Macro Notes - American University · Jones Chapter 6: Romer Model Combining the simple Solow and Romer ModelsMalthusian ModelReferences Macro Notes Alan G. Isaac American University

Jones Chapter 6: Romer Model Combining the simple Solow and Romer Models Malthusian Model References

Macro Notes

Alan G. Isaac

American University


Every generation has perceived the limits to growththat finite resources and undesirable side effectswould pose if no new recipes or ideas werediscovered. And every generation has underestimatedthe potential for finding new recipes and ideas. Weconsistently fail to grasp how many ideas remain tobe discovered.– Paul Romer (1993), as cited in Jones (2011, p.130)


Within thirty years, we will have the technologicalmeans to create superhuman intelligence. Shortlyafter, the human era will be ended.– Vinge (1993)


Ideas

Paul Romer (1990 JPE) proposes a fundamental distinctionbetween objects and ideas, especially ideas about how to makeobjects.

instructions

recipes

designs

blueprints


Nonrivalry

Ideas are

costly to generate

nonrivalrous in use

Nonrivalry: my use of an idea does not limit your use.

Rivalry is the basis of scarcity. Objects are generally highlyrivalrous: one person’s use of an object reduces its usefulnessto another person.

In contrast, if I implement a production design, you canequally well implement the design. In this sense, the design isnonrivalrous.(But, any particular implementation will be rivalrous.)


Nonrivalry vs. Nonexcludability

We are interested here in the idea, not the medium oftransmission. (We might transmit instructions or a recipe or adesign or a blueprint on paper or on a flash drive.)

Nonrivalry does not imply nonexcludability.

Excludability: the extent to which there are enforceableproperty rights in a good, allowing the restriction of access oruse.


Scarcity

Economics is

“the science which studies human behaviour as arelationship between ends and scarce means whichhave alternative uses.”Lionel Robbins (1932)

The scarcity of “means” forces us to make trade-offs betweentheir alternative uses.This notion of scarcity invokes rivalry in use.For economists, the term scarcity is used to indicate arequirement to make trade-offs.Even so, the term has important ambiguities, and economistsare more likely to use more precise terms, such as rivalry orexcludability.


Scarcity

Many ordinary economic goods are inherently scarce in thesense of being rivalrous.If I consume a peach, you cannot also consume it.Some economic goods are less scarce in this sense.If I watch a TV, you might also be able to watch it. But 50 ofus cannot watch it at the same time.In a “knowledge-based economy”, some economic goods arebarely scarce in this sense. If I watch a film on cable TV, youcan too, and so can many others.Of course such goods are still produced with scarce resources.


Excludability

A good that is not rivalrous may be excludable. One may seeexclusion as a way to create scarcity of a nonrivalrous good.

E.g., cable TV.

Naturally the incentives of the cable TV company to provideprogramming are affected by its ability to exclude viewers whodo not pay.

Ideas are perfectly nonrivalrous. In this sense, any existing ideais not inherently scarce. However it may be rendered “scarce”(i.e., exculdable) through patent or copyright.


Scarcity of New Ideas

What about new ideas?

These are hard to come by: to produce new ideas we give upother production.

In this sense, new ideas are scarce.


Ideas and Returns to Scale

Ideas → nonrivalry → increasing returns → problems withpure competition


Sunk Costs and Increasing Returns


Yt = AtK1/3t L

2/3t

We still have constant returns to scale in capital and labor.Think about the technological parameter as another input toproduction: if we double all inputs (A,K , L), then wequadruple output.That is, we have increasing returns to scale in all inputs.


Let’s work for a moment with a simplified version: a singleinput, which we will call labor.

Yt = AtLyt

So we have constant returns to scale in labor alone, butincreasing returns to scale in all inputs (A, L).


Addition to our theory: tell a story about A.The growth rate of technology (gA) depends on the amount oflabor we allocate to the production of new ideas.

Yt = AtLyt

∆At/At = zLαt

Lyt + Lαt = N

Lαt = ¯N

(Here, ∆At ≡ At+1 − At .)


