The Long-Run Growth Model

**PLEASE NOTE:** These basic lecture notes on Hall and Taylor
(HT) Chapter 4 are required for Econ 302. If possible, students
should read these notes prior to attending class lectures on HT
chapter 4. A more advanced detailed version of these notes is also
available on this web site; the advanced notes are recommended but
not required.

- Basic References:
- HT Chapter 4;
- Study Guide, Chapter 4.

In Chapter 4, HT give a more detailed discussion of the concept
of potential GDP. Specifically, HT interpret potential GDP as the
good and services that would be produced if the current labor force
were fully employed, making allowances for the natural (normal) rate
of unemployment, and if the current capital stock were being used to
full (normal) capacity. Thus, potential GDP in any period T
measures an economy's *capabilities* for producing output
rather than its actual realized output. In the remaining portions
of the chapter, HT discuss various factors affecting the growth of
potential GDP over time.

Recall from HT3 that the *real wage* in any period T is
the nominal wage W(T) divided by the cost of living as measured by
some appropriate price index. In Chapter 4, HT take this price
index to be the GDP implicit price deflator P(T), so that

W(T) (4.1) ------ = real wage for period T . P(T)

For any period T, the *full employment level* is then
defined to be the volume of employment N*(T) that would obtain,
given existing institutional incentives (e.g., fringe benefits), if
real wages were fully flexible and responsive to demand and suply
pressures.

Suppose the maximum real GDP Y that can be produced from any
given levels (N,K,A) of labor input N, capital input K, and
technology A is described by an *aggregate production
function* of the form

(4.2) Y = F(N,K,A) .

HT make the economically reasonable assumption that the production function F(N,K,A) is an increasing function of N, K, and A --- that is, given fixed values for any two of these three inputs, the value of F increases with increases in the remaining input. Moreover, HT also assume diminishing marginal returns, meaning that the output increases obtained with successive increases in any one input, holding all remaining inputs fixed, steadily diminish in size.

In particular, for any given K and A, HT assume that F(N,K,A) is an increasing function of N in the sense that

(4.3) F(N',K,A) > F(N'',K,A) if and only if N' > N'' .[see HT Figure 4-1.] Relation (4.3) simply states that, for any given levels of K and A, the maximum obtainable output increases with increases in labor input. This is a reasonable assumption, at least for realistic levels of labor input. Of course, if a firm with a fixed-size work site keeps hiring workers, eventually a point will be reached where congestion effects cause the output of the firm to begin to decline; economists call this a point of "satiation."

Diminishing returns to labor means that, as each additional worker is added into the production process, the associated incremental increase in output becomes smaller and smaller in size. This diminishing marginal returns assumption captures the idea that a production process will eventually tend toward a point of saturation if any one input is increased without bound for fixed levels of other inputs.

Assuming F(N,K,A) is a differentiable function of N, another
way to state property (4.3) is that the partial derivative (change)
of F(N,K,A) with respect to N is positive. This partial derivative
of F(N,K,A) with respect to N, referred to as the *marginal
product of labor*, will henceforth be denoted by F_N(N,K,A).
Diminishing marginal returns to labor means that the marginal
product of labor F_N(N,K,A) is itself a decreasing function of N.
This latter property will be used, below, to establish that the
labor demand curve is downward sloping in the N-W/P plane.

The level Y*(T) of real GDP that would result in period T if
employment N were equal to full employment N*(T) and if the
existing capital stock K(T) were used to full (normal) capacity is
called *potential GDP*. Potential GDP can be more formally
defined in terms of the aggregate production function (4.3) as
follows:

(4.4) Y*(T) = F(N*(T),K(T),A(T)), where N*(T) = full employment for period T; K(T) = existing stock of capital in period T; A(T) = state of technology in period T.The unemployment rate that prevails when real GDP coincides with potential GDP, denoted by U*(T), is called

As discussed in HT3, the reason the natural rate of
unemployment is not zero is that there are always some people who
continue to engage in job search by choice rather than by strict
necessity. For this reason, some economists refer to U*(T) as the
rate of *voluntary* or * voluntary frictional* unemployment.

FULL EMPLOYMENT AND POTENTIAL GDP

We will now undertake a more detailed understanding of the market equilibrium interpretation for N*(T) and Y*(T) that HT and other macroeconomists ascribe to.

For any given period T, the *labor demand curve* shows the
maximum amount of labor N that profit-maximizing firms would be
willing to hire at each different real wage W/P, taking as given the
existing capital stock K(T) and technology A(T). As depicted in HT
Figure 4-2, this profit-maximizing level of N is determined by the
condition that the marginal product of labor should equal the real
wage W/P. That is, the profit-maximizing level of N is the
particular level of N that satisfies the relation

(4.5) F_N(N,K(T),A(T)) = W/P .

