production and growth - american university · 2021. 2. 25. · the neoclassical growth model of...

Production and Growth

Neoclassical Growth Model Analytical Solution, Numerical Solution, Linearization, and Relaxation

Continuous Time Growth Model The neoclassical growth model of the 1950s is often called the Solow growth model or the Solow-Swan growth model. The canonical citation is [Solow-1956-QJE]_.

Fundamental Dynamic Equation

This section derives the fundamental dynamic equation of the neoclassical growth model and character-izes the implied steady state.

Deriving the Fundamental Dynamic Equation

Define the effective capital-labor ratio to be k = K /L, where K is the capital stock and L is the effective labor supply. (Effective labor is not just labor hours; it includes the level of labor productivity.) Let a dot indicate a time derivative and apply the quotient rule.

k=

K

L-

L

K

L2(7)

Check this using computer algebra. When using WL, user defined symbols should begin with a lower-case letter. For convenience, this presentation nevertheless uses K and L.

In[124]:= ClearAll[f, F, K, L, S, k, kk, nn, λ, s, n, r, w, y, kdot]

kdot = DK[t] L[t], t

Out[125]=K′[t]

L[t]-K[t] L′[t]

L[t]2

The change in the capital stock (K

) equals net saving, which is gross saving (S) less the depreciation of the existing capital stock (d K). Substituting the accounting relationship K

= S - d K produces

In[126]:= kdot = kdot /. {K'[t] → S[t] - d K[t]}

Out[126]=-d K[t] + S[t]

L[t]-K[t] L′[t]

L[t]2

Let a hat indicate a proportional rate of change, and rewrite the rate of change of the effective capital-

2 Solow_continuous.nb

labor ratio as follows.

k=

S - d + LK

L(8)

This numerator of this expression focuses our attention on the determinants of the change in the effective capital labor ratio. A first simplifying assumption of this growth model is that population growth rate is constant (L = n). (Alert: this notation diverges slightly from MRW, discussed below.) Make this substitution as follows.

In[127]:= kdot = kdot /. {L'[t] → n L[t]} // Simplify

Out[127]=

-d + n K[t] + S[t]

L[t]

k=

S - d + nK

L(9)

A second simplifying assumption is that saving rate out of income is constant: S = s Y.

In[128]:= kdot = kdot /. {S[t] → s Y[t]}

Out[128]=

-d + n K[t] + s Y[t]

L[t]

k=

sY - d + nK

L(10)

Traditionally, we write these in terms of k = K /L and y = Y /L.

In[129]:= kdot = kdot /. {K[t] → L[t] × k[t], Y[t] → L[t] × y[t]} // Simplify

Out[129]= -d + n k[t] + s y[t]

k= s y - d + n k (11)

If it is possible to express y as a function of k, say f [k], this produces the fundamental dynamic equation of the Solow model: a first-order differential equation in k.

In[130]:= kdot = kdot /. {y → f[k]}

Out[130]= -d + n k[t] + s f[k][t]

k= s f k - d + n k (12)

Particularly interesting is the possibility of a steady state, where k= 0. For a nonzero steady state, this

implies

0 = s f k - d + n k (13)

f k

k=

n + d

s(14)

Solving this equation determines the steady state value(s) of k.

CRTS Production Technology

Solow_continuous.nb 3

Let Y = F[K, L]. Constant returns to scale in capital and efficiency labor imply that scaling the inputs proportionally scales the output.

In[131]:= Clear[F, K, L]

crts = λ F[K, L] ⩵ F[λ K, λ L];

An immediate implication is that L F[K /L, 1] ≡ F[K, L].

In[133]:= crts /. λ → 1 L

Out[133]=F[K, L]

L⩵ F

K

L, 1

Recalling the notation y = Y /L, and k = K /L, we can write production in per capita terms. (For simplicity, we often call this per capita, although the scaling is by effective labor.)

Y = F[K, L]Y / L = F[K / L, 1]

y = f[k]

(15)

Give constant returns in K and L, the fundamental dynamic equation of the Solow-Swan model is a non-linear first-order differential equation in k. Express this in WL as the following function.

In[134]:= ClearAll[kdot, s, f, d, n, k, y, B, α]

kdot[k_] := s f[k] - d + n k

Given a particular function form for f , we can then solve for the steady state value of k.

Cobb-Douglas Production Function

A popular special case of such technology is the Cobb-Douglas production function.

Y = B Kα L1-α

y = B kα(16)

In this section, B will just be a constant scale factor. (When it is allowed to change, its growth is referred to as Hicks-neutral technical change.)

Illustrating this technology graphically depends on specific values for the model parameters. Empiri-cally, many estimates of α are around 1 /3; use 0.36 for now. (A subsequent section will produce an estimate.) The value for B depends on the units chosen for K and L; for now set it rather arbitrarily to 5.0. These two values will be our first set of production parameters. Substituting these values (with the /. operator) produces a production function we can plot.


In[136]:= Clear[f]

prms`cd01 = {B → 5, α → 0.36}; (* prms`cd01 matches NFR *)

f[k_] := EvaluateB kα /. prms`cd01 (* parameterized prodn *)

plot`cd = Plot[f[k], {k, 0, 60},

PlotLabel → "Cobb-Douglas Production Technology", AxesLabel → {"k", "y"}]

Out[139]=

10 20 30 40 50 60k

5

10

15

20

yCobb-Douglas Production Technology

Marginal Products

In[140]:= Clear[f, F, k, K, L, s, n, w, r]

Recall that with constant returns to scale, the following equality expresses an identity between the two sides: F[λK, λ L]⩵λ F[K, L]. Since this is an identity, we may differentiate the two sides and still pro-duce an equality. Differentiate with respect to λ.

In[141]:= D[crts, λ]

Out[141]= F[K, L] ⩵ L F(0,1)[K λ, L λ] + K F(1,0)[K λ, L λ]

The resulting identity holds true for any value of λ. (The right side is unaffected by λ because linear homogeneity of the function F implies the partial derivatives are homogeneous of degree 0.) For example, letting λ→ 1:

In[142]:= D[crts, λ] /. {λ → 1}

Out[142]= F[K, L] ⩵ L F(0,1)[K, L] + K F(1,0)[K, L]

Rewrite this with more familiar mathematical notation, using subscripts to indicate partial derivatives.

L FL + K FK ⩵ F[K, L]

Next, we express these marginal products in terms of f instead of F.


In[143]:= crts /. λ → 1 L

Assuming[L > 0, MultiplySides[%, L]]

Out[143]=F[K, L]

L⩵ F

K

L, 1

Out[144]= F[K, L] ⩵ L FK

L, 1

Define f [k] = F[k, 1] so that under constant returns to scale F[K, L] = L f [K /L], and use the Grad com-mand to produce the gradient (i.e., the vector of first-order partial derivatives)

In[145]:= Grad[L * f[K / L], {K, L} ] //. {K / L → k}

Out[145]= {f′[k], f[k] - k f′[k]}

Next, do the same thing by hand:

F[K,L] ≡ L f[K/L]FK[K, L] = L f '[K /L]* 1

L= f '[k]

FL[K, L] = f [K /L] + L f '[K /L] * -K

L2 = f [k] - k f '[k]

With competitive factor markets, marginal products are equal to real factor prices. Let r denote the real return to a unit of capital, often called the rental rate, and let w denote the real wage. This implies that factor payments exactly exhaust production.

