Nature, Society, and Mathematical Models
Combining dynamic optimization with macro-economic theory to study the interaction between flooding and economic growth.
A guest article written by student Natasha Patnaik, edited by Tyler Perini.
The 2016 paper “Modeling the Interaction Between Flooding Events and Economic Growth” [1], published in Ecological Economics, is a great example of how mathematical models can be used to precisely describe complex relationships. This Austrian study takes an interdisciplinary approach, drawing upon principles of macroeconomics and operations research to design a model that describes how flooding events and economic growth relate to one another.
Taking a minute to think about all the underlying natural phenomena, societal activity, and time-dependent components of such a large-scale system, we can appreciate how incredibly complicated it is to try and capture all of this nuance in just a couple of equations! With this in mind, we’ll look through some highlights of the paper, with specific attention given to the formulation of the model and its most important components.
Background and Motivation.
Firstly, before we delve into the context of our model of interest, we should take a moment to appreciate how operations research and macroeconomics have been used in-tandem here, thus giving rise to a mathematical model that draws upon standard practices from both fields. In framing exogenous (meaning “explicitly controlled by the researchers”) economic variables and parameters within an objective function and set of constraints, we end up with an optimization problem. Then as a direct consequence, finding the optimal solution to this problem means that we have also solved for our endogenous (meaning “implicit components within the model”) economic variables. With this in mind, we can move onto exploring the model presented in the current paper of interest.
Economic growth can be thought of as an increase in the value of goods and services that an economy produces over time; this is often measured as a percentage change of “real” gross domestic product (GDP), where “real” means accounting for inflation overtime. Economic growth is traditionally considered to be an important and desirable objective to pursue, because it can pave the way for increased standards of living and development. Of course, a prevalent threat to such goals of community well-being, stability, and advancement are natural disasters or environmental shocks, such as flooding events.
Socio-hydrology models are mathematical representations of how community development, flooding damage, and long-term economic recovery relate to one another. To connect the environment and the economy in this manner, dynamic optimization models and standard macroeconomic growth models have been creatively combined in order to analyze these relationships. This operates under the assumption of some “decision maker” or “social planner” deciding how much investment should be allocated towards traditional physical capital (machines, tools, buildings, infrastructure, etc.) versus defense capital (flood control basins, levees, dams, warning systems, etc. to protect from future flooding). In this allocation process, our social planner is ultimately guided by the canonical economic objective of maximizing “utility”- an idea that encapsulates the pleasure, satisfaction, or happiness derived from consumption. Here, consumption refers to the usage of any sort of good or service (for example: the food you ate for lunch and the apartment you currently rent are both examples of economic consumption).
The catch here is that there is a tradeoff between choosing to invest more in physical capital versus defense capital. By taking the pre-emptive measure of investing in flood prevention/ mitigation strategies, an economy is better positioned to minimize flood damage, thereby protecting its ability to produce more output and at least maintain the current standard of living. But on the other hand, investment in traditional physical capital is in-and-of-itself necessary to produce a greater quality and quantity of total output. So which is best: investing in production and consumption that will make us better-off now in the current time period, or investing in protection against a potential threat to the standard of living in a future time period? How do we strike a n optimal balance? And how might this balance change as time goes on, depending on factors such as our current capital stock levels, or the frequency and prevalence of flooding events?
We can summarize this problem as follows:
What is the optimal combination of consumption and investment for maximizing utility, subject to shocks and damage from flooding events?
Construction of a Model
Below is a schematic representation of the paper’s model (modified from a diagram in the paper itself), with additional labelling, color-coding, and connections to aid in our discussion.
Before we can specifically discuss the objective function, decision variables, and key constraints in mathematical terms, we stand to benefit from first considering a big-picture view of how this model works. Let’s now deconstruct the schematic from above into its component parts:
The social planner chooses the level of consumption cons [referred to in figures as c(t)], the amount of investment into physical capital inv_phys [referred to in figures as i_y(t)], and the amount of investment into defense capital inv_def [referred to in figures as i_d(t)], guided by the objective of maximizing utility. Note that consumption cons is considered a direct input of the utility-based objective function. The economy is assumed to be closed, such that all the produced economic output ec_out [referred to in figures as Y(t)] will be used up (no trade with foreign economies).
