of capital) and *w* (wage). Technology at period *t* is represented by the production function *Y* = *F*
^{t}(*K*, *L*), which exhibits constant returns to scale. Given factor prices and output, the period *t* cost function of a firm is as follows:

$$ {C}^t\left(r,w,Y\right)= \max \left\{rK+wL:Y={F}^t\left(K,L\right)\right\} $$

(1)

Since we assume constant-returns-to-scale technology, the cost function is a multiplication of the unit cost function and output, such as *C*
^{t}(*r*, *w*, *Y*) = *C*
^{t}(*r*, *w*, 1) ⋅ *Y*. Applying Shephard’s lemma (Shephard 1970), we derive the following unit labor cost function as the function of factor prices and time:

$$ {\mathrm{ULC}}^t\left(r,w\right)=\frac{w\cdot \partial {C}^t\left(r,w,Y\right)/\partial w}{Y}=w\cdot \partial {C}^t\left(r,w,1\right)/\partial w $$

(2)

This is the key equation for determining changes in ULC. Let us compare ULC for two periods, 0 and 1. First, we look at the comprehensive impact of the change in wage on the ULC. The wage effect is measured by the ratio ULC^{t}(*r*, *w*
^{1})/ULC^{t}(*r*, *w*
^{0}). It indicates the change in ULC induced by the change in wage going from period 0 to 1, using the technology that is available during the reference period *t* and facing the reference user cost of capital, *r*. Since each choice of the reference vector (*t*, *r*) might generate a different measure, we calculate two measures using different reference vectors (0, *r*
^{0}) and (1, *r*
^{1}) which, in fact, are observed in each period and thus are equally reasonable. Then, following Fisher (1922) and Diewert (1976), we use the geometric mean of these measures as a theoretical measure of the wage effect, *Wage*, as follows:

$$ \mathrm{Wage}=\sqrt{\frac{{\mathrm{ULC}}^0\left({r}^0,{w}^1\right)}{{\mathrm{ULC}}^0\left({r}^0,{w}^0\right)}\cdot \frac{{\mathrm{ULC}}^1\left({r}^1,{w}^1\right)}{{\mathrm{ULC}}^1\left({r}^1,{w}^0\right)}} $$

(3)

Second, we consider the comprehensive impact of the change in user cost of capital on ULC. The user cost effect is measured by the ratio ULC^{t}(*r*
^{1}, *w*)/ULC^{t}(*r*
^{0}, *w*). It indicates the change in ULC induced by the change in user cost going from period 0 to 1, using the technology that is available during the reference period *t* and facing the reference wage, *w*. As each choice of the reference vector (*s*, *w*) might generate a different measure, we calculate two measures using different reference vectors (0, *w*
^{0}) and (1, *w*
^{1}) which, in fact, are observed in each period and thus are equally reasonable. Then, we use the geometric mean of these measures as a theoretical measure of user cost effect, *User cost*, as follows:

$$ \mathrm{User}\ \mathrm{cost}=\sqrt{\frac{{\mathrm{ULC}}^0\left({r}^1,{w}^0\right)}{{\mathrm{ULC}}^0\left({r}^0,{w}^0\right)}\cdot \frac{{\mathrm{ULC}}^1\left({r}^1,{w}^1\right)}{{\mathrm{ULC}}^1\left({r}^0,{w}^1\right)}} $$

(4)

Lastly, we consider the impact of a technical change on the ULC. Technical change is measured by the ratio ULC^{1}(*r*, *w*)/ULC^{0}(*r*, *w*). It indicates the change in ULC induced by technical change going from period 0 to 1, facing the reference factor prices *r* and *w*. Since each choice of the reference vector (*r*, *w*) might generate a different measure, we calculate two measures using different reference vectors (*r*
^{0}, *w*
^{0}) and (*r*
^{1}, *w*
^{1}) which, in fact, are observed in each period and thus are equally reasonable. Then, we use the geometric mean of these measures as a theoretical measure of technical change effect, *Technology*, as follows:

$$ \mathrm{Technology}=\sqrt{\frac{{\mathrm{ULC}}^1\left({r}^0,{w}^0\right)}{{\mathrm{ULC}}^0\left({r}^0,{w}^0\right)}\cdot \frac{{\mathrm{ULC}}^1\left({r}^1,{w}^1\right)}{{\mathrm{ULC}}^0\left({r}^1,{w}^1\right)}} $$