In[146]:= D[crts, λ] /. {λ → 1} /. F(1,0)[K, L] → r, F(0,1)[K, L] → w

Out[146]= F[K, L] ⩵ K r + L w

r K + w L==Y

r k + w==y

Plotting the function f requires a specific functional form and a specification of the function parame-ters. Based on the derived expressions, a simple plot can illustrate the marginal products. As a prelimi-nary, recall that the tangent line to a real function f at a point x0 can be described by a first-order Taylor expansion around x0.

In[147]:= Clear[f]

f[k_] := EvaluateB kα /. prms`cd01

tangent[f_, x0_] := x f[x0] + x - x0 f'[x0] (* f is a parameter *)


Out[150]=

k0k

w0

y0

yFactor Prices

r0=slopef[k]

Steady State of the Solow-Swan Model

Recall our fundamental dynamic equation for the Solow-Swan model under CRTS.

k= s f k - d + n k

Characterize a steady state of the Solow-Swan growth model by k= 0.

0 ⩵ss

s f k - d + n k

Cobb-Douglas Technology

Given a Cobb-Douglas production technology, the growth rate of the capital stock is fully parameter-ized. With a Cobb-Douglas production technology, the fundamental dynamic equation becomes

k= s B kα - d + n k (17)

Given the Cobb-Douglas production technology, the equation k⩵ 0 describes a unique positive steady-

state value of k. (By inspection, we can see that k ⩵ 0 also describes a steady state; we will return to this.)

In[151]:= Clear[f, B, α, s, d, n, k]

f[k_] := B kα

ss`keq = 0 == kdot[k]

Out[153]= 0 ⩵ -k d + n + B kα s

Solve for the steady-state value by hand in the usual way to find k ⩵ B s

d+n

11-α in the steady state. Check

this with Solve.

In[154]:= soln = Solve[ss`keq, k] // Quiet

Out[154]= k →B s

d + n

1

1-α

Remember that this produces a list of solutions. Use the first (and only) solution in this list as assign to a variable.


In[155]:= ss`k = k /. First@soln

Out[155]=B s

d + n

1

1-α

Graphical Illustration of the Steady State

We already introduced values for the production function parameters, in order to graph the production function. A graphical characterization of the steady state requires specification of a few more parame-ters. The following values are crudely plausible in magnitude; we will experiment with them later.

In[156]:= prms`ss20 = Join[prms`cd01, {d → 0.075, n → 0.01, s → 0.2}];

Out[158]=

kssk

s y(d+n)k

Equilibrium in the Solow-Swan Model

s y

(d+n)k

Solving with Mathematica (Solve vs. Reduce)

To produce this steady-state solution in WL, we set k⩵ 0 and then used Solve. (The Quiet command

silenced a warning; we return to this later.)

In[159]:= Clear[k, s, d, n, B, α]

solns = Solve0 ⩵ s B kα - d + n k, k // Quiet

Out[160]= k →B s

d + n

1

1-α

Notice that Mathematica returns a list of solutions, where each solution is a list of rules. In this case, there is a single solution in the list (which ignores the trivial steady state at k ⩵ 0). Bind ss`k to the steady-state solution we have found.

In[161]:= ss`k = k /. First[solns]

Out[161]=B s

d + n

1

1-α

Here the Solve command gave us what we want. Nevertheless, it is informative to turn to Reduce, which looks for a fuller characterization of the possible solutions. Use the optional third argument to


restrict the domain of possible solutions to the real numbers.

In[162]:= Reduce0 ⩵ s B kα - d + n k && k ≥ 0, k, Reals // Simplify

Out[162]= k ⩵ 0 ||d + n

B s

1

-1+α⩵ k

Note that Reduce characterizes a collection of solutions as equalities, in contrast to the rules returned by Solve. If we would rather work with rules, we can convert with ToRules.

In[163]:= {ToRules[%]}

Out[163]= {k → 0}, d + n

B s

1

-1+α→ k

In this model, we are seldom be interested in the trivial solution k ⩵ 0. Nevertheless, it is good to be reminded that it exists. It is also a good reminder that there may be more than one steady state in a dynamic model. However, in this case, the second solution yields the result we were looking for.

In sum, the non-trivial steady state of the traditional neoclassical growth model can be characterized in terms of this steady-state value of k.

Transformations Can Help Solve

To help Mathematica along, look for ways to rewrite the equality we are trying to solve. One approach is to ensure that a single term involves the unknown we are solving for. For example, for k ≠ 0, we might take advantage of the fact that k

⩵ 0 iff k

/k ⩵ 0. The latter has a single term in k.

In[164]:= kdot[k_] := s B kα - d + n k

In[165]:= transformed01 = Simplifykdot[k] k

Out[165]= -d - n + B k-1+α s

We can then solve

In[166]:= Solve[0 ⩵ transformed01, k] // Quiet

Out[166]= k →d + n

B s

1

-1+α

Another salient transformation is to divide by kα, which again yields a single term in k.

In[167]:= transformed02 = Simplifykdot[k] kα

Out[167]= -k1-α d + n + B s

We can again use Solve to find the steady state value of k.

In[168]:= Solve[0 ⩵ transformed02, k] // Quiet

Out[168]= k →d + n

B s

1

-1+α

Find such transformations was important as recently as Mathematica 9, but in more recent versions they are required much less often. Nevertheless, we will illustrate one much approach that can be very


useful: transformation of variables. Define z = k1-α . Then we can solve for the nonzero steady state by solving a linear equation in z, which is a trivial exercise for any computer algebra system.

In[169]:= solns`z = Solves B - d + n * z ⩵ 0, z

Out[169]= z →B s

d + n

This give us our steady-state value of z:

In[170]:= ss`z = z /. First[solns`z]

Out[170]=B s

d + n

Reverse the transformation (k = z1

1-α ) to find the implied solution for kss.

In[171]:= ss`z1

1-α ⩵ ss`k

Out[171]= True

Consumption in the Steady State

A steady-state value of k naturally implies a steady-state value for y.

In[172]:= ss`y = f[k] /. {k → ss`k} // Simplify

Out[172]= BB s

d + n

α

1-α

Correspondingly, it is common to define c = (1 - s) y, the consumption per unit of efficiency labor.

In[173]:= ss`c = 1 - s ss`y;

Let us look at this graphically:

Out[174]=

kssk

yss

s y(d+n)k

Equilibrium in the Neoclassical Growth Model

(1-s)y y

s y

(d+n)k

Steady-State Predictions of the Model

In order to explore the predictions of the neoclassical growth model, it is useful to distinguish between the labor input N and the effective labor input L = A N. (Here, changes in A are labor-augmenting or


Harrod neutral technical change.) Relate their growth rates using standard growth algebra. Two common notations for the growth rate of x are x and gx. When convenient, we will additionally write g[x].

L= A+ N= gA + gN (18)

Review: growth rates are the rates of change of logarithms.

In[175]:= Dt[Log@L]

Out[175]=Dt[L]

L

Use this fact to compute the growth rate of a product.

In[176]:= With[{L = A * Ν}, Dt[Log@L]] // Apart

Out[176]=Dt[A]

A+Dt[Ν]

Ν

Correspondingly, the fundamental dynamic equation is often written as:

k= s f k - d + gA + gN k (19)

The neoclassical growth model typically treats gA and gN as exogenous constants. Since y = Y /L = Y / (A N) is constant in the steady state, technological change implies that Y /N must be growing constantly. This is a core prediction of the traditional neoclassical growth model.