The graph of relation (4.5) in the N-W/P plane is depicted in HT Figure 4-3 as a straight line for expositional simplicity. Given the usual restrictions on the aggregate production function, however, the labor demand curve would normally have more of a bowl-shaped appearance, as depicted in Figure 1 below.

The crucial property of the labor demand curve captured both in HT Figure 4-3 and in Figure 1 below is that the labor demand curve is downward sloping. The reason for this is the assumption of diminishing marginal returns to labor. As explained above, this means that the marginal product of labor appearing on the left hand side of (4.5) is a decreasing function of N. Consequently, starting at any point (N,W/P) where relation (4.5) is satisfied, if the real wage W/P is increased (decreased), then N must be decreased (increased) in order for relation (4.5) to continue to hold. That is, higher real wages correspond to lower labor demands, and vice versa.

W/P | . | . | . | . | . | . ------------------------------ N 0 Figure 1: Downward Sloping Labor Demand Curve

For any given period T, the *labor supply curve* shows the
maximum amount of labor N that households would be willing to supply
at each different real wage W/P. Hereafter this relation will
be denoted by

(4.6) N = h(W/P) .For reasons that will now be clarified, HT assume that the labor supply curve [the graph of (4.6) in the N-W/P plane] is nearly vertical, with a large positive slope, because empirical studies have shown that a change in the real wage has two opposite effects on labor supply --- a price effect and a wealth effect --- that offset each other in the aggregate. This implies that aggregate labor supply is not very sensitive to changes in the real wage. See HT Figure 4-4.

The real wage W/P measures the price (opportunity cost) of an
hour of leisure. Consequently, in analogy to other goods, one might
expect to see the demand for leisure fall (and hence the supply of
labor rise) as the real wage W/P increases, implying an upward
sloping labor supply curve. Call this the *price effect* of an
increase in W/P.

On the other hand, for any given amount of supplied labor, an
increase in the real wage W/P also represents a direct increase in
income, and this increase in income is more substantial the longer
the increase in W/P is expected to last. Indeed, since wealth is
simply accumulated income, an increase in the real wage also implies
an increase in wealth. If leisure is a *normal good*, in the
sense that the demand for leisure increases with increases in income
or wealth, then an increase in W/P would tend to increase the demand
for leisure and hence to decrease the supply of labor. Call this
the *wealth effect*.

The price effect and the wealth effect thus tend to move the
supply of labor in opposite directions, implying that the *net*
change in labor supply following upon a real wage change is
theoretically ambiguous. Labor supply might increase in response to
an increase in the real wage, resulting in an upward sloping labor
supply curve (price effect outweighs wealth effect), or it might
decrease, resulting in a backward bending labor supply curve (wealth
effect outweighs price effect). In Figure 2 below, for example, the
depicted labor supply curve becomes backward bending (wealth effect
outweighs price effect) for real wages higher than (W/P)', implying
that labor supply actually decreases with increases in the real wage
above (W/P)'.

W/P | s N = h(W/P) | s | (W/P)'|. . . . . . . s | s . | s . ----------------------------- N 0 N' Figure 2: Backward Bending Labor Supply Curve

As previously mentioned, this theoretical ambiguity regarding whether the labor supply curve is upward sloping or backward bending appears to be reflected in the empirical finding for the U.S. that estimated labor supply curves tend to be fairly steeply sloped. In particular, in the aggregate, labor supply appears to be fairly insensitive to changes in the real wage, suggesting that wealth and price effects are offsetting one another. As we will see later on, HT make implicit use of this empirical finding by assuming that labor supply (hence potential GDP) is unaffected by changes in the income tax rate t.

as Equilibrium Levels:

For any given period T, full employment N*(T) and potential GDP Y*(T) can now be characterized in terms of labor market equilibrium as follows.

Let the labor demand curve and the labor supply curve be graphed together in the N-W/P plane. [See HT Figure 4-5.] The point of intersection of these two curves then represents equilibrium in the labor market, in the sense that labor demand equals labor supply.

At this point of intersection, two quantities are determined:
an amount of labor; and a real wage. Let the amount of labor be
termed the *full employment level* for period T, denoted by
N*(T) and let the real wage be termed the *full employment real
wage* for period T, denoted by (W/P)*(T). See Figure 3, below.

s W/P N^S = h(W/P) | d | d s | d | d s (W/P)*(T) |.....................d | s . d F_N(N,K(T),A(T)) = W/P | s . d | s . d ------------------------------------------------- N 0 N*(T) Figure 3: Labor Market Equilibrium

Once N*(T) is determined, potential GDP for period T, denoted by Y*(T), is determined as the maximum GDP that can be produced in period T using N*(T), taking as given the existing capital stock K(T) and the existing technology A(T). Thus, Y*(T) is the level of GDP generated by the aggregate production function F(N,K(T),A(T)) evaluated at N=N*(T). See Figure 4, below.

