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Intertemporal capital substitution and Hayekian booms

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Abstract

Hayek’s business cycle theory portrays monetary expansion and monetary contraction with counterintuitive asymmetry. On the one hand, it suggests that they both change relative prices and cause costly reallocations of production factors. At the same time, the theory predicts that while a monetary contraction causes the economic crisis, the monetary expansion comes with the boom. I argue that what I call intertemporal capital substitution in industries close to final consumption explains why there is a boom in spite of the costly reallocations. More specifically, monetary expansion only gradually increases the demand for nonspecific factors of production by industries that are temporally remote from final consumption. Responding to the expected higher cost of nonspecific factors, consumer-goods industries temporarily increase output and depreciate specific durable production factors faster than they planned.

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Notes

  1. Garrison (2004, pp. 326–337) points out that Hayek was not able to explain the increase in consumption during the boom. Other authors, such as Garrison (2001, 2004) and Koppl (2014) whom I mentioned above, try to address the problem. As I noted, these solutions are only informal.

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Acknowledgements

I would like to thank to Harry David, Steven Horwitz, Pavel Kuchař, G.P. Manish, Caryn Werner, and participants of APEE 2015 conference for valuable comments and suggestions on the earlier draft of this paper. I am also thankful to Stephen Williamson, whose textbook inspired me to think of the tools that I used in the paper. I gratefully acknowledge the financial help that I received from Academic Support Committee at Allegheny College, and Center for the History of Political Economy at Duke University while working on this project. I am responsible for all errors.

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Correspondence to Simon Bilo.

Appendix to “Intertemporal capital substitution and Hayekian booms”

Appendix to “Intertemporal capital substitution and Hayekian booms”

This appendix is a formal companion to section 4 of the paper. It shows the circumstances under which an increase in the wage rate in the second period increases a firm’s capital utilization and demand for labor in the first period. It also shows how this change in the wage rate decreases the two variables in the second period. Lastly, it spells out the necessary conditions for the change in the real interest that are consistent with the results of my model.

This appendix has four sections. The first one overlaps with the discussion in the main paper so that it can be read as a stand-alone document. The second shows the equilibrium conditions for the allocations of capital and labor in the two periods. The third section concerns the equilibrium responses to the increase in the wage rate in the second period, which is the main driver of my model. The fourth section incorporates the decrease in the real interest rate and spells out the conditions under which the qualitative conclusions of my model still hold.

1.1 The underlying model

The firm in this two-period model maximizes the present value of its profits, π. Denoting Y as real output, w as the wage rate, L as the quantity of labor, and the subscripts 1 and 2 as the indicators of the respective two periods, one can say that

$$ \uppi ={\uppi}_1+{\uppi}_2={\mathrm{Y}}_1-{w}_1{L}_1+\frac{Y_2-{w}_2{L}_2}{1+ r}. $$
(14)

The firm’s production function does not change over the two periods, so that

$$ {Y}_1={L}_1^{\alpha}{K}_1^{\beta}, $$
(15)
$$ {Y}_2={L}_2^{\alpha}{K_2}^{\beta}, $$
(16)

where K measures the usage of durable capital and α + β < 1, which means that the returns to scale are diminishing. The firm allocates across the two periods a fixed amount of capital, the total quantity of which is standardized to unity. The corresponding production functions are then

$$ {Y}_1={L}_1^{\alpha}{K}_1^{\beta}, $$
(17)
$$ {Y}_2={L}_2^{\alpha}{\left({1- K}_1\right)}^{\beta}. $$
(18)

Substitution of (Eqs. 17) and (18) into (Eq. 14) shows how the firm maximizes its profits:

$$ \uppi ={L}_1^{\alpha}{K}_1^{\beta}-{w}_1{L}_1+\frac{L_2^{\alpha}{\left(1-{K}_1\right)}^{\beta}-{w}_2{L}_2}{1+ r}. $$
(19)