Kaldor (1957, 1961) offered six famous stylized facts about the growth of advanced industrialized economies. In this subsection, the function g yields the proportional rate of change of its argument. Here N is the input of labor hours (not effective labor).

◼ g[Y /N] > 0 and relatively constant (i.e., exponential growth in living standards, as measured by income per worker)

◼ g[K /N] > 0 and relatively constant (i.e., stable growth in capital-labor ratio)

◼ g[Y /K] ≈ 0 (i.e., fairly stable capital-output ratio)

◼ g[Π /K] ≈ 0 (i.e., no trend in the rate of return to capital) given the investment share, which is correlated with profit share

◼ g[Y] and g[Y /N] vary a lot across countries

◼ income shares relatively constant, with high Π /Y correlated with high I /Y

In the steady state, standards of living keep up with technological change.

0 = y = g[Y / (AN)] = g[Y / N] - g[A]

g[Y / N] = g[A](20)

Similarly, in a steady state

0 = g[K / (AN)] = g[K / N] - g[A]

g[K / N] = g[A](21)

Note that the steady-state capital-output ratio will have to be constant since y and k are constant:


Y / K ⩵Y

A N

K

A N⩵ y / k (22)

A steady state has a trendless capital-output ratio along with a trendless rate of return to capital. These imply the shares of capital and labor will be roughly constant. The profit rate is constant, and the capital output ratio is constant, so correspondingly profits are a constant share of output (since the capital-output ratio is constant.)

Π ⩵ r K ⩵ f'[kss] K (23)

Π

K⩵

Π

Y

Y

K(24)

Kaldor’s fifth stylized is relatively constant income shares, with a high profit share being correlated with high investment share. This also requires more discussion. It says that, across steady states, we should see a high profit share when we have a high saving rate. However, give a Cobb-Douglas produc-tion technology and competitive factor markets, the profit share is determined by the capital-elasticity of output (α).

Π

Y⩵

r K

F[K, L]==

r k

f[k]⩵ f'[k]

k

f[k]⩵ α (25)

(Make sure you can prove the last step.)

The last stylized fact is that labor productivity and output growth vary a lot across countries. This is more problematic. We can get any variation we wish in Y /N by varying s and n, but varying the growth rate requires more discussion. We will return to this.

Steady-State Comparative Statics

Recall solution for the steady-state capital-stock, given our Cobb-Douglas technology.

kss = B s

d+n

1

1-α (26)

In[177]:= ss`k =B s

d + n

1

1-α;

Recall the power rule from calculus: ⅆxn /ⅆx ⩵ n xn-1.

In[178]:= D[x^n, x]

Out[178]= n x-1+n

When the exponent is 1 / (1 -α), this power rule implies

In[179]:= D[x^n, x] /. n -> 1 1 - α // Simplify

Out[179]=x-

α

-1+α

1 - α

Now we consider the case where x = B s

d+n. This means that changes in x are caused by the changes in

the other variables. For example, ⅆx /ⅆs ⩵ B / (d + n). Consider all of the possible changes at one go. This is summarized by the gradient.


In[180]:= GradB s

d + n, {s, d, n, B}

Out[180]= B

d + n, -

B s

d + n2, -

B s

d + n2,

s

d + n

This makes it easy to understand the response of the steady-state k.

In[181]:= Grad[ss`k, {s, d, n, B}]

Out[181]= B B s

d+n-1+

1

1-α

d + n 1 - α, -

B s B s

d+n-1+

1

1-α

d + n2 1 - α, -

B s B s

d+n-1+

1

1-α

d + n2 1 - α,

s B s

d+n-1+

1

1-α

d + n 1 - α

Another perspective arises if we simplify this expression:

In[182]:= ss`exogs = {s, d, n, B};

dks = Grad[ss`k, {s, d, n, B}] // Simplify

Out[183]= - B s

d+n

1

1-α

s -1 + α,

B s

d+n

1

1-α

d + n -1 + α,

B s

d+n

1

1-α

d + n -1 + α, -

B s

d+n

1

1-α

B -1 + α

Now we see the steady-state capital stock in each expression. Substitute to make this explicit.

In[184]:= dks /. B s

d + n→ kss^1 - α // Simplify

Out[184]= kss

s - s α,

kss

d + n -1 + α,

kss

d + n -1 + α,

kss

B - B α

Rewrite this as follows, which is readily signed as {+, -, -, +}.∂kss

∂{s,d,n,B}= kss

s (1-α), -kss

(d+n) (1-α), -kss

(d+n) (1-α), kss

B (1-α) (27)

Graphical illustration of these comparative static results:

Out[187]=10 20 30 40 50 60

k

1

2

3

4

5

s f[k]increase s

10 20 30 40 50 60k

1

2

3

4

5

s f[k]increase B

10 20 30 40 50 60k

1

2

3

4

5

s f[k]increase d

10 20 30 40 50 60k

1

2

3

4

5

s f[k]increase n

Since y = f [k], we expect exactly the same qualitative comparative statics for y. Specifically, for parame-ters that do not change f , dyss = f '[kss]dkss. Since f ' > 0, the change in yss is in the same direction as the


change in kss. The following algebra confirms this. (Note that B is a production function parameter, so we leave it out here for simplicity.)

In[188]:= ss`y = f[ss`k] // Simplify

Out[188]= BB s

d + n

α

1-α

In[189]:= dys = Grad[ss`y, {s, d, n}] // Simplify

(* check that the chain rule produces the same result: *)

dys ⩵ B * α * ss`k^α - 1 * Grad[ss`k, {s, d, n}] // Simplify

Out[189]= -B B s

d+n

α

1-α α

s -1 + α,

B s

d+n

1

1-α α

s -1 + α,

B s

d+n

1

1-α α

s -1 + α

Out[190]= True

It should be no surprise that once again, we notice kss everywhere in this expression. Make this explicit by substitution.

In[191]:= dys /. B s


Out[191]= B kssα α

s - s α,

kss α

s -1 + α,

kss α

s -1 + α

By inspection, we can simplify even further.

In[192]:= dys ⩵α

s 1 - α{ss`y, -ss`k, -ss`k} // Simplify

Out[192]= True

Now, do all of these comparative statics at one go.

In[193]:= Grad[{ss`k, ss`y}, {s, d, n}] // Simplify;

ss`dkydsdn = % /. B s


TableForm[ss`dkydsdn,

TableHeadings → {{k, y}, {s, d, n}}]

Out[194]= kss

s - s α,

kss

d + n -1 + α,

kss

d + n -1 + α,

B kssα α

s - s α,

kss α

s -1 + α,

kss α

s -1 + α

Out[195]//TableForm=

s d n

k kss

s-s α

kss

(d+n) (-1+α)

kss

(d+n) (-1+α)

y B kssα α

s-s α

kss α

s (-1+α)

kss α

s (-1+α)

Exercise: explain why dyss /dk ≠ df[k] /dk. (Hint: compare each element, and review the definition of y[].)


In[196]:= TableForm[Sign[ss`dkydsdn] // Simplify,

TableHeadings → {{k, y, c}, {s, d, n}}, TableAlignments → Center]


s d n

k 1 -1 -1y 1 -1 -1

Golden Rule

This section shows that the sustainable rate of per capita consumption is not monotonically related to the saving rate. Recall that we defined css = (1 - s) yss.

