Optimal investment and location decisions of a firm in a flood risk area using impulse control theory

Abstract

Flooding events can affect businesses close to rivers, lakes or coasts. This paper provides an economic partial equilibrium model, which helps to understand the optimal location choice for a firm in flood risk areas and its investment strategies. How often, when and how much are firms willing to invest in flood risk protection measures? We apply Impulse Control Theory and develop a continuation algorithm to solve the model numerically. We find that, the higher the flood risk and the more the firm values the future, i.e. the more sustainable the firm plans, the more the firm will invest in flood defense. Investments in productive capital follow a similar path. Hence, planning in a sustainable way leads to economic growth. Sociohydrological feedbacks are crucial for the location choice of the firm, whereas different economic settings have an impact on investment strategies. If flood defense is already present, e.g. built up by the government, firms move closer to the water and invest less in flood defense, which allows firms to generate higher expected profits. Firms with a large initial productive capital surprisingly try not to keep their market advantage, but rather reduce flood risk by reducing exposed productive capital.

Keywords: Flood; Impulse control theory; Location choice; Optimal investment; Socio-hydrology; Sustainability.

Fig. 1

The firm chooses where to build its production plant by choosing the distance to the water

Fig. 2

State dynamics in the planning period [0, T]. Firm’s capital K (a) increases only when the flood risk is low as a result of a high flood risk protection standard (b)

Fig. 3

Capital investment $I_K$

Fig. 4

Solution structure given by the objective value $V^{\ast }$ depending on D for no impulse investments (grey), one impulse investment (black), two impulse investments (blue) and three impulse investments (green). Note, that the objective values do not exist for every value of D for each case (color figure online)

Fig. 5

State dynamics for ${\delta }_S=0$ (dashed blue line), ${\delta }_S=0.1$ (dotted dark blue line) and ${\delta }_S=0.25$ (solid light blue line) in the planning period [0, T]. If the salvage value of the firm is important (a) firm’s capital K increases towards the end of the planning horizon and (b) firms invest higher amounts and more often in flood risk protection measures

Fig. 6

Solution structure given by the objective value $V^{\ast }$ depending on the planning horizon T for no impulse investments (grey), one impulse investment (black), two impulse investments (blue) and three impulse investments (green) (color figure online)

Fig. 7

State dynamics for planning horizon $T=50$ (dashed dark green line), $T=100$ (dotted blue line) and $T=150$ (solid light green line). If the planning horizon T is longer (a) firms invest more in capital K even at the beginning and (b) firms invest higher amounts and more often in flood risk protection measures

Fig. 8

State dynamics for different economic situations: benchmark model with $H_0=0$ and $K_0=10$ (dotted blue line), protected flood plain $H_0=200$ (dashed red line), and capital intense firm $K_0=500$ (solid orange line)

Fig. 9

The net present value $V^{\ast }$ of the firm increases for higher initial capital stocks. The colors indicate no impulse investments (grey), one impulse investment (black), two impulse investments (blue) and three impulse investments (green) (color figure online)

Fig. 10

The net present value $V^{\ast }$depending on D for no impulse investments (grey), one impulse investment (black), two impulse investments (blue) and three impulse investments (green). For high feedbacks (${\xi }_H$) it does not pay off to invest in flood defense and will be more profitable to build the production plant further away from the water (color figure online)