1.2 The equilibrium allocations of production factors

I can turn now to the underlying question of the optimal allocation of factors across the two periods. The first-order condition for capital is

$$ \frac{\partial \pi}{\partial {K}_1}={\beta L}_1^{\alpha}{K}_1^{\beta -1}-\frac{\beta {L}_2^{\alpha}{\left(1-{K}_1\right)}^{\beta -1}}{1+ r} $$
(20)
$$ 0={\beta L}_1^{\alpha}{K}_1^{\beta -1}-\frac{\beta {L}_2^{\alpha}{\left(1-{K}_1\right)}^{\beta -1}}{1+ r} $$
(21)

The first-order condition for the quantity of labor L1 is

$$ \frac{\partial \uppi}{\partial {L}_1}={\alpha L}_1^{\alpha -1}{K}_1^{\beta}-{w}_1 $$
(22)
$$ {\alpha L}_1^{\alpha -1}{K}_1^{\beta}={w}_1 $$
(23)
$$ {L}_1={\left(\frac{w_1}{\alpha {K}_1^{\beta}}\right)}^{\frac{1}{\alpha -1}} $$
(24)

And the first-order condition for the quantity of labor L2 is

$$ \frac{\partial \uppi}{\partial {\mathrm{L}}_2}=\frac{{\alpha L}_2^{\alpha -1}{\left(1-{K}_1\right)}^{\beta}-{w}_2}{1+ r}=0 $$
(25)
$$ {\alpha L}_2^{\alpha -1}{\left(1-{K}_1\right)}^{\beta}={w}_2 $$
(26)
$$ {L}_2={\left(\frac{w_2}{\alpha {\left(1-{K}_1\right)}^{\beta}}\right)}^{\frac{1}{\alpha -1}}. $$
(27)

I now combine the first-order conditions and substitute (Eqs. 24) and (27) into (21) to find out the optimal quantity of capital in each of the two periods.

$$ 0=\upbeta {\left(\frac{w_1}{\alpha {K}_1^{\beta}}\right)}^{\frac{\alpha}{\alpha -1}}{K}_1^{\beta -1}-\frac{\beta {\left(\frac{w_2}{\alpha {\left(1-{K}_1\right)}^{\beta}}\right)}^{\frac{\alpha}{\alpha -1}}{\left(1-{K}_1\right)}^{\beta -1}}{1+ r} $$
(28)
$$ 0={\left(\frac{w_1}{K_1^{\beta}}\right)}^{\frac{\alpha}{\alpha -1}}{K}_1^{\beta -1}-\frac{{\left(\frac{w_2}{{\left(1-{K}_1\right)}^{\beta}}\right)}^{\frac{\alpha}{\alpha -1}}{\left(1-{K}_1\right)}^{\beta -1}}{1+ r} $$
(29)
$$ 0={w_1}^{\frac{\alpha}{\alpha -1}}{K}_1^{\frac{\alpha \beta}{1-\alpha}}{K}_1^{\beta -1}-\frac{{w_2}^{\frac{\alpha}{\alpha -1}}}{1+ r}{{\left(1-{K}_1\right)}^{\frac{\alpha \beta}{1-\alpha}}\left(1-{K}_1\right)}^{\beta -1} $$
(30)
$$ 0={w_1}^{\frac{\alpha}{\alpha -1}}{K}_1^{\frac{\alpha \beta}{1-\alpha}+\beta -1}-\frac{{w_2}^{\frac{\alpha}{\alpha -1}}}{1+ r}{\left(1-{K}_1\right)}^{\frac{\alpha \beta}{1-\alpha}+\beta -1} $$
(31)
$$ \frac{{w_2}^{\frac{\alpha}{\alpha -1}}}{1+ r}{\left(1-{K}_1\right)}^{\frac{\alpha \beta}{1-\alpha}+\beta -1}={w_1}^{\frac{\alpha}{\alpha -1}}{K}_1^{\frac{\alpha \beta}{1-\alpha}+\beta -1} $$
(32)
$$ {\left(\frac{1-{K}_1}{K_1}\right)}^{\frac{\alpha \beta +\beta -\alpha \beta -1+\alpha}{1-\alpha}}=\left(1+ r\right){\left(\frac{w_1}{w_2}\right)}^{\frac{\alpha}{\alpha -1}} $$
(33)
$$ {\left(\frac{1-{K}_1}{K_1}\right)}^{\frac{\alpha +\beta -1}{1-\alpha}}=\left(1+ r\right){\left(\frac{w_1}{w_2}\right)}^{\frac{\alpha}{\alpha -1}} $$
(34)
$$ \left(\frac{1}{K_1}-1\right)={\left(1+ r\right)}^{\frac{1-\alpha}{\alpha +\beta -1}}{\left(\frac{w_2}{w_1}\right)}^{\frac{-\alpha \left(1-\alpha \right)}{\left(\alpha -1\right)\left(\alpha +\beta -1\right)}} $$
(35)
$$ \frac{1}{K_1}-1={\left(1+ r\right)}^{\frac{1-\alpha}{\alpha +\beta -1}}{\left(\frac{w_2}{w_1}\right)}^{\frac{\alpha}{\left(\alpha +\beta -1\right)}} $$
(36)
$$ {K}_1^{-1}={\left(1+ r\right)}^{\frac{1-\alpha}{\alpha +\beta -1}}{\left(\frac{w_2}{w_1}\right)}^{\frac{\alpha}{\left(\alpha +\beta -1\right)}}+1 $$
(37)
$$ {K}_1={\left({\left(1+ r\right)}^{\frac{1-\alpha}{\alpha +\beta -1}}{\left(\frac{w_2}{w_1}\right)}^{\frac{\alpha}{\left(\alpha +\beta -1\right)}}+1\right)}^{-1} $$
(38)