In[197]:= ss`y = BB s

d + n

α

1-α;

ss`c = 1 - s ss`y

Out[198]= B 1 - sB s

d + n

α

1-α

This means that the effects of increase in d or n on css are obvious: since they lower yss, they must lower css. However, an increase in s is harder to analyze: it increases yss, but it reduces the proportion of yss consumed. An increase in s may either raise or lower consumption. On the one hand, more saving means less consumption out of any give level of income. On the other hand, more saving means more capital accumulation and thereby higher income. This seems to suggest an elasticity will be required for a determinant result.

In[199]:= Grad[ss`c, {s, d, n}] // Simplify

% /. B s


Sign[%] // Simplify

Out[199]= B B s

d+n

α

1-α (s - α)

s -1 + α, -

-1 + s B s

d+n

1

1-α α

s -1 + α, -

-1 + s B s

d+n

1

1-α α

s -1 + α

Out[200]= -B kssα (-s + α)

s -1 + α, -

kss -1 + s α

s -1 + α, -

kss -1 + s α

s -1 + α

Out[201]= {-Sign[s - α], -1, -1}

The golden rule of capital accumulation sets steady-state consumption as high as possible. Given our Cobb-Douglas functional form, we can solve quickly for the optimal saving rate. How saving should change depends on whether α is greater than or smaller than s.


Out[204]=

0.2 0.4 0.6 0.8 1.0s

5

10

15

css

Naturally, we would like to find the top of this curve. The value of s that maximizes c is called the golden rule saving rate.

In[205]:= ss`c

ss`dcds = D[ss`c, s] // Simplify

Solve[0 ⩵ ss`dcds, s] (* solve FOC for s *)

Out[205]= B 1 - sB s

d + n

α

1-α

Out[206]=

B B s

d+n

α

1-α (s - α)

s -1 + α

Out[207]= {{s → α}}

Maximum sustainable consumption requires s ⩵α. This is the golden rule of saving. Note that an economy can save too much! If saving exceeds the golden rule, more saving will raise the capital stock and thus income, but it will still lower consumption. But when saving less than the golden rule, more saving will mean reduced consumption in the short-run but will raise consumption in the long run. Finally, when saving less than the golden rule, if we further reduce our saving, there will be a short-run increase in consumption. But eventually we pay for it with permanently lower consumption.

Approaching this more slowly may offer a gain in intuition. Consider picking s to maximize s f [k], subject to the steady-state constraint.

s f[k]=(d+n)k

The long-run comparative-statics analysis demonstrates that increasing the saving rate raises the steady-state per capita capital stock (and thus income per capita). Of course, if the saving rate is higher and income is higher, the effect on consumption is unclear. Since consumption per capita is (1 - s) f [k], in the steady-state, we want to

maxs,k (1 - s) f[k]

s.t. s f[k] = (d + n) k(28)

Note how the constraint restricts the covariation in s and k (since the steady state k depends on s). Set up the Lagrangian for this constrained optimization, and then produce first-order necessary conditions from the gradient. (Assume the constraint is binding.)

ℒ=(1-s)f[k]+λ(s f[k]-(d+n)k) (29)

Using this Lagrangian, find the optimal s and k in the usual way. Here is how we might approach this in


WL,.

In[208]:= Clear[f, s, k, λ]

ℒ = 1 - s f[k] + λ s f[k] - d + n k;

gradient = Grad[ℒ, {s, k, λ}];

gr`focs = Thread[gradient ⩵ 0];

Labeled[

TableForm[gr`focs,

TableHeadings → {{s, k, λ}, None}],

"first-order conditions", Top]

Out[212]=

first-order conditionss -f[k] + λ f[k] ⩵ 0

k 1 - s f′[k] + λ -d - n + s f′[k] ⩵ 0

λ -k d + n + s f[k] ⩵ 0

Although Mathematica will not currently solve this system as a whole, we can solve this problem recursively. First, since the constraint binds, λ ≠ 0, and the first equation therefore gives us λ⩵ 1.

In[213]:= Solve[gr`focs[[1]], λ]

Out[213]= {{λ → 1}}

Since λ⩵ 1, second equation then tells us that f '[k] = n + d, and the third just reiterates our constraint. We can therefore find k such that the slope of the production function matches the slope of the depreci-ation curve, and then choose s to make that level of k the steady state level. That is, first find k so that f '[k] = d + n, and then find s such that s f [k] = k f '[k]. (So the golden-rule s equals the elasticity of f .)

In[214]:= Solve[gr`focs[[2]] /. {λ → 1}, f'[k]]

Solve[gr`focs[[3]] /. {d + n → f'[k]}, s]

Out[214]= {{f′[k] → d + n}}

Out[215]= s →k f′[k]

f[k]

Recall that in the particular case of our Cobb-Douglas function, this elasticity is α.

In[216]:= gr`s =k f′[k]

f[k]/. f[k] → B kα, f'[k] → α B kα-1

Out[216]= α

Similarly, a Cobb-Douglas production technology allows us to solve for the golden-rule level of the capital stock and income.

In[217]:= gr`k = k /. First@Solvef′[k] ⩵ d + n /. f'[k] → α B kα-1, k // Simplify // Quiet

gr`y = B gr`kα // Simplify

Out[217]=d + n

B α

1

-1+α

Out[218]= Bd + n

B α

α

-1+α

If we specify numerical values for the parameters, we can illustrate the solution process graphically.


Just for the sake of the graphical display, our plots will use a much higher saving rate.

Out[223]=

50 100 150 200 250 300

10

20

30

40

Find k

50 100 150 200 250 300

10

20

30

40

Find s

A More Traditional Presentation

Log-Linear Formulation for Empirical Application

Recall the steady state value of the Solow-Swan model with Cobb-Douglas production.

In[231]:= ss`y = BB s

d + n

α

1-α;

A logarithmic transformation results in

In[232]:= ss`lny = Log[ss`y] // PowerExpand // Simplify

Out[232]= -Log[B] + α Log s

d+n

-1 + α

Rewrite this by hand as follows.

In[233]:= ss`lny ⩵Log[B]

1 - α+

α

1 - αLog[s] -

α

1 - αLog[d + n] // Simplify

Out[233]= True

This rewritten formulation suggests a possible linear regression equation. To emphasize this, substi-tute coefficients that can be estimated linearly.

In[234]:= betas = β00 →Log[B]

1 - α, β1 →

α

1 - α, β2 → -

α

1 - α;

ss`lny ⩵ β00 + β1 Log[s] + β2 Log[d + n] /. betas // Simplify

Out[235]= True

Unfortunately, estimation of this regression equation requires data on y = Y /L = Y / (A N). The MRW solution is to move the technology variable to the right-hand side, as follows.

Assume the efficiency of labor grows at a constant rate: At = A0 ⅇgA t. Since Y /N = A y, we have Log[Y /N] = Log[A] + Log[y]. This is also true in a steady state yss, where

Log[Y /N]⩵ Log[A0] + gA t + Log[yss]⩵ Log[A0] + gA t + β00 + β1 Log[s] + β2 Log[d + n]


[Mankiw.Romer.Weil-1992-QJE]_ bring this to the data as follows. First, add a country-specific stochas-tic element to the model by specifying that for each country i,

Log[Ai,0] = a + ϵi

Here the constant a is shared across countries but ϵ is a country-specific shock. Putting this all together yields a linear regression specification where for each country

Log[Y /N]⩵β0 + β1 Log[s] + β2 Logd + gA + gN + ϵ (30)

Recall β1 =α

1-α and β2 = - α1-α , and note that all constant terms move into the intercept:

β0 = a + gA t + Log[B]1-α .)