Y | Y = F(N,K(T),A(T)) | p | p Y*(T) |....................p | p . | p . | p . | p . | p . --------------------------------------------------- N 0 N*(T) Figure 4: Determination of Y*(T) from N*(T)

*Some Key Points to Note:*- (a) By construction, N*(T), (W/P)*(T), and Y*(T) are
*hypothetical*equilibrium levels for labor, real wage, and output. These equilibrium levels depend*only*on the following four factors: (1) the form of the labor supply curve h(); (2) the form of the aggregate production function F() [including the properties of its partial derivative F_N()]; (3) the current level of capital K(T); and (4) the current state of technology A(T). Note in particular that, ceteris paribus (i.e., everything else held fixed), Y*(T) does not vary in response to changes in the period T market price P(T). - (b) Unanticipated disturbances to the economy in period T
that affect Y*(T) in period T are referred to as
*period T supply-side shocks*. Examples of period T supply side shocks would be an unanticipated increase in A(T) (e.g., a new invention) in period T, or a sudden change in household tastes in period T that shifts out the period T labor supply curve (for each real wage, a higher labor supply). In each case, the value of Y*(T) in period T would increase. - (c) However, Y*(T) can also change
*over time*in a more gradual anticipated way. For example, as will be more carefully discussed in subsequent chapters, a positive net investment level I(T) in period T translates into a capital level K(T+1) in period T+1 that is higher than the capital level K(T) for period T. Given the assumptions imposed above on the aggregate production function, this increase in capital input in period T+1 causes the aggregate production function to shift up in period T+1 (for each level of N, a higher Y). Ceteris paribus, this results in a potential GDP level Y*(T+1) for period T+1 that is higher than the potential GDP level Y*(T) for period T.

Potential GDP tends to change over time due to three key supply-side events:

Measuring changes in the quantity of labor and capital inputs, and to some extent in the quality of labor and capital inputs, is possible in principle; but measuring changes in total factor productivity is more problematic. Many researchers attempt to measure changes in total factor productivity --- or "productivity change" for short --- by defining productivity change as a residual change in actual measured real GDP after changes in the quantity and quality of labor and capital inputs have been accounted for. In schematic terms:

(4.7) [Change in Total Factor Productivity] = [Total change in real GDP] - [Change in real GDP due to change in N quantity and quality] - [Change in real GDP due to change in K quantity and quality].

[Important: Note that the total factor productivity concept
currently under discussion differs from the concept of *labor*
productivity discussed in HT3 and in Chapter 2 of the ERP (pp. 56-59); labor
productivity is measured as output per hour of labor.]

Relation (4.7) is the approach to "growth accounting" undertaken by Robert Solow, a 1987 Nobel prize winner in economics. Solow's work on growth accounting is now so well known, and so extensively used in both empirical and policy research, that it is worthwhile to present it in some detail.

Suppose the period T aggregate production function relating maximum possible output Y to labor usage N, conditional on the capital stock K(T) and technology A(T), takes the particular form

(4.8) Y = A(T)f(N,K(T)) .The technology factor A(T) is then a measure of

Also, the representation (4.8) for the aggregate production function implies that any growth in Y from one period to the next must be attributable either to a change in total factor productivity, or to a change in labor usage, or to a change in capital usage.

Robert Solow used the basic form (4.8) for the aggregate production function to derive a formula connecting the percentage rate of change in Y with the percentage rates of change in A, N, and K. In the discussion of this formula given in the appendix to HT Chapter 4, HT use discrete time changes [Y(T+1)-Y(T)] rather than time derivatives dY(T)/dT, resulting in a heuristic rather than a rigorous derivation. Derivatives will be used in the following presentation of the Solow growth formula so that the formula is in exact rather than approximate form. [For a rigorous derivation of the Solow growth formula, see the advanced class lecture notes for HT4 linked to the Econ 302 home page.]

*Notational Note:* For expositional simplicity, given any
variable V(T) that depends on time, the expression DV(T) will be
used to denote dV(T)/dT, the derivative of V with respect to time.
Also, R^K(T) will be used to denote the (imputed) rental price of
capital in period T. Finally, it will be assumed that the only two
sources of income in the economy are wage income and (imputed)
capital rents, so that real GDP Y(T) can be expressed as the sum
of real wage payments to labor and real rent payments to capital.