The Romer model on the material standard of living, Y /N .

Y

N= At

Lyt

N(1)

= At(1− ¯) (2)


So, at point in time, allocating labor to the production ofknowledge is costly. But over time, it pays off in new knowlegeand higher levels of production:

∆At

At= zLαt

= z ¯N

= g

(3)

At = (1 + g)At−1 = (1 + g)tA0 (4)


Source: ERP (2010, Figure 10-3)


The growth of technology determines the growth rate of ourmaterial standard of living:

yt =Yt

Nt

= At(1− ¯)

= (1 + g)tA0(1− ¯)

= A0(1− ¯)(1 + g)t

(5)


Output per Person over Time


Increase in Population


Increase in R& D


Word Processor Production Function


Growth Accounting

Consider the production function.

Yt = AtK1/3t L

2/3yt (6)

We can express this in rates of growth:

gYt = gAt +1

3gKt +

2

3gLyt (7)

So to understand the growth rate of output, we need tounderstand the three growth rates on the right.


Similarly

gYt − gLt︸︷︷︸growth of Y /L

= gAt +1

3(gKt − gLt)︸︷︷︸change in K/L

+2

3(gLyt − gLt)︸︷︷︸labor composition

(8)



Note the productivity slowdown after the first oil price shock.(A decline is R&D spending has also been blamed.)

The productivity recovery in the 1990s is sometimes referredto as the new economy. This change has been linked to ITinvestment.

BUT: We cannot observe gA. We compute it as a residual.


Our simple endogenous (Romer) growth model implies agrowth rate for technology:

gAt =∆At

At= zLαt = z ¯N = g (9)


Our simplest neoclassical (Solow) growth model implies agrowth rate for capital:

gKt = ∆Kt/Kt = sYt/Kt − d (10)

which tells us that if we are on a balanced growth path, suchthat Yt/Kt constant, then gKt is constant over time.Along a balanced growth path we have

g ∗K = g ∗

Y (11)

Let us return to this in a bit.


Kaldor (1958) offered six famous “stylized facts” about thegrowth of advanced industrialized economies:

1 Y /N > 0 relatively constant in LR (exponential growth)

2 K/N > 0 and K roughly constant

3 Y /K ≈ 0 in LR: capital output ratio fairly constant

4 π/K ≈ 0 in LR, so the rate of return to capital istrendless. For example, the real interest rate ongovernment debt in the U.S. is trendless.

5 Y and Y /N vary a lot across countries

6 high π/Y associated with high I/Y


The number of production workers does not change in oursimple growth models. (Production workers are a constantfraction of a constant population.)

gLyt = 0 =⇒ gY = g +1

3gK +

2

30 (12)

or, along a blanced growth path,

g ∗Y = g +

1

3g ∗Y +

2

30 (13)

or

g ∗Y =

3

2g =

3

2z ¯N (14)

This is the growth rate along a balanced growth path.


Contrast this with our simplest Romer model, where g ∗Y = g .

The difference arises from our reintroduction of capitalaccumulation. Now, in addition to the direct effect of higherproductivity, we have an indirect effect: higher productivityleads to higher capital stock, which also raises output.

It is still the case that capital is not the engine of growth.But capital accumulation amplifies the effect of productivitygrowth.


Solow II: Combing Romer and Solow I

Econ 301 students are not responsible for the Solow-Romeralgebra that follows.


Two equations from our simplest neoclassical (Solow) growthmodel:

Yt = AtK1/3t L

2/3yt (15)

∆Kt = sYt − dKt (16)

Changes: At can vary over time; Lyt instead of all labor.Three equations from Romer model:

∆At/At = zLαt (17)

Lyt + Lαt = N (18)

Lαt = ¯N (19)

At any time t, the five endogenous variables (“unknowns”)are: Yt , Kt+1, At+1, Lyt , and Lαt .