MRW claim that data on factor shares suggest α ≈ 1 /3 so that we should expect β1 = -β2 ≈ 1 /2.

Use the data they provide for 75 “intermediate” countries to estimate this with ordinary least squares. The model is

yN = X.β + ϵ

Here the LHS variable is (the log of) per-capita GDP and the matrix X of exogenous variables has the following columns: all ones (for the intercept), the log of the saving rate, and the log of 0.05 + gN (where the unobserved d + gA is approximated by 0.05 for all countries).

Review of Least Squares

Get MRW Data

MRW provide their data in an appendix, which was imported by optical scan from the PDF of their paper. (The replication results reported below are close but not perfect, so the scan is not perfect or the article contains a typo.) I provide the data online.

Data details: Real national accounts of [Summers.Heston-1988]_, covering 1960-1985. (s I /GDP, and

gN is the average growth 1960-85 of 15–64 age group (“working age”). They set gA +δ⩵ 0.05 and claim plausible changes do not affect the results.) Subtleties: In practice investment rates are not constant, so MRW average them over the period.


In[236]:= mrwDataLocation = "https://subversion.american.edu/aisaac/hw/data/mrw1992.csv";

(* online data *)

(* mrwDataLocation=dataFolder<>"mrw1992.csv"; *) (* for downloaded data *)

SetOptions[Dataset, MaxItems → 10];

dsAll = SemanticImport[mrwDataLocation]

Out[238]=

Country N I O Y1960 Y1985 GDP agepop IoY

Algeria 1 1 0 2485 4371 4.8 2.6 24.1

Angola 1 0 0 1588 1171 0.8 2.1 5.8

Benin 1 0 0 1116 1071 2.2 2.4 10.8

Botswana 1 1 0 959 3671 8.6 3.2 28.3

Burkina Faso 1 0 0 529 857 2.9 0.9 12.7

Burundi 1 0 0 755 663 1.2 1.7 5.1

Cameroon 1 1 0 889 2190 5.7 2.1 12.8

Central African Republic1 0 0 838 789 1.5 1.7 10.5

Chad 1 0 0 908 462 -0.9 1.9 6.9

Republic of the Congo 1 0 0 1009 2624 6.2 2.4 28.8

rows 1–10 of 121

Scrolling through the data reveals quite a few missing values. Filter out such records with the Select command, which takes a boolean function as its argument. Missing values have a head of Missing, so the MemberQ command can look for missing field values. For example

In[239]:= MemberQ[_Missing]@{1, 2, Missing[], 4}

Out[239]= True

Since we want to keep the records that have no missing fields, we reverse the by composing with the Not command.


In[240]:= dsNoMissing = Select[dsAll, Not@*MemberQ[_Missing]]

Out[240]=

Country N I O Y1960 Y1985 GDP agepop IoY

Algeria 1 1 0 2485 4371 4.8 2.6 24.1

Angola 1 0 0 1588 1171 0.8 2.1 5.8

Benin 1 0 0 1116 1071 2.2 2.4 10.8

Botswana 1 1 0 959 3671 8.6 3.2 28.3

Burkina Faso 1 0 0 529 857 2.9 0.9 12.7

Burundi 1 0 0 755 663 1.2 1.7 5.1

Cameroon 1 1 0 889 2190 5.7 2.1 12.8

Central African Republic1 0 0 838 789 1.5 1.7 10.5

Chad 1 0 0 908 462 -0.9 1.9 6.9

Republic of the Congo 1 0 0 1009 2624 6.2 2.4 28.8

rows 1–10 of 104

The cross-country distribution of income has shifted substantially over time. One way to see this is to use a histogram. Notice the substantial flattening and rightward shift of the distribution.

In[241]:= all`Y1960 = dsNoMissing[All, "Y1960"];

all`Y1985 = dsNoMissing[All, "Y1985"];

GraphicsRow[{

Histogram[all`Y1960, {3000}, PlotRange → {{0, 21 000}, {0, 70}}],

Histogram[all`Y1985, {3000}, PlotRange → {{0, 21 000}, {0, 70}}]

}, ImageSize → Large, PlotLabel → "Per Capita Income, 1960 and 1985"]

Out[243]=

0 5000 10000 15000 20000

10

20

30

40

50

60

70

0 5000 10000 15000 20000

10

20

30

40

50

60

70

A more refined window on the same distribution is provided by a kernel-density plot. The SmoothKernelDistribution command does not yet accept dataset objects, so as a preliminary, use the Normal command to turn it into a normal list. We can plot both distributions in a single chart.


In[244]:= Plot[{

PDF[SmoothKernelDistribution[Normal@all`Y1960], x],

PDF[SmoothKernelDistribution[Normal@all`Y1985], x]},

{x, 0, 20 000},

PlotLegends → {"1960", "1985"}]

Out[244]=

5000 10000 15000 20000

0.00005

0.00010

0.00015

0.00020

1960

1985

Next, construct the three subsets considered by MRW by selecting based on their dummy variables. If you look up the Function command, you will find that is has an ampersand postfix shorthand, which is commonly used by WL aficionados. We use it here. Note that we are able to prefix an field name with an octothorpe (#) in order to refer to the field in our function.

In[245]:= dsN = Select[#N ⩵ 1 &]@dsNoMissing; (* nonoil *)

dsI = Select[#I ⩵ 1 &]@dsNoMissing; (* intermediate *)

dsO = Select[#O ⩵ 1 &]@dsNoMissing; (* OECD *)

As a check on our work, let us use the Length command to count the number of observations in each dataset. (The /@ operator is an infix shorthand for the Map command, which applies a function to each element of a list.)

In[248]:= Length /@ {dsN, dsI, dsO}

Out[248]= {98, 75, 22}

Let us get the income data for the “intermediate” countries.

The basic neoclassical growth model predicts that steady-state income (per worker) depends positively on the saving rate and negatively on population growth, log-linearly. Postpone the logarithmic transfor-mation of the RHS variables for reasons that will soon be evident.

Next, let us look at the correlation between the saving rate, population growth, and per-capita income. First, convert our dataset into a matrix by stripping out the keys (with the Values command) and then convert it to a normal matrix (with the Normal command). (As of Mathematica version 12, this prelimi-nary step remains necessary.) Then use the Correlation command to produce the correlations. (We also use TableForm here, to format it nicely.


In[249]:= With[{

rawdata = Normal@Values@dsI[All, {"IoY", "agepop", "Y1985"}],

headings = {"s", "n", "Y/N"}

},

TableForm[Correlation@rawdata,

TableHeadings → {headings, headings}]]


s n Y/Ns 1. -0.393421 0.665647n -0.393421 1. -0.606623Y/N 0.665647 -0.606623 1.

Next produce the related scatter plots, first for the saving rate and per capita income, and then for the growth rate of the working-age population and per capita income.

In[250]:= ListPlot[dsI[All, {"IoY", "Y1985"}],

AxesLabel → {"saving (%)", "income"}]

Out[250]=

5 10 15 20 25 30 35saving (%)

5000

10000

15000

20000

income

In[251]:= ListPlot[dsI[All, {"agepop", "Y1985"}],

AxesLabel → {"pop. growth (%)", "income"}]

Out[251]=

1 2 3 4pop. growth (%)

5000

10000

15000

20000

income

We can also use a three-dimensional scatter plot to get an overall view of these correlations.