Physical capital cap_phys [referred to in figures as k_y(t)] includes implements such as machines or infrastructure. Economic output ec_out depends on the amount of physical capital cap_phys available for production. In turn, this output can be used either for consumption cons or investment inv_phys and inv_def. Physical capital cap_phys is augmented by investment inv_phys.
Defense capital cap_def [referred to in figures as k_d(t)] includes flood prevention technologies and mitigation measures, such as dams, levees, and flood control basins. The defense capital cap_def can reduce the damage caused by flooding events, and is augmented by investment inv_def.
Although consumption cons is a direct input of the utility function, investment inv_phys and inv_def also affects utility via augmenting/ diminishing the capital stock cap_phys and cap_def. This impacts the economy’s ability to produce more economic output ec_out for consumption cons.
The occurrence of flooding events depends on the water level W(t), which is assumed to be an exogenous function. Thus, damage is a function of both water level W(t) and defense capital cap_def, and can be denoted d(W(t), cap_def). Flooding events damage both types of capital cap_phys and cap_def.
Decision Variables, Objectives, and Key Constraints
Our three decision variables (written in pink in Figure 2) are as follows:
(I) Measured in billions of dollars, cons ∈ [0, ec_out]
(II) Because inv_def is originally measured in meters (for compatibility with units used in the damage function), we need to apply transformation Q to get a cost measurement in billions of dollars. Thus, Q(inv_def) ∈ [0, ec_out − cons].
(III) Measured in billions of dollars inv_phys ∈ [0, ec_out — cons − Q(inv_def)].
As mentioned previously, our objective is to maximize utility. It should be noted that the actual objective function to be maximized contains an exponential expression, which means our mathematical program is nonlinear:
In observing the upper bounds of our three decision variables, we see that their sum is bounded by the value of aggregate economic output ec_out. Logically, because the economy is closed (no foreign trade) and all output is used up, the money flows generated from production of goods and services must equal total spending on consumption and investments (this principle is commonly referred to as circular flow of income). Thus arises one of the key constraints of this model: the budget constraint.
Two other key constraints of interest include the equation which constrains the level of physical capital cap_phys, as well as the equation which constrains the level of defense capital cap_def.
Essentially, these two constraints form a dynamical system (constantly changing until a steady state or equilibrium is reached). For further reading on the implication of this, we can refer to other texts on the standard Ramsey–Cass–Koopmans [2] model for economic growth.
Along with an exogenously determined water function and damage function, all of these constraints complete the formulation of our optimization problem. A long run optimal solution for this problem can then be calculated analytically or solved numerically. By achieving this, we can better conceptualize and discuss relevant policies for flood mitigation strategies. Of course, no mathematical model is perfectly representative of any given real-world scenario or community’s specific circumstances. Yet, through using both dynamic optimization and a macroeconomic framework, we can still arrive at meaningful and useful conclusions. As such, this paper and its findings are a great example of how to formulate a mathematical model that illuminates and clarifies complex relationships between natural phenomena and societal activities.
[1] Grames, J., Prskawetz, A., Grass, D., Viglione, A., & Blöschl, G. (2016). Modeling the interaction between flooding events and economic growth. Ecological Economics, 129, 193–209. https://doi.org/10.1016/j.ecolecon.2016.06.014
[2] Wikimedia Foundation. (2021, May 2). Ramsey–Cass–Koopmans model. Wikipedia. Retrieved November 24, 2021, from https://en.wikipedia.org/wiki/Ramsey%E2%80%93Cass%E2%80%93Koopmans_model#:~:text=The%20Ramsey%E2%80%93Cass%E2%80%93Koopmans%20model,the%20work%20of%20Frank%20P.&text=The%20Ramsey%E2%80%93Cass%E2%80%93Koopmans%20model%20differs%20from%20the%20Solow%E2%80%93,so%20endogenizes%20the%20savings%20rate.