1.3 Equilibrium responses of the firm to the increase in w2

Assuming there is an increase in the real wage in the second period, which is one of the main shocks in my model, I want to find out what happens to the optimal quantity of capital K1.

$$ \frac{\partial {K}_1}{\partial {w}_2}=-{\left({\left(1+ r\right)}^{\frac{1-\alpha}{\alpha +\beta -1}}{\left(\frac{w_2}{w_1}\right)}^{\frac{\alpha}{\alpha +\beta -1}}+1\right)}^{-2}\frac{\alpha}{\alpha +\beta -1}\left({\left(1+ r\right)}^{\frac{1-\alpha}{\alpha +\beta -1}}\ {\left(\frac{w_2}{w_1}\right)}^{\frac{\alpha}{\alpha +\beta -1}-1}\right)\frac{1}{w_1} $$
(39)

Because α + β < 1, one can say that

$$ \frac{\alpha}{\alpha +\beta -1}<0, $$
(40)

This means that the increase in the wage in the second period leads to an increase in the amount of capital employed in the first period and to a decrease in the amount of capital employed in the second period, or

$$ \frac{\partial {K}_1}{\partial {w}_2}>0. $$
(41)

The change in the wage rate also influences the employment of labor in each of the two periods. In the first period,

$$ \frac{\partial {L}_1}{\partial {w}_2}=\frac{1}{\alpha -1}{\left(\frac{w_1}{\alpha {K}_1^{\beta}}\right)}^{\frac{1}{\alpha -1}-1}\left(-\beta \right)\left(\frac{w_1}{\alpha {K}_1^{\beta +1}}\right)\frac{\partial {K}_1}{\partial {w}_2}. $$
(42)

Since α-1 < 0, β > 0, and ∂K1/∂w2 > 0, the right-hand side of (Eq. 42) is positive and the firm therefore increases the amount of labor in the first period.

I turn to the consequences of the change in the wage rate on the labor employed in the second period.

$$ \frac{\partial {L}_2}{\partial {w}_2}=\frac{1}{\alpha -1}{\left(\frac{w_2}{\alpha {\left(1-{K}_1\right)}^{\beta}}\right)}^{\frac{2-\alpha}{\alpha -1}}\left(\frac{\alpha {\left(1-{K}_1\right)}^{\beta}-{w}_2\alpha \beta {\left(1-{K}_1\right)}^{\beta -1}\left(-\frac{\partial {K}_1}{\partial {w}_2}\right)}{{\left(\alpha {\left(1-{K}_1\right)}^{\beta}\right)}^2}\right) $$
(43)

As α-1 < 0, the right-hand side is negative and the amount of labor employed in the second period decreases.