In[252]:= ListPlot3D[dsI[All, {"IoY", "agepop", "Y1985"}],

AxesLabel → {"s", "gN", "Y/N"}]

Out[252]=

Recode the dataset by creating a transformation to map across each record.

In[253]:= dsI`recode = <|

"ln`s" → Log#"IoY" 100,

"ln`n" → Log0.05 + #agepop 100, (* recall MRW notation difference (n=gA+gN) *)

"ln`y" → Log[#Y1985]

|> & /@ dsI

Out[253]=

ln`s ln`n ln`y

-1.42296 -2.57702 8.38275

-1.26231 -2.50104 8.20822

-2.05573 -2.64508 7.69166

-2.91877 -2.6173 6.41017

-2.08747 -2.37516 7.44073

-1.7487 -2.47694 7.19218

-2.64508 -2.63109 6.88244

-2.02495 -2.60369 6.71296

-2.6173 -2.63109 6.56526

-2.48891 -2.59027 7.76132

rows 1–10 of 75

If we plot this data, the positive correlation with the saving rate is evident but the negative correlation with the population growth rate somewhat less so.


In[254]:= ListPlot3D[dsI`recode, AxesLabel → {"Log[s]", "Log[d+n]", "Log[Y/N]"},

Mesh → None, InterpolationOrder → 3, ColorFunction → "SouthwestColors"]

Out[254]=

Next, produce a fitted model with the LinearModelFit command. Currently, the input to this command must be a matrix of values, where each row represents an observation and the last value in each row is the dependent variable. (Unfortunately, this command does not yet work directly with dataset objects.) Convert this dataset into a matrix by stripping out the keys with the Values command and then converting it to a normal matrix with the Normal command.

In[255]:= dsI`recodeM = Normal@Values@dsI`recode; (* convert to matrix *)

dsI`fit = LinearModelFit[dsI`recodeM, {s, n}, {s, n}]; (* fitted model *)

Normal@dsI`fit (* view the fit *)

Out[257]= 5.34587 - 2.0172 n + 1.31755 s

Ordinarily, we want to take a look at the parameter table for a fitted model.

In[258]:= dsI`fit["ParameterTable"]

Out[258]=

Estimate Standard Error t-Statistic P-Value1 5.34587 1.54308 3.46442 0.00089895s 1.31755 0.170943 7.70758 5.38321×10-11n -2.0172 0.533866 -3.77848 0.000322373

Alternatively, we may prefer to look at confidence intervals. (These are 95% intervals by default.)

In[259]:= dsI`fit["ParameterConfidenceIntervalTable"]

Out[259]=

Estimate Standard Error Confidence Interval1 5.34587 1.54308 {2.2698, 8.42193}s 1.31755 0.170943 {0.976785, 1.65832}n -2.0172 0.533866 {-3.08144, -0.952957}

As a crude measure of goodness of fit, we may look at the adjusted R2. Roughly speaking, this is the proportion of the variability in the dependent variable that is explained by variation in the independent variables. For such a simple model of such a complex phenomenon, this measure looks pretty good.

In[260]:= dsI`fit["AdjustedRSquared"]

Out[260]= 0.587767

One problem with the estimates is that they are not clearly related as the algebraic derivation of the model leads us to expect. So, take a look at the restricted model.


In[261]:= dsI`rfit = LinearModelFit[dsI`recodeM, {ln`s - ln`n}, {ln`s, ln`n}];

dsI`rfit["ParameterTable"]

Out[262]=

Estimate Standard Error t-Statistic P-Value

1 7.09292 0.145614 48.7106 2.04004×10-57

-ln`n+ ln`s 1.43096 0.139123 10.2855 7.57589×10-16

In[263]:= r`fittedI["AdjustedRSquared"]

Out[263]= 0.586111

Despite the restriction, the residuals do not grow much and the adjusted R2 barely budges. Therefore, expect not to reject this constraint. Nevertheless, we should test it.

To test the constraint, first compute the F-statistic for the model comparison. Get the sum of squares from each model, and then use the standard formula (the proportional change in the residuals scaled by the proportional change in the degrees of freedom).

In[264]:= TableForm

{"unrestricted SSE", sse = fittedI["ANOVATableSumsOfSquares"][[-2]]},

{"restricted SSE", r`sse = r`fittedI["ANOVATableSumsOfSquares"][[-2]]},

"F statistic", nI = Length[dsIRecode];

k = 3; (* number of regressors, unconstrained case *)

fstat =r`sse - sse 1

sse nI - k


unrestricted SSE 26.8475restricted SSE 27.3298F statistic 1.29333

Then find the p-value:

In[265]:= 1 - CDF[FRatioDistribution[1, nI - k], fstat]

Out[265]= 0.259206

This is too big to reject the restriction. We (naturally) get the same result if we use the HypothesisTesting package.

In[266]:= << HypothesisTesting`

In[267]:= FRatioPValue[fstat, 1, nI - k]

Out[267]= OneSidedPValue → 0.259206

In sum, things look ... not bad. We get a reasonable amount of the variance explained, given that we are using cross section data, and we cannot reject a constraint implied by theory. But the key coeffi-cient estimate of around 3 /2 implies a value for α of around 0.6, which is almost twice the share of capital in the national income accounts.

And there is another problem. If we restrict the analysis to just their OECD subsample, the estimates are less encouraging. The large p-values mean we lack confident that saving rates and population growth rates have predictive power.


In[268]:= dsORecode = <|

"s" → #"IoY" 100,

"d+n" → 0.05 + #agepop 100, (* recall notation difference (n=gA+gN) *)

"lny" → Log[#Y1985]

|> & /@ dsO;

fittedO = LinearModelFit[Normal@Values@dsORecode, {Log[s], Log[dn]}, {s, dn}];

fittedO["ParameterTable"]

Out[270]=

Estimate Standard Error t-Statistic P-Value1 8.02061 2.51789 3.18545 0.00487043Log[s] 0.49989 0.433896 1.1521 0.263572Log[dn] -0.741921 0.852195 -0.870601 0.394839

Note that the model predicts both the signs and the magnitudes of s and n on Y /N , since α is capital’s

share and is known to be about 1 /3 (implying [α /(1-α)]≈ .5. Their cross section estimates of (1) support the sign prediction, and they also accept the equal size of the coefficients on n and s. But yield

α≈ .59, which is much too large. (And the s.e. is small enough to easily reject α 1 /3.)

Transition Dynamics

Neoclassic Growth Model: Linearization

Solution by Discretization

Mankiw, Romer, and Weil (1992 QJE) [Knight-1946-AER]_ notes the need for a broad concept of capital—broader than that used in the standard Solow model. [Mankiw.Romer.Weil-1992-QJE]_ suggest that many shortcomings of the Solow model can be easily overcome by adding human capital. They augment the Solow model with human capital and an assumption that countries share the same rate of efficiency growth. Across

countries, the initial level of efficiency, A(0), is assumed to vary randomly (due to local factors like climate), and this may motivate the randomness. MRW’s work received a lot of attention for several reasons:

◼ There had been a tendency to presume that the Solow model “ ‘explains’ cross country variation in labor productivity largely by appealing to exogenous variations in technologies”, but this feels unsatisfactory.

◼ Most provocatively, they argue that saving and population growth differences do a good job in explaining differences in per capita income

◼ Economists were also interested in it as a counterthrust to the endogenous growth literature, which tends to assume constant returns to scale in capital inputs (possibly physical and human capital inputs).


