Why the structure of capital and the useful lives of its components matter: A test based on a model of Austrian descent

Abstract

This paper centers on the structure of capital and the useful lives of its components by considering an economy with two representative firms, one producing a necessity and another producing a luxury. This difference determines their reinvestment opportunities. Therefore, while the one applies replacement, the other adopts scrapping. However, as these capital policies lead to different service lives, the analysis confronts the issues raised by Miller (Review of Income and Wealth 29:284–296, 1982, Review of Income and Wealth 36:67–82, 1990) and deals with them by drawing on Haavelmo’s (A study in the theory of investment, Chicago: The University of Chicago Press, 1960) suggestions regarding the aggregation of capital. Among other findings, it turns out that the simulation results are highly robust, thus demonstrating that real-world implications may be even stronger than strictly suggested by the model.

Appendices

Appendix A

Suppose that embodied technological progress reduces the capital–output coefficient b _X (t) at the constant and exogenous rate μ. This rate scales with time because it depends on the interval over which it is considered. Therefore, suppose that we are at time t and we observe a vintage of capital that was built υ periods ago. On average, from υ to t, the capital coefficient must have declined as follows:

$$\frac{{b_X \left( t \right) - b_X \left( \upsilon \right)}}{{b_X \left( \upsilon \right)}} = \mu \left( {t - \upsilon } \right),$$

(26)

with μ < 0.

From this, we get:

$$b_X \left( t \right) = b_X \left( \upsilon \right)\left[ {1 + \mu \left( {t - \upsilon } \right)} \right].$$

(27)

Now, assume that the process of technological progress begins at t = υ. At that moment, the capital coefficient will be equal to b(υ). Next, divide the interval t − υ into m equal time steps. At the end of the first time step, the capital coefficient will be:

$$b_X \left( {\upsilon + \frac{{t - \upsilon }}{m}} \right) = b_X \left( \upsilon \right)\left[ {1 + \mu \frac{{t - \upsilon }}{m}} \right].$$

(28)

At the end of the second timestep the capital coefficient will be:

$$b_X \left( {\upsilon + 2\frac{{t - \upsilon }}{m}} \right) = b_X \left( {\upsilon + \frac{{t - \upsilon }}{m}} \right)\left[ {1 + \mu \frac{{t - \upsilon }}{m}} \right] = b_X \left( \upsilon \right)\left[ {1 + \mu \frac{{t - \upsilon }}{m}} \right]^2 .$$

(29)

And at the end of the m time step, the capital coefficient will have declined to:

$$b_{X} {\left( t \right)} = b_{X} {\left( {\upsilon + {\left( {m - 1} \right)}\frac{{t - \upsilon }}{m}} \right)}{\left[ {1 + \mu \frac{{t - \upsilon }}{m}} \right]}^{m} .$$

(30)

Now, let m→∞. Then, the time step (t − υ)/m will go to zero, and we will have:

$$b_X \left( t \right) = b_X \left( \upsilon \right)\left[ {1 + \mu \frac{{t - \upsilon }}{m}} \right]^m = b_X \left( \upsilon \right)e^{\mu \left( {t - \upsilon } \right)} ,$$

(31)

which is Eq. 3 in the text.

Appendix B

Inserting Eqs. 10 into 9 and rearranging, we obtain:

$$P_X \left( 0 \right) = \frac{{\eta _X }}{{1 + \eta _X }}\frac{{\sigma - \mu }}{\sigma }\frac{{1 - e^{ - \mu T_X } + \beta \sigma }}{{1 - e^{ - \left( {\mu - \sigma } \right)T_X } }}b_X \left( 0 \right)w.$$

(32)

In turn, if this is introduced into Eq. 7, it yields:

$$n_X \left( 0 \right) = - \frac{1}{{1 + \eta _X }}\frac{{1 - e^{\mu T_X } + \beta \sigma }}{\sigma }w.$$

(33)

However, the representative firm is presumed to defend its monopoly by restricting itself through pricing rule Eq. 5 to making only normal profits, i.e., profits that would result in an industry in perfectly competitive long-run equilibrium. Therefore, it will let the price elasticity of demand η _X approach to minus infinity and use the approximation:

$$\frac{{\eta _X }}{{1 + \eta _X }} = 1.$$

(34)

Now, if we deduct 1 from both sides of this equation and insert the result into Eq. 33, we obtain:

$$n_X \left( 0 \right) = 0.$$

(35)

This proves that under the adopted pricing rule, the unit net worth of equipment in the initial vintage is set equal to zero and corroborates the claim made in the text.

Appendix C

At υ, the worth of revenue minus the labor cost of operating a unit of equipment per small fraction dt of a year located at time t is given by:

$$\left[ {\frac{{P\left( \upsilon \right)}}{{b\left( \upsilon \right)}}e^{\mu \left( {t - \upsilon } \right)} - w} \right]e^{ - \sigma \left( {t - \upsilon } \right)} {\text{d}}t$$

(36)

where w and σ denote, respectively, the economy-wide rates of wages and interest.

