Economic Models and Optimization

The first is the “ceteris paribus” assumption. The second is that decision-making agents try to optimize their goals. The third is the need for distinction between positive and normative nature of the questions.

Ceteris paribus means holding other things constant. In economics, while examining the impact of a variable on any other variable, the other factors are assumed to be fixed or constant. This assumption allows us to separate the effect of variable of interest from other possible variables related with dependent variable.

In the broad sense, a model is a simplification of reality or system showing cause and effect relation between variables under certain assumptions. These simplifications may utilize either mathematics (symbolic logic) or verbal logic, which are also formal language to express researchers themselves.

A variable is a number whose value can change over time or across the units, or both. The simplest model in functional form may be written as follows: y=f(x)

In the equation above, y is the dependent variable whose value is identified within model. x is the independent variable treated as an external shock. In this model, y is determined by x. In other words, y is a function of x.

a is a constant parameter, which is also called intercept whose value and magnitude remain unchanged. b, c, d are defined as slope terms, i.e. shows direction and magnitude of the relationship between y and each independent variable: x, w, and z. While signs of these parameters show the direction, parameter itself gives us information on what extent x, w, and z affects y separately.

One of the properties of economic modeling is that the economic agents, which are basically firms and households, are optimizing their actions. Households maximize (minimize) their utility (expenditure) and firms maximize (minimize) their profit (cost). Economists construct their models based on optimizing behaviors. In addition to these agents, we can assume that the government also acts in accordance with optimizing behavior aiming at maximizing public welfare. This principle is called the “optimization principle”.

Optimization brings us toward equilibrium so that actions of agents adjust the outcomes of the market. For example, due to behaviors of households and firms, quantities that demand and supply would be equalized via price mechanism. This aspect of the framework is defined as “the equilibrium principle”.

According to Allen (2010), a good model should have three features: Being formal, testable, and simple.

Models that are formal: Models should be explicit about its assumptions. This feature allows the researcher to develop a model when it fails to explain a phenomenon.

Models that are testable: a useful model should be tested with data and produce some results.

Models that are simple: Models that explain something in a simple way are always preferred to complicated ones since in economics we try to explain a social phenomenon including complexities and interdependencies.

The slope tells us how much y will change when x changes from x1 to x2. The sign of the slope will tell us the direction of the relation between y and x. While negative slope means a negative relation between y and x, positive slope will mean positive relation.

Elasticity is the responsiveness of the dependent variable to the change in the independent variable. To put it differently, it answers what percentage y changes when x changes 1%. Note that elasticity is equal to marginal relation (slope) multiplied by x / y ratio.

ε_y,x= %change in y / % change in x

ε_y,x= +50/-25

ε_y,x= -2

Derivative of a function, f(x), is said to be the limit of the function for a small change in the dependent variable, x.

Optimization may be done under some constraint or not. If there is no constraint when optimization is an aim, this is called unconstrained optimization. Suppose that we want to maximize our happiness and we have no constraint such as time or money. This is called unconstrained optimization. In our daily life, however, we have many constraints. We cannot go to vacation for 365 days in a year for example. We will either not have enough money or time for this. We will probably have to decide how many days to go to vacation by considering our budget and time we have. This will be a constrained optimization.

In a two-variable function, if second order derivatives exist and are themselves continuous, cross partial derivatives are equal. This is known as Young’s theorem.

The Lagrangian multiplier (λ) measures the responsiveness of the objective function to the change in the constraint function. Lagrange multiplier, which reflects the marginal impact of the constraint on the objective.