◼ Finally, it offers an answer to a question raised by Romer: how to reconcile relatively similar growth rates with quite different per capita incomes.

The key move will be to augment the Solow growth model to include the accumulation of human capital.

Predictions:

Recall the Solow model predicted

◼ increased s increases the steady state Y /N

◼ increased n decreases the steady state Y /N

As we saw above, MRW show that cross section data support these predictions of the Solow model, explaining 50%+ of the variation in cross section data. But the actual correlations in the data are larger

than those predicted by the Solow model. (However, MRW don’t reject the implication that effects of s

and n are of equal magnitude but opposite in sign.)

Mankiw et al. remedy this by adding a proxy for human capital (H) to the production function. Their

argument is as follows. Let hH /(AN), we have

y=Y/(A N)==(1/(A N))F[AN,K,H]==F[1,k,h]==f[k,h] (49)

K sk Y -δK

H sh Y - δH

(50)

Model human capital accumulation just like physical capital accumulation. For notational simplicity, let n = d + gA + gN. Then

h sh y - n h (51)

The model is otherwise unchanged, so we still have

k sk y - n k (52)

Out[271]=


Steady State: Algebraic Details

Start with YF[K, H, AN]AN F[K /AN , H /AN , 1] f [k, h] kα hβ. Maintaining the standard accumulation form, with exogenous savings rates, we have

sh kα hβ ⩵ nh ⇒ kα hβ-1 ⩵

n

sh

sk kα hβ ⩵ nk ⇒ kα-1 hβ ⩵

n

sk

(53)

In log form, this becomes

α Logk - (1 - β) Logh ⩵ Logn

sh

β Logh - (1 - α) Logk ⩵ Logn

sk

(54)

Express this as

α -(1 - β)

-(1 - α) βLog[k]Log[h]

⩵ Log[n / sh]Log[n / sk]

(55)

Find the inverse of the coefficient matrix:

In[272]:= mA =α -1 - β

-1 - α β;

Inverse[mA] ⩵-1

1 - α - β

β 1 - β

1 - α α// Simplify

Out[273]= True

Finally, solve for Log[k] and Log[h] as

Log[k]Log[h]

⩵-1

1 - α - β

β (1 - β)(1 - α) α

Log[n / sh]Log[n / sk]

(56)

Equivalently,

Log[k]Log[h]

⩵-1

1 - α - ββ Log[n / sh] + (1 - β) Log[n / sk](1 - α) Log[n / sh] + α Log[n / sk]

(57)

Since Log[y]α Log[k]+ β Log[h], we have

ln y-α

(1 - α - β)β ln

n

sh+ (1 - β) ln

n

sk -

β

(1 - α - β)(1 - α) ln

n

sh+ α ln

n

sk (58)

which can be rewritten as

Log[y]β

(1 - α - β)Log[sh] +

α

(1 - α - β)Log[sk] -

(α + β)

(1 - α - β)Log[n] (59)

Results

MRW (p.419) proxy the rate of human-capital accumulation by the fraction of the eligible population (aged 12 to 17) enrolled in secondary school multiplied by the fraction of the working-age population


that is of school age (aged 15 to 19). This is the SCHOOL column of their data appendix (p.436).

In[274]:= With[{

rawdata = Normal@Values@dsI[All, {"IoY", "SCHOOL", "agepop", "Y1985"}],

headings = {"sk", "sh", "n", "Y/N"}

},

TableForm[Correlation@rawdata,

TableHeadings → {headings, headings}]]


sk sh n Y/Nsk 1. 0.589224 -0.393421 0.665647sh 0.589224 1. -0.380738 0.700287n -0.393421 -0.380738 1. -0.606623Y/N 0.665647 0.700287 -0.606623 1.

In[275]:= dsIRecode02 = <|

"sk" → #"IoY" 100,

"sh" → #"SCHOOL" 100,

"d+n" → 0.05 + #agepop 100, (* recall notation difference (n=gA+gN) *)

"lny" → Log[#Y1985]

|> & /@ dsI;

fittedI02 =

LinearModelFit[Normal@Values@dsIRecode02, {Log[sk], Log[sh], Log[dn]}, {sk, sh, dn}];

fittedI02["ParameterTable"]

Out[277]=

Estimate Standard Error t-Statistic P-Value

1 7.79131 1.19242 6.53401 8.30147×10-9Log[sk] 0.700367 0.150583 4.65105 0.0000148757Log[sh] 0.730549 0.0952292 7.67148 6.79223×10-11Log[dn] -1.49978 0.403216 -3.71955 0.00039564

◼ sh significant

◼ increases R2 to .78 (but R2 .24 for OECD, and the coefficients are not significant!).

◼ lowers estimated α to ∼ 1 /3

◼ still does not reject restriction that coefficients sum to zero

Conclusion: adding human capital improves the empirical performance of the Solow model and calls into question the need for the endogenous growth models. (You will cover endogenous growth in your macro courses.)

But: somewhat dubious proxies for aggregate human capital. The focus is almost exclusively on school-ing rather than training. This is mainly due to data limitations, but less excusable is the attention frequently paid to school enrollment rates. These rates were initially regarded as one of the more robust and satisfactory variables in the growth literature, but it is worth remembering that they are likely to be positively correlated with initial efficiency, so the results could often be spurious. Certainly it has been much harder to find an effect of human capital in panel data studies, although it is also true that too few researchers think carefully about the specification. (Rather optimistically, they tend to expect a change in school enrollments to raise growth almost instantly.) Equally importantly, there are some conceptual difficulties with the use of school enrollment data. Only rarely do these rates corre-


spond well to the human capital variables highlighted in theoretical models. In many empirical growth papers it is not clear whether school enrollment rates are intended to represent a Àow of investment in human capital, or its stock. In practice these rates may be a poor proxy for either, and given that data on average years of schooling in the population or labour force is now available, the continuing use of school enrollment figures has little to recommend it.

Understanding the Change in Results

Recall

hs s skα sh

1-α n1/(1-α-β) (60)

or

(1 - α - β) ln hs s α ln sk + (1 - α) ln sh - ln n (61)

So that

ln sh -[(1 - α - β) / (1 - α)] ln h -α

1 - αln sk +

1

1 - αln n (62)

is a steady state relationship. Recalling our regression equation

ln (Y /N) at + [α / (1 - α - β)] ln sk + [β / (1 - α - β)] ln sh - [(α + β) / (1 - α - β)] ln n (63)

and substituting for sh yields

ln (Y /N) at + [α / (1 - α)] ln sk - [α / (1 - α)] ln n + [β / (1 - α)] ln hs s (64)

Note that this is identical to the original production function except for the last term. So the original estimation had an omitted regressor.

But our steady state solution for h implies that it is positively correlated with sk and negatively corre-

lated with n. Omitting h would therefore tend to increase the magnitude of the coefficients on the other two regressors. This bias can explain our earlier puzzle about the coefficient magnitudes when we estimate (1).

A focus on potential bias can be problematic for their results as well, however. For example, as noted

by Grossman and Helpman (1994), if low Y /N causes high n this can explain their negative coefficient

on n. (Similarly if investment is encouraged by growth.)

Problem: If the sample is restricted to only the 22 OECD countries, the adjusted R2 falls to .28. The

differences in n and I /Y between rich and poor countries drives their results.