THEORETICAL FORM OF THE SOLOW GROWTH FORMULA: DY(T) DA(T) W(T)N(T) DN(T) R^K(T) K(T) DK(T) (4.9) ------- = ----- + -------- ----- + ----------- ----- . Y(T) A(T) P(T)Y(T) N(T) P(T)Y(T) K(T) labor share capital share of nominal GDP of nominal GDP

The ratio W(T)N(T)/P(T)Y(T) consisting of the nominal wage bill W(T)N(T) divided by the nominal level P(T)Y(T) of GDP is the share of nominal GDP that is paid out to labor. Similarly, the ratio R^K(T)K(T)/P(T)Y(T) is the share of nominal GDP that is paid out to capital. The empirical measurements of these shares for the U.S. have been remarkably stable over time, equalling approximately 0.7 and 0.3, respectively. Substituting these empirical share estimates into (4.9), and omitting the time period T for expositional simplicity, one obtains:

EMPIRICALLY ESTIMATED FORM OF THE SOLOW GROWTH FORMULA: DY DA DN DK (4.10) ---- = ---- + .7 ---- + .3 ---- . Y A N K[Compare HT equation (4-7).] Finally, re-arranging terms in (4.10), one obtains:

DA DY DN DK (4.11) ---- = ---- - .7 ---- - .3 ---- . A Y N K

Since all terms on the right-side of (4.11) are measurable in principle, (4.11) gives an empirically operational way to measure the growth rate DA/A of total factor productivity.

According to the Solow growth formula (4.10), a 3.3 percent increase in the growth rate DK/K of capital translates into only a 1 percent increase in the growth rate DY/Y of real GDP. That is:

(4.12) 1% increase in DK/K ---> .3% increase in DY/Y ; hence (4.13) (1/.3)% = 3.3% increase in DK/K --> 1% increase in DY/Y.

On the other hand, also according to (4.10), employment growth has over twice the leverage of capital growth. Specifically:

(4.14) 1% increase in DN/N ----> .7% increase in DY/Y or (4.15) (1/.7)% = 1.43% increase in DN/N --> 1% increase in DY/Y.

Finally, the most leverage of all is obtained from a change in productivity growth. Specifically:

(4.16) 1% increase in DA/A -------> 1% increase in DY/Y .

In HT Figure 4-7, estimates are given for the factor contributions DN/N, DK/K and DA/A to real GDP growth DY/Y for the U.S. from 1961 to 1990. In order to smooth out business cycle fluctations, the estimated factor contributions are averaged over ten-year periods. The data reveal a relatively increasing contribution to GDP growth from both capital investment, DK/K, and labor growth, DN/N, over this time period. However, there appears to have been a marked decrease in the relative contribution from total factor productivity growth, DA/A.

Recently a consensus seems to be forming across all sectors of the U.S. economy that a large part of the measured productivity slowdown can be attributed to a decine in educational quality, leading to a sharp decline in the human capital embodied in the labor force. One very important form of human capital in decline seems to be entrepreneurial skill.

Moreover, there also seems to be widespread agreement that the deteriorating nature of the U.S. economy's infrastructure (i.e., its highways, bridges, dams, airports, and other publicly owned capital) is also a significant factor in the productivity slowdown. Over the past twenty years, U.S. government investment in infrastructure has taken place at a low rate, relative both to American experience and to rates of infrastructure investment in other developed countries.

The primary way to stimulate output growth through
increased labor *quality* is through increased expenditures on
human capital, e.g., education, training, health and safety
measures, etc. An increase in labor *usage* might be
encouraged by increasing labor supply at each real wage by lowering
marginal tax rates on wage income, and/or by increasing labor demand
at each real wage by reducing private firm sector responsibility for
provision of key worker benefits, e.g., health insurance.

The primary way to stimulate output growth through
increased capital *quality* is through increased expenditure on
research and development (R&D). An increase in capital *usage*
(as opposed to capital quality) might be stimulated by various
types of tax credits or deductions (e.g., an investment tax
credit).

Finally, a stimulation of output growth through increased productivity growth (increases in DA/A) might be achieved by more efficient exploitation of economies of scale, better organization of the workplace, better product distribution systems, better allocation of resources (e.g., job availability services), an improved work ethic, and so forth.

What is the proper role of government, if any, in encouraging productivity growth? As stressed throughout the ERP, government could increase public expenditures on human and physical capital investment, including basic research and development, and/or it could encourage the private sector to increase such expenditures by implementing appropriate tax and transfer policies. Alternatively, a minimalist government could focus exclusively on traditional government objectives (e.g., national security), leaving the determination of productivity growth largely to the private sector.

This issue of "devolution" (the proper division of economic responsibilities among the Federal government, State and Local governments, and the private sector) is the focus of Chapter 3 of the ERP. It will be explored at a later point in the course after the complete HT model has been presented.