Consequently, at υ, the worth of the sum total of revenue minus operating labor cost of a unit of such equipment over its entire useful life T is:

$$n_X \left( 0 \right) = \int_\upsilon ^{\upsilon + T} {\left[ {\frac{{P\left( \upsilon \right)}}{{b\left( \upsilon \right)}}e^{\mu \left( {t - \upsilon } \right)} - w} \right]e^{ - \sigma \left( {t - \upsilon } \right)} } {\text{d}}t - \beta w = \frac{{P_X \left( \upsilon \right)}}{{b_X \left( \upsilon \right)}}\frac{{1 - e^{\left( {\mu - \sigma } \right)T} }}{{\sigma - \mu }} - w\frac{{1 - e^{ - \sigma T} }}{\sigma }.$$

(37)

Now, let P be the purchase price of a new unit of electricity generators. Assuming its salvage value upon retirement is zero, its net worth is:

$$n\left( \upsilon \right) = \frac{{P\left( \upsilon \right)}}{{b\left( \upsilon \right)}}\frac{{1 - e^{\left( {\mu - \sigma } \right)T} }}{{\sigma - \mu }} - w\frac{{1 - e^{ - \sigma T} }}{\sigma } - P.$$

(38)

Finally, since by Eq. 4 the minimum labor required to build a new unit of electricity-generating capacity is β, under perfectly competitive transfer prices from the capital building to the capital using department of the representative firm, plus the assumption that electricity-generating capacity is built solely by means of labor, it will hold that:

$$P = \beta w.$$

(39)

Thus, substituting Eq. 39 into Eq. 38 gives Eq. 7.

Appendix D

Let a capital stock be replaced forever every υ years. At time t = 0, the firm acquires the vintage zero capital stock $K_X \left( 0 \right) = b_X \left( 0 \right)X\left( 0 \right)$. At time t = T _X , the vintage zero capital stock is retired and replaced by vintage υ capital stock. However, vintage T _X capital stock is more efficient due to technological progress. How more efficient it is relative to vintage zero capital stock is given by $b_X \left( {T_X } \right) = b_X \left( 0 \right)e^{\mu T_X }$. Hence, to keep the capacity constant from vintage to vintage, at t = υ, the capital stock that is needed for replacement is less. In particular, its quantity is given by $K_X \left( {T_X } \right) = b_X \left( {T_X } \right)X\left( 0 \right) = b_X \left( 04 \right)X\left( 0 \right)e^{\mu T_X }$. At t = 2T _X , the capital stock that will be needed for replacement will be $K_X \left( {2T_X } \right) = b_X \left( {2T_X } \right)X\left( 0 \right) = b_X \left( 0 \right)X\left( 0 \right)e^{2\mu T_X }$, at time t = jT _X , the capital stock that will be needed is $K_X \left( {jT_X } \right) = b_X \left( {jT_X } \right)X\left( 0 \right) = b_X \left( 0 \right)X\left( 0 \right)e^{j\mu T_X }$, and so on.

Now, consider the net worth of the acquisitions. At time t = 0, the net worth of the capital stock is $n_X \left( 0 \right)K_X \left( 0 \right) = b_X \left( 0 \right)n_X \left( 0 \right)X\left( 0 \right)$. At time t = υ, the net worth of the acquired capital stock is $n_X \left( {T_X } \right)K_X \left( {T_X } \right) = b_X \left( 0 \right)n_X \left( 0 \right)X\left( 0 \right)e^{\mu T_X }$. Notice in this expression that n(υ) has been substituted for n(0). This has been done on account of the expressions 6 and 7 in the text. Therefore, if the net worth of the capital stock acquired at t = υ is discounted to t = 0 using the discounting term $e^{ - \sigma T_X }$, we obtain $b_X \left( 0 \right)n_X \left( 0 \right)X\left( 0 \right)e^{\left( {\mu - \sigma } \right)T_X }$. Finally, repeating the preceding steps for the net worth of capital stock acquired at t = jT _X gives $b_X \left( 0 \right)n_X \left( 0 \right)X\left( 0 \right)e^{\left( {\mu - \sigma } \right)jT_X }$. Consequently, since in addition to the initial acquisition there take place j replacements, the net present value of the 1 + j acquisitions of capital stock is given by:

$$b_X \left( 0 \right)n_X \left( 0 \right)X\left( 0 \right)\left[ {1 + e^{\left( {\mu - \sigma } \right)T_X } + \ldots + e^{\left( {\mu - \sigma } \right)jT_X } } \right].$$

(40)

The expression in the brackets is a geometric progression with 1 + j terms, each of which is equal to the preceding one multiplied by $e^{\left( {\mu - \sigma } \right)jT_X }$, where (μ − σ)T _X  < 0. This implies that the bracketed expression constitutes a geometric progression with declining terms. Thus, if we set j→∞, at t = 0, the sum of the infinite series of replacements K _X (0), K _X (T _X ), K _X (2T _X ),... is given by:

$$A_X \left( 0 \right) = \frac{{b_X \left( 0 \right)n_X \left( 0 \right)X\left( 0 \right)}}{{1 - e^{\left( {\mu - \sigma } \right)T_X } }}.$$

(41)

This is Eq. 8 in the text.