The Economic Impact of Civil Wars: A Production Function Approach

Abstract

The economic cost of a civil war arises from the destruction of production factors, the rise of uncertainty, and the misallocation of resources. An endogenous growth model shows how the re-allocation of resources from productive to unproductive activities suppresses the growth rate of the economy. The direct impact on output is evaluated by employing the growth-accounting framework on a production function with constant returns to scale. The Chapter describes how data can be used to provide estimates of the destruction of production factors and the loss of output in the Greek Civil War.

Appendix

8.1.1 The Endogenous Growth Rate in an Economy with Civil War

The growth rate of output is endogenously derived for a closed economy along the lines described by Barro and Sala-i-Martin (1995, Ch. 5). To conform to the time framework presented in Chaps. 5 and 7, the model is specified in discrete time. For simplicity, population growth is assumed away. Hence, we consider an economy populated by a constant number (N) of identical households, each owning a firm with a capital stock (k) that depreciates at rate (δ). The government imposes a distortionary tax rate (τ) and uses collected revenues to finance productive infrastructure and enforce property rights, as described in Sect. 2.1.

Production per firm is characterized by constant returns to scale and given by the linear function:

$$ y=A\bullet k\bullet \pi \left(\omega \tau \right)\bullet h\left[\left(1 - \omega \right)\tau \right] $$

(8.17)

The net real rate of return to capital is (ρ _k ), thus the rental cost of the firm’s stock is (ρ _k  + δ). Profit maximization requires that the after-tax marginal product of capital equals the rental cost, thus

$$ \left(1-\uptau \right)\left(\partial y/\partial k\right)=\left(1-\tau \right)A\pi h={\rho}_k+\delta \Longrightarrow {\rho}_k=\left(1-\tau \right)A\pi h-\delta $$

(8.18)

Output, consumption, and capital stock per household grow at the same rate that is endogenously derived below. Households choose consumption in order to maximize a discounted utility function given by

$$ \underset{c_t}{ \max }U={\sum}_{t=1}^{\infty}\frac{1}{{\left(1+\rho \right)}^t}\bullet \frac{c_t^{1-\varphi }}{1-\varphi } $$

(8.19)

where ρ is the rate of discounting the future, and φ is a risk aversion parameter.

Non-consumed output is invested in the firm. The capital accumulation constraint is:

$$ {k}_{t+1}=\left(1-\delta \right)\bullet {k}_t+\left(1-\tau \right)\bullet {y}_t-{c}_t $$

(8.20)

The transversality condition requires that

$$ \underset{t\to \infty }{ \lim }{k}_t{\left(1+\rho \right)}^{-t}=0 $$

(8.21)

The Hamiltonian at period (t) is defined as:

$$ {\mathcal{H}}_t=\frac{c_t^{-\varphi }}{1-\varphi }+{\upsilon}_{t+1}\bullet \left[\left(1-\delta \right){k}_t+{y}_t-{c}_t\right] $$

(8.22)

where υ _t+1 is the Lagrange multiplier for next period’s capital stock. The first-order conditions for maximization are

$$ \frac{\partial {\mathcal{H}}_t}{\partial {c}_t}=0\Longrightarrow {\upsilon}_{t+1}={c}_t^{-\varphi } $$

(8.23a)

and

$$ \left(1+\rho \right)\bullet {\upsilon}_t=\frac{\partial {\mathcal{H}}_t}{\partial {k}_t}={\upsilon}_{t+1}\bullet \left[1-\delta +\left(1-\tau \right)\bullet \frac{{\partial y}_t}{\partial {k}_t}\right] $$

(8.23b)

The second order condition is satisfied as a consequence of decreasing marginal utility, i.e.

$$ \frac{\partial^2{\mathcal{H}}_t}{\partial {c}_t^2}=-\varphi {c}_t^{-\varphi -1}<0 $$

(8.23c)

Lagging (8.23a) one period and substituting into (8.23b), we obtain:

$$ \frac{\upsilon_t}{\upsilon_{t+1}}={\left[\frac{c_t}{c_{t-1}}\right]}^{\varphi } $$

(8.24)

The growth rate of consumption per household—and similarly that of output and capital stock—is finally obtained as

$$ g=\frac{c_t-{c}_{t-1}}{c_{t-1}}={\left[\frac{\left(1-\tau \right)A\pi h+1-\delta }{1+\rho}\right]}^{1/\varphi }-1 $$

(8.25)

For relatively small values of ρ and δ, the following approximation is obtained.

$$ g\approx \frac{1}{\varphi}\bullet \left[\left(1-\tau \right)A\pi h-\delta -\rho \right] $$

(8.26)

This is the same as that obtained by Barro & Sala-i-Martin for continuous-time models.

8.1.2 Proof of Propositions 1, 2, 3

Taking the total differential of Eq. (8.8a) we get

$$ \left(1+\beta \right)\bullet d\tau =-\frac{d\tau }{1+\lambda \left(1-\tau \right)}-\left\{\frac{ln\left[1+\lambda \left(1-\tau \right)\right]}{\lambda^2}-\frac{\left(1-\tau \right)}{\lambda \bullet \left[1+\lambda \left(1-\tau \right)\right]}\right\}\bullet d\lambda $$

or

$$ \left(1+\beta +\frac{1}{1+\lambda \left(1-\tau \right)}\right)\bullet d\tau =-\frac{1}{\lambda^2}\left\{ ln\left[1+\lambda \left(1-\tau \right)\right]-\frac{\lambda \left(1-\tau \right)}{1+\lambda \left(1-\tau \right)}\right\}\bullet d\lambda $$

$$ =-\frac{1}{\lambda^2}\bullet \varPsi (x)\bullet d\lambda $$

(8.27)

where for brevity it was set x = λ(1−τ) and

$$ \varPsi (x)= ln\left[1+x\right]-\frac{x}{1+x}\Longrightarrow {\varPsi}^{\prime }(x)=\frac{x}{{\left(1+x\right)}^2}>0,\mathrm{f}\mathrm{o}\mathrm{r}\;x>0 $$

(8.28)

Thus function Ψ(x) is increasing and, given that Ψ(0) = 0, it is positive for every x>0. Then from (8.27) it easily follows that

$$ \frac{d\tau }{d\lambda }=-\frac{1}{\lambda^2}\bullet \left(1+\beta +\frac{1}{1+x}\right){\bullet}^{-1}\varPsi (x)<0 $$

(8.29)

and Proposition 1 is established. From (8.8b) we obtain

$$ \omega \tau =\left(1+\beta \right)\tau -1 $$

(8.30)

Differentiating w.r.t. (τ), we get $\partial \left(\omega \tau \right)/\partial \tau >0$ and $\partial \omega /\partial \tau >0$, hence Propositions 2 and 3 are readily obtained.