Problems: There are still some problematic stylized facts. As Lucas (1988) notes, international wage differentials and patterns of migration don’t seem to fit the neoclassical framework. To continue Romer’s Philippines example, a worker should not be able to earn a higher wage by migrating to the US. Further, empirically human capital migrates from places where it is scare (and, in a neoclassical setting, should therefore be highly rewarded) to places where it is abundant.

MRW draw attention to one particularly dramatic finding. They claim that around 80% of the interna-tional variation in per capita incomes can be explained using just three variables: population growth,


and investment rates for physical and human capital. The corollary is that differences in technical efficiency can have only a small role to play in explaining cross-country income variation. However, their approach makes two controversial assumptions, that investment rates are exogenous to the level of income and uncorrelated with efficiency. More recently, Peter J. Klenow and Andrés Rodríguez-Clare (1997) have drawn attention to another underlying problem with the MRW finding. The human capital variable that MRW use only captures variation in secondary schooling. Since it ignores primary school-ing, it tends to exaggerate the variation in human capital across countries. When Klenow and Rodríguez-Clare correct for this, they find that MRW’s model only explains around half the variation in incomes, leaving a central role for technology differences.

If one follows them in assuming that TFP has grown at a rate of 2% a year in all countries since 1960, what does one make of the many countries which have grown at less than 2% a year over this period? The model implies that unless these countries have received a large negative shock to technology they are converging to their steady state from above. (I.e., such a country must have exceeded its steady state capital stock in the early 1960s and have been running it down subsequently.) This seems unlikely. There are other problems. Economic miracles, such as Japan’s post-war growth, are hard to explain by capital accumulation alone. In most frameworks, such a process would have to be accompa-nied by a steeply falling real interest rate, something which has not been observed. Temple (1999) argues that such problems suggest that the neoclassical revival in growth economics went too far. Understanding the reasons for differences in technical efficiency and TFP growth is essential to the empirical growth project.

Convergence

Background: Gerschenkron (1952) and Abramovitz (1986 JEHist) hypothesize that less developed countries should be able to grow faster than developed countries, which is known as convergence.

Broadly speaking: convergence must be false.

Throughout most of human history, almost everyone lived near subsistence. Now some countries are rich, while others remain near subsistence. This is extreme divergence.

Why do economists search for convergence in contemporary data?

◼ policy concern that poor countries “catch up” with rich countries

◼ suspicion that technology transfers could close most of the gap between rich countries and poor countries. (These need not be policy based: business and engineering journals are available online.)

◼ the neoclassical growth model has convergence as a basic prediction,

An early effort to shed empirical light on historical convergence trends was offered by [Baumol-1986-AER]_. Jones (fig 3.3, p.57) plots similar data over the period 1870–1994 for Germany, Japan, the US, and the UK. Appearance: clearly shows narrowing gaps between the countries.


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For the OECD countries (fig 3.5, p.59), he presents an analogous scatter plot for 1960–90. Shows a strong negative correlation.

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Note that n and δ will vary across countries, but conventional estimation of the model ignores this subtlety.

Note also that the convergence equation is obtained by taking a Taylor series approximation around a deterministic steady state. Binder and Pesaran (1996) argue that, when growth is stochastic, such linearization is potentially misleading.

Why does initial income affect growth? The reason is ‘transitional dynamics’. The relatively poor econ-omy must have lower stocks of physical and human capital. The marginal product of extra capital is correspondingly higher in this economy: for a given rate of investment its growth will be faster. Thus in


a regression that controls for the determinants of steady states, like investment ratios, the initial income will take a negative sign.

This is called conditional convergence. It is conditional in that it appears after we control for determi-nants of the steady state levels of income.

In this sense, conditional convergence does not imply that poorer countries will catch up with rich ones. For example, anything that drives apart investment rates in rich and poor countries will tend to lead to increased income dispersion.

In principle the level of technology should be included in the regression. But this variable is unob-served. Omitted variable problem: the other parameter estimates are biased if one or more regressors are correlated with the level of technology. One proposed solution: panel data methods.

Using a rather remarkable data set, Jones (fig 3.4, p.58) also presents a scatter plot of average annual growth rates vs. initial income from 1885–1994 for a larger set of industrialized countries. (Similar to your HW, but larger time span.) Again: a strong negative correlation.

Out[280]=

On the face of it, a simple convergence hypothesis does remarkably well in this qualitative sense. However, De Long (1988 AER) pointed to an important difficulty with Baumol’s results (and all these figures): only countries that were rich at the end of the sample were included. This means that some countries that were relatively rich at the beginning of the period were not included, biasing the appar-ent results toward convergence. Jones’s OECD data (as he recognizes) has the same problem.

Jones (fig. 3.6, p.60) therefore produces an analogous scatter plot for most countries in the world for 1960–1990. In this graph, there is no clear tendency toward convergence. (Baumol had also reported that in large samples of countries there is no evidence of convergence.)


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Quah (1993) identified another problem: Galton’s fallacy. (See the posted HW exploring this.)

Convergence in the MRW Model

The MRW model seems to do pretty well in the explanation of steady state income differences. Now consider the ability of the model to explain differences in growth rates.

The MRW rate of convergence is

λ (n + g + δ) (1 - α - β) (65)

The debate about convergence rates can be linked to another ongoing dispute: the importance of differences in technology across countries. Recall that MRW show that income and growth differences can be almost completely explained using a model in which technology is a public good, freely avail-able to individuals in all countries. They attribute differences in BGP incomes largely to steady state levels of human and physical capital.

Most development economists, economic historians, and theorists, have not been satisfied with this. They stress technological and institutional factors in the problems of developing countries. Capital accumulation is important because it is part of the important task of adopting appropriate foreign technology.

Researchers in this latter tradition emphasise the role of IP protection and trade secrets. I find this somewhat implausible, especially when it comes to patents, which expire after 20 years. Twenty-year old technology is more than adequate to make poor countries rich. Of course there is a learning by doing problem, but the NICs demonstrated that this can be done very swiftly.

ReferencesBarro, Robert J. “Notes on Growth Accounting.” Journal of Economic Growth, vol. 4, no. 2, 1999, pp.


119–137. JSTOR: www.jstor.org/stable/40216002.

Kaldor, Nicholas (1957). “A Model of Economic Growth”. The Economic Journal. 67 (268): 591–624.

Kaldor, Nicholas, “Capital Accumulation and Economic Growth,” in F.A. Lutz and D.C. Hague,eds., The Theory of Capital, St.Martins Press, 1961, pp. 177–222.

Quah, Danny, 1993, "Galton’s Fallacy and Tests of the Convergence Hypothesis" Scandinavian J.Econ 95, Dec, 427–43.

Binder, M. & Pesaran, M.H. Journal of Economic Growth (1999) 4: 139. https://-doi.org/10.1023/A:1009802421114

(It is also worth noting that conditional convergence does not imply a rejection of ‘AK’ models of growth. On this see Narayana Kocherlakota and Kei-Mu Yi (1995) and Charles Leung and Quah (1996).) KOCHERLAKOTA,NARAYANA R. AND YI,KEI-MU. “Can convergence regressions distinguish between exogenous and endogenous growth models?” Economics Letters, August 1995, 49(2), 211–215.

LEUNG,CHARLES AND QUAH,DANNY T. “Convergence, endogenous growth, and productivity distur-bances,” Journal of Monetary Economics, December 1996, 38(3), 535–547.


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