In introductory treatments of economics the time dimension of the problem is often ignored, or supressed for
simplicity. In reality, the time dimension is always present. In the examples of market equilibrium already discussed,
we should think of the quantities as flows per period of time, so that qD is the quantity demanded of a good per
period, however short or long that period may be in terms of calendar time. Models where we explicitly or implicitly
consider a situation within a single period of time we refer to as static models. The economic activities studied in
static models do not take into account the history of these activities and do not consider the future consequences of
these activities. Static problems are solved independently of the passage of time. Put this way, it seems that the static
framework for economic anal- ysis is extremely restrictive. However, many useful models of economic behavior
have been developed in the static framework, and we will be studying several such examples in the book. Many
problems, though, are necessarily dynamic in nature. The theories of economic growth, inflation, and resource
depletion, for example, are impossible to model without explicit consideration of the time dimension.
Explicit consideration of time opens up new challenges as well as new oppor- tunities for the modeler, and in the
later chapters of the book we will develop the more important techniques for studying dynamic models. It is useful
at this point, however, to highlight some important concepts that emerge when we explicitly account for the passage
of time.
Quantities and values that recur each period in a dynamic model continue to be referred to as flows. Income,
investment, saving, production, worker hires, and purchase and consumption of goods are examples. Some of these
flows may be
8 CHAPTER 1
INTRODUCTION
stored or accumulated into stocks. Flows of investment in machinery become the accumulated capital stock.
Accumulated savings are assets.
Accumulated hires are the workforce. Unsold production accumulates into inventories. Accumulated deficits are
total debt, and so on.
Mathematically we must specify the relationships between the flows and the stocks as part of any dynamic model.
The relationship between investment and the capital stock is a useful example. Denote by It the amount of
investment during period t , and denote the capital stock at the begining of period t by Kt . Then we can define the
flow of investment in terms of the change in the capital stock between periods (ignoring the depreciation of capital):
It =Kt+1−Kt
The flow of investment is simply the change in the stock of capital. Note that we use time subscripts to date the
stocks and the flows that we are interested in.
From the modeling point of view, we have certain choices when we are decid- ing on the appropriate mathematical
structure for a dynamic model. One of these is the modeling of time itself. So far we have thought of time as being
divided up into intervals or “periods.” In these models of discrete time, all relevant economic fac- tors are allowed to
change between periods but not within periods. Of course, since the length of a period is arbitrary, this condition is
not very restrictive. Different mathematical techniques are required if we wish to think of time as evolving continuously. In continuous time models, we date the stocks and flows by instants in time, and we can invoke calculus
to define the relationships between flows and suc- cessive instantaneous values of stocks. We show how this is done
later in the book.
In introductory treatments of economics the time dimension of the problem is often ignored, or supressed for
simplicity. In reality, the time dimension is always present. In the examples of market equilibrium already discussed,
we should think of the quantities as flows per period of time, so that qD is the quantity demanded of a good per
period, however short or long that period may be in terms of calendar time. Models where we explicitly or implicitly
consider a situation within a single period of time we refer to as static models. The economic activities studied in
static models do not take into account the history of these activities and do not consider the future consequences of
these activities. Static problems are solved independently of the passage of time. Put this way, it seems that the static
framework for economic anal- ysis is extremely restrictive. However, many useful models of economic behavior
have been developed in the static framework, and we will be studying several such examples in the book. Many
problems, though, are necessarily dynamic in nature. The theories of economic growth, inflation, and resource
depletion, for example, are impossible to model without explicit consideration of the time dimension.
Explicit consideration of time opens up new challenges as well as new oppor- tunities for the modeler, and in the
later chapters of the book we will develop the more important techniques for studying dynamic models. It is useful
at this point, however, to highlight some important concepts that emerge when we explicitly account for the passage
of time.
Quantities and values that recur each period in a dynamic model continue to be referred to as flows. Income,
investment, saving, production, worker hires, and purchase and consumption of goods are examples. Some of these
flows may be
8 CHAPTER 1
INTRODUCTION
stored or accumulated into stocks. Flows of investment in machinery become the accumulated capital stock.
Accumulated savings are assets.
Accumulated hires are the workforce. Unsold production accumulates into inventories. Accumulated deficits are
total debt, and so on.
Mathematically we must specify the relationships between the flows and the stocks as part of any dynamic model.
The relationship between investment and the capital stock is a useful example. Denote by It the amount of
investment during period t , and denote the capital stock at the begining of period t by Kt . Then we can define the
flow of investment in terms of the change in the capital stock between periods (ignoring the depreciation of capital):
It =Kt+1−Kt
The flow of investment is simply the change in the stock of capital. Note that we use time subscripts to date the
stocks and the flows that we are interested in.
From the modeling point of view, we have certain choices when we are decid- ing on the appropriate mathematical
structure for a dynamic model. One of these is the modeling of time itself. So far we have thought of time as being
divided up into intervals or “periods.” In these models of discrete time, all relevant economic fac- tors are allowed to
change between periods but not within periods. Of course, since the length of a period is arbitrary, this condition is
not very restrictive. Different mathematical techniques are required if we wish to think of time as evolving continuously. In continuous time models, we date the stocks and flows by instants in time, and we can invoke calculus
to define the relationships between flows and suc- cessive instantaneous values of stocks. We show how this is done
later in the book.