(5)

These three measures are theoretical ones. Thus, even though we know the factor prices prevailing at each period, we cannot compute these measures, which are defined by the unknown ULC functions. There are multiple ways of implementing these measures. Here, we adopt the index number approach and derive the index number formula that approximates the theoretical measures proposed above. Our purpose is to propose a tractable way of investigating the sources of the change in ULC, replacing the conventional decomposition.^{Footnote 7}

We implement them by assuming the following production functions for *t* = 0, 1:

$$ {F}^t\left(K,L\right)={A}^t{K}^{\alpha^t}{L}^{\left(1-{\alpha}^t\right)} $$

(6)

It is a variant of the Cobb–Douglas production function allowing output elasticity of capital *α*, which is known to be equal to capital share, to vary in each period. Technology of a firm in period *t* is represented by a combination of *A*
^{t} and *α*
^{t}. Under this specification, the three theoretical measures coincide with a formula for factor input prices and quantities observed at two periods 0 and 1 as follows^{Footnote 8}:

$$ \mathrm{Wage}={\left(\frac{w^1}{w^0}\right)}^{\frac{1}{2}\left({s}_L^0+{s}_L^1\right)} $$

(7)

$$ \mathrm{User}\ \mathrm{cost}={\left(\frac{r^1}{r^0}\right)}^{\frac{1}{2}\left({s}_K^0+{s}_K^1\right)} $$

(8)

$$ \mathrm{Technology}=\left(\left(\frac{w^1{L}^1}{Y^1}\right)/\left(\frac{w^0{L}^0}{Y^0}\right)\right)/\left({\left(\frac{w^1}{w^0}\right)}^{\frac{1}{2}\left({s}_L^0+{s}_L^1\right)}\times {\left(\frac{r^1}{r^0}\right)}^{\frac{1}{2}\left({s}_K^0+{s}_K^1\right)}\right) $$

(9)

where \( {s}_K^t=\frac{r^t{K}^t}{\left({r}^t{K}^t+{w}^t{L}^t\right)} \) and \( {s}_L^t=\frac{w^t{L}^t}{\left({r}^t{K}^t+{w}^t{L}^t\right)} \) are the capital and labor compensation share defined for *t* = 0, 1.

Our measure of the wage effect is smaller than the conventional measure of wage effect *w*
^{1}/*w*
^{0}. A higher wage directly increases the ULC by raising the labor compensation. However, it induces a firm to substitute labor by employing more capital. Less labor raises labor productivity, lowering the ULC. Thus, the direct impact of a wage increase on labor compensation is somewhat mitigated. That is what the measure of wage effect proposed by this study incorporates. Three measures are independently proposed to capture the distinct effect on the ULC. Under the assumption of Eq. (6), the change in ULC is completely decomposed into these factors, as follows:

$$ \frac{{\mathrm{ULC}}^1\left({r}^1,{w}^1\right)}{{\mathrm{ULC}}^0\left({r}^0,{w}^0\right)}=\left(\frac{w^1{L}^1}{Y^1}\right)/\left(\frac{w^0{L}^0}{Y^0}\right)=\mathrm{Wage}\times \mathrm{User}\ \mathrm{cost}\times \mathrm{Technology} $$

(10)

Under this decomposition, labour productivity effect is captured by \( User\ cost\times Technology=\left(\left(\frac{w^1{L}^1}{Y^1}\right),/,\left(\frac{w^0{L}^0}{Y^0}\right)\right)/\left({\left(\frac{w^1}{w^0}\right)}^{\frac{1}{2}\left({s}_L^0+{s}_L^1\right)}\right) \), which contrast the conventional measure of labour productivity effect \( \left(\left(\frac{w^1{L}^1}{Y^1}\right),/,\left(\frac{w^0{L}^0}{Y^0}\right)\right)/\left(\frac{w^1}{w^0}\right)=\left(\frac{L^1}{Y^1}\right)/\left(\frac{L^0}{Y^0}\right) \).