Output per Person

yt = Yt/L

= Yt(1− `)/Lyt

= Atk1/3t (1− `)

(20)

where k = K/Ly .


The capital output ratio along a balanced growth path isconstant.

g ∗Y = g ∗

K

g ∗Y = s(Y ∗

t /K ∗t )− d

(21)

Solve for (K/Y )∗:

(K/Y )∗ = s/(g ∗Y + d) (22)


Next let us solve for income per worker. Start from theproduction relation

Yt = AtK1/3t L

2/3yt (23)

we can rewrite this as(Yt

Lyt

)2/3

= At

(Kt

Yt

)1/3

(24)

orYt

Lyt= A

3/2t

(Kt

Yt

)1/2

(25)

orYt

N= (1− `)A

3/2t

(Kt

Yt

)1/2

(26)

Along a balanced growth path we therefore have(Yt/N

)∗= (1− `)A

3/2t

(s

g ∗Y + d

)1/2

(27)


Transition Dynamics: Increase in s

Source: Jones (2011)


Saving and Growth

So we see:raising the saving rate can only temporarily raise growth.But now we have another way to save.As in the simplest Romer model, we can allocate more labor toresearch and development. This still raises the growth rate,even in the long run.


Delong offers a simple Malthusian variant. http://delong.

typepad.com/sdj/2008/02/econ-101b-feb-1.html

Resources R are fixed in quantity but augmented bytechnological progress.

ER = E = gR (28)

Resources are an input into the production function:

Y = Kα(ER)βL1−α−β (29)

orY /L = (K/L)α(ER/L)β (30)

Steady state factor use per “capita” will be constant, so Y /Lwill too.

http://delong.typepad.com/sdj/2008/02/econ-101b-feb-1.html

http://delong.typepad.com/sdj/2008/02/econ-101b-feb-1.html


This suggestsLss = gR (31)

Population growth must move to a rate justified bytechnological progress. What is the mechanism?


Adopt a Malthusian mechanism: population grows fasterabove subsistence level y ∗.

L = γ(Y /L− y ∗) (32)


In the steady state, the population growth rate is constant.

Lss = γ(Y /L− y ∗) (33)

so(Y /L)ss = y ∗ + Lss/γ (34)

is also constant. This means

Yss = Lss (35)

Furthermore(Y /L)ss = y ∗ + gR/γ (36)

We live about subsistence only to the extent that technicalchange allows.


Now from the production function

Y = αK + βER + (1− α− β)L (37)

so0 = αK/L + βER/L (38)

which holds as the factor ratios are constant in the steadystate.


Capital is accumulated in standard fashion:

K = sY /K − δ (39)

Since (K/L)ss = 0, we must have

Lss = sY /K − δ (40)

orγ((Y /L)ss − y ∗) = s(Y /K )ss − δ (41)


We can solve forY

L= y ∗ + gR/γ (42)

http://delong.typepad.com/delongslides/2008/02/

the-malthusian.html Only technological change holds usabove the subsistence level.

http://delong.typepad.com/delongslides/2008/02/the-malthusian.html



How do we escape from the Malthusian trap? DeLong saysthe most obvious way is to raise y ∗, but he wonders about theeffectiveness of this strategy. http://delong.typepad.com/

delongslides/2008/02/the-malthusian.html




Escape from the Trap: Permanent or Temporary

Source:http://delong.typepad.com/delongslides/2008/02/

greg-clarks-wor.html

http://delong.typepad.com/delongslides/2008/02/greg-clarks-wor.html

http://delong.typepad.com/delongslides/2008/02/greg-clarks-wor.html


Romer, Paul (1990, October). “Endogenous TechnologicalChange.” Journal of Political Economy 98(part 2),S71–S102.

Vinge, Vernor (1993, Winter). “The TechnologicalSingularity.” Whole Earth Review 81.

Documents

Macro Notes - American University · Jones Chapter 6: Romer Model Combining the simple Solow and Romer ModelsMalthusian ModelReferences Macro Notes Alan G. Isaac American University