1.4 Equilibrium responses of the firm to the increase in w2 and the decrease in r

After discussing the consequences of the change in w2 on its own, I extend the analysis by the decrease in the real interest rate that I have to account for because I assume that the interest rate changes through the liquidity effect. I assume in the paper that the effect of the decrease in r countervails the effects of the increase in w2; therefore, I have to spell out the conditions under which the effects of the change in w2 are stronger than those of the change in r so that the qualitative conclusions of my model hold.

First, I show that a decrease in r decreases K1, by taking the first derivative with respect to (Eq. 38).

$$ {K}_1={\left({\left(1+ r\right)}^{\frac{1-\alpha}{\alpha +\beta -1}}{\left(\frac{w_2}{w_1}\right)}^{\frac{\alpha}{\left(\alpha +\beta -1\right)}}+1\right)}^{-1} $$
(44)
$$ \frac{\partial {K}_1}{\partial r}={-\left({\left(1+ r\right)}^{\frac{1-\alpha}{\alpha +\beta -1}}{\left(\frac{w_2}{w_1}\right)}^{\frac{\alpha}{\left(\alpha +\beta -1\right)}}+1\right)}^{-2}\frac{1-\alpha}{\alpha +\beta -1}\left({\left(1+ r\right)}^{\frac{1-\alpha}{\alpha +\beta -1}-1}{\left(\frac{w_2}{w_1}\right)}^{\frac{\alpha}{\left(\alpha +\beta -1\right)}}\right) $$
(45)

Since α + β < 1 and α < 1, one can say that

$$ \frac{1-\alpha}{\alpha +\beta -1}<0, $$
(46)

This implies that a decrease in the interest rate leads to a decrease in the amount of capital utilized in the first period.

$$ \frac{\partial {K}_1}{\partial r}>0. $$
(47)

Given the result, K1 is pushed in two opposite directions when w2 rises and r decreases. My model, where K1 increases, therefore rests on the assumption that

$$ d{K}_1=\frac{\partial {K}_1}{\partial {w}_2} d{w}_2+\frac{\partial {K}_1}{\partial r} d r>0. $$
(48)

I will now show that it is also the case that when (Eq. 48) holds, the amount of labor hired increases in the first period in its response to the rise in w2 and the decrease in r, and thereby follows the prediction of my model. (Eq. 42) above shows that L1 is in a positive relationship with w2 and (Eq. 49) below shows the positive correlation with r.

$$ \frac{\partial {L}_1}{\partial r}=\frac{1}{\alpha -1}{\left(\frac{w_1}{\alpha {K}_1^{\beta}}\right)}^{\frac{1}{\alpha -1}-1}\left(-\beta \right)\left(\frac{w_1}{\alpha {K}_1^{\beta +1}}\right)\frac{\partial {K}_1}{\partial r}>0. $$
(49)

The expression (Eq. 49) is positive because α-1 < 0, β > 0, and ∂K1/∂r > 0. To find when L1 increases in the setting of my model, one has to find out when the following inequality holds.

$$ d{L}_1=\frac{\partial {L}_1}{\partial {w}_2} d{w}_2+\frac{\partial {L}_1}{\partial r} d r>0. $$
(50)

I substitute (Eqs. 42) and (49) into (Eq. 50).

$$ d{L}_1=\frac{1}{\alpha -1}{\left(\frac{w_1}{\alpha {K}_1^{\beta}}\right)}^{\frac{1}{\alpha -1}-1}\left(-\beta \right)\left(\frac{w_1}{\alpha {K}_1^{\beta +1}}\right)\left[\frac{\partial {K}_1}{\partial {w}_2} d{w}_2+\frac{\partial {K}_1}{\partial r} d r\right]>0 $$
(51)

Since the expression in front of the squared bracket is positive, the amount of labor employed in the first period, L1, then increases when

$$ d{K}_1=\frac{\partial {K}_1}{\partial {w}_2} d{w}_2+\frac{\partial {K}_1}{\partial r} d r>0. $$
(52)

One can confirm that the necessary condition (Eq. 52) is identical to the necessary condition (Eq. 48).

