Baumol's Dynamic Model:

The dynamic model is an improvement over the static single-period model. The most serious weakness of the static model is the short time-horizon of the firm and the treatment of the profit constraint as an exogenously determined variable. These problems have been solved in the dynamic model by incorporating an extended time-horizon and endogenising the profit constraint.

Assumption:

1) The firm maximises the rate of growth of sales over its lifetime. 

2) Profit is the main source of finance for growth of sales. It is an instrumental variable whose value is endogenously determined.  

3) Demand is downward sloping and costs curves are U-shaped. The Multi-period Model

Let us assume that the sales revenue (R) grows at a rate of g percent. Over its lifetime, the stream of revenues would be,

R,R (1 + g),R (1 + g)² ......,R (1 + g)ⁿ .

Suppose i is the subjective rate of discount of the firm. It is exogenously given by the expectations and risk-preferences of the firm. The discount rate is higher than any form of market interest rate because it includes subjective assessment of risk.

The present value of the stream of future revenues is estimated by using 'i' in the discount factor as follows:

The firm attempts to maximise the present value of the stream of sales revenue over its lifetime by choosing appropriate values for the current (initial) level of sales revenue (R) and its growth rate (g). So, it is obvious that S is positively related to both R and g. 

The growth function of the firm is given by

Expansion of the firm depends on the current level of profit because the retained portion of  π is the primary source of growth. Consequently, the highest attainable growth rate (g) will be at the point of maximum profit. 

Beyond the level of sales revenue where profits are maximised, say, R_πM , the growth rate g will decline as  π declines. Beyond that point, however, current sales revenue continues to increase but the rate of growth declines. This feature is represented in the following Figure.

Thus, beyond R_πM sales revenue and growth become competing goals. The firm has to choose between higher current revenue growing at a lower growth rate over time and lower current sales growing faster over time. Although there are infinite combination of values of g and R that is available to the firm.

It would choose those, which maximise the present value of the future stream of sales S. To arrive at the equilibrium, let us consider an iso-present value curve which shows all combinations of g and R, which yield the same S. such that S is positively related to both R and g and negatively to i. Given i (which is exogenously determined),  the simplest relation that can be postulated is as follows:

S = b₁g +b₂R

where b₁, b₂ are constants and can be estimated from past performance and can be written as

Plotting this in the g - R plane would give us a downward sloping straight line. The further away an iso-present value curve is from the origin, the higher is the value of S. This is depicted as follows:

Depending on the postulate we make regarding the relation between S, g and R, that is, whether it is linear or non-linear, we accordingly would derive a linear or a non-linear iso-present value curve.

Superimposing the growth curve of Figure,  we arrive at the equilibrium level of g*, R* in Figure, where the growth-curve is tangent to the highest iso-present value curve.

Given R*, the equilibrium output X* is obtained from the TR curve. The equilibrium price P* is given by OR*/OX*. From the total profit curve, the amount of profit that accrues to the firm in equilibrium is given by  π*.