In the last step of this appendix, I turn to L2 to find the conditions under which it behaves according to the model in the paper, that is, when L2 decreases with the rise in w2 and the decrease in r. To do this, I first show the negative relationship between the real interest rate r and L2.

$$ \frac{\partial {L}_2}{\partial r}=\frac{1}{\alpha -1}{\left(\frac{w_2}{\alpha {\left(1-{K}_1\right)}^{\beta}}\right)}^{\frac{2-\alpha}{\alpha -1}}\left(-{w}_2\alpha \beta {\left(1-{K}_1\right)}^{\beta -1}\left(-\frac{\partial {K}_1}{\partial r}\right)\right)<0 $$
(53)

(Eq. 53) is negative because α-1 < 0 and ∂K1/∂r > 0. Moving on to find out the assumptions under which L2 decreases when w2 rises and r declines, I look at the value of the total differential for L2.

$$ {dL}_2=\frac{\partial {L}_2}{\partial {w}_2} d{w}_2+\frac{\partial {L}_2}{\partial r} d r<0 $$
(54)

Substituting (Eqs. 43) and (53) into (Eq. 54), one gets

$$ {dL}_2=\frac{1}{\alpha -1}{\left(\frac{w_2}{\alpha {\left(1-{K}_1\right)}^{\beta}}\right)}^{\frac{2-\alpha}{\alpha -1}}\left(\frac{\alpha {\left(1-{K}_1\right)}^{\beta}}{{\left(\alpha {\left(1-{K}_1\right)}^{\beta}\right)}^2} d{w}_2+{w}_2\alpha \beta {\left(1-{K}_1\right)}^{\beta -1}\left[\frac{\frac{\partial {K}_1}{\partial {w}_2}}{{\left(\alpha {\left(1-{K}_1\right)}^{\beta}\right)}^2} d{w}_2+\frac{\partial {K}_1}{\partial r} d r\right]\right)<0 $$
(55)

I now look at the segments of (Eq. 55) and determine whether they are positive or negative. Because α-1 < 0,

$$ \frac{1}{\alpha -1}{\left(\frac{w_2}{\alpha {\left(1-{K}_1\right)}^{\beta}}\right)}^{\frac{2-\alpha}{\alpha -1}}<0. $$
(56)

This means that for the inequality (Eq. 55) to hold, the remainder of the (Eq. 55) expression has to be positive. I also know that

$$ \frac{\upalpha {\left(1-{K}_1\right)}^{\upbeta}}{{\left(\upalpha {\left(1-{\mathrm{K}}_1\right)}^{\upbeta}\right)}^2}{dw}_2>0 $$
(57)

and that

$$ {w}_2\upalpha \upbeta {\left(1-{K}_1\right)}^{\upbeta -1}>0 $$
(58)

This means with the rise in w2 and decline in r, dL2 is negative and consistent with my model, when

$$ \frac{\frac{\partial {K}_1}{\partial {w}_2}}{{\left[\upalpha {\left(1-{\mathrm{K}}_1\right)}^{\beta}\right]}^2}{dw}_2+\frac{\partial {K}_1}{\partial r} d r\ge 0, $$
(59)

where (Eq. 59) is a sufficient rather than a necessary condition. In other words,

$$ \frac{\partial {K}_1}{\partial {w}_2}{dw}_2\ge {\left[\upalpha {\left(1-{\mathrm{K}}_1\right)}^{\beta}\right]}^2\frac{\partial {K}_1}{\partial r} d r. $$
(60)

Since

$$ 0<\left[\upalpha {\left(1-{\mathrm{K}}_1\right)}^{\upbeta}\right]<1 $$
(61)

(Eq. 55) is already satisfied by the previous necessary condition stating that

$$ \frac{\partial {K}_1}{\partial {w}_2}{dw}_2+\frac{\partial {K}_1}{\partial r} dr>0. $$
(62)

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Bilo, S. Intertemporal capital substitution and Hayekian booms. Rev Austrian Econ 31, 277–300 (2018). https://doi.org/10.1007/s11138-017-0379-y

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