Comparing Growth Models With Policy Shocks

Why Compare Growth Models?

A policy can raise output today without changing the economy’s long-run growth rate. It can also change a growth rate, so that the output advantage accumulates over time. These are different outcomes, and a policy that changes both the resources available for production and investment may have both effects. To analyze a proposed intervention, first identify the model’s growth mechanism, then trace which parameter or initial condition the policy changes.

This tutorial compares three mechanisms: physical-capital accumulation in an AK model, knowledge accumulation in a Romer-style research model, and human-capital accumulation in a human-capital model. The equations below use discrete annual periods. They are deliberately simplified to isolate each mechanism; conclusions apply under the stated assumptions, not automatically to every richer version of these models. The comparisons are aligned with the SOA VEE Economics (Validation by Educational Experience) syllabus.

Three Models and Their Growth Mechanisms

Definition: A level effect changes the amount of output or productivity at a given date without changing its long-run proportional growth rate. A growth-rate effect changes the proportional rate at which output or productivity grows while the relevant policy or parameter change remains in place. A permanent policy can have a level effect, a growth-rate effect, or both.

In a discrete-time model, if a variable $z_t$ grows by the same factor $q$ each year, then $z_{t+1}=qz_t$. Its annual proportional growth rate is $q-1$. If $q$ changes permanently at a policy date, the growth rate changes from that date onward; the level at the policy date is still determined by the path up to that date.

AK model: Let $k_t$ be capital per worker, $y_t$ output per worker, and $A$ a constant productivity parameter. Production is $y_t=Ak_t$. With a constant saving rate $s$, depreciation rate $\delta_K$, and no population growth, capital evolves as $k_{t+1}=(1-\delta_K)k_t+sy_t$. This model treats the reproducible capital stock broadly enough to sustain growth without diminishing returns to that broad capital.

k_{t+1}=[1-\delta_K+sA]k_t, \qquad g_k=sA-\delta_K.

Here $g_k$ is the annual proportional growth rate, because the capital growth factor is $1+g_k$. Output per worker grows at the same rate as capital per worker when $A$ is constant. The formula assumes the parameters stay fixed and the growth factor is positive. If a policy changes $s$ permanently, it changes this model’s long-run growth rate; if it only raises $k_t$ once, it raises the level of the future path without changing its growth rate.

Romer-style research model: Let $A_t$ be the stock of ideas or productivity, $\eta$ research productivity, and $r$ the share of a fixed, normalized labor force devoted to research. Assume knowledge evolves as $A_{t+1}=(1+\eta r)A_t$. For the examples below, output per worker is $y_t=(1-r)A_t$: research labor is not producing current goods, while the stock of ideas raises production productivity.

g_A=\eta r, \qquad y_t=(1-r)A_t.

Under these assumptions, a permanent change in research productivity $\eta$ or research share $r$ changes the growth rate of knowledge. Changing $r$ also changes the share of labor producing current goods, so it affects the current output level. A one-time grant that raises $A_t$ but leaves $\eta$ and $r$ unchanged raises productivity’s level, not its subsequent growth rate. This simple specification abstracts from other research-model features, such as changes in labor supply or the details of research incentives.

Human-capital model: Let $h_t$ denote human capital per worker, $e$ the share of time devoted to education, and $\phi$ the effectiveness of education. Assume $h_{t+1}=(1+\phi e)h_t$. In the reduced-form production relation $y_t=A(1-e)h_t$, $A$ is constant and $1-e$ is time available for current production. The relation isolates the trade-off between education investment and current production; it does not represent every feature of a full Lucas model.

g_h=\phi e, \qquad g_y=\phi e \quad\text{when } A \text{ and } e \text{ are fixed.}

A permanent change in the education share changes the growth rate of human capital and, under these assumptions, output. It also changes the share of time available for current production. Thus, an education policy can lower output initially yet raise its subsequent growth rate. The output-growth conclusion requires a fixed education share after the policy change; if the share continues to vary, output growth also reflects those changing labor-time allocations.

Deriving the AK Growth Result

The AK result follows directly from the capital accumulation equation. Substitute production $y_t=Ak_t$ into the assumed evolution of capital. This expresses next year’s capital as a constant multiple of this year’s capital.

Start with capital accumulation.
$k_{t+1}=(1-\delta_K)k_t+sy_t$. The term $(1-\delta_K)k_t$ is capital left after depreciation, and $sy_t$ is saving invested as new capital.

Substitute production.
Since $y_t=Ak_t$, $k_{t+1}=(1-\delta_K)k_t+sAk_t=[1-\delta_K+sA]k_t$.

Convert the growth factor to a growth rate.
The proportional growth rate is $k_{t+1}/k_t-1$, so it equals $[1-\delta_K+sA]-1=sA-\delta_K$, provided $k_t>0$.

This derivation also shows why it is not enough to look only at the saving contribution $sA$: depreciation must be subtracted. The conclusion depends on the AK production assumption and a constant saving rate. With diminishing returns to the relevant capital, the same equation would not generally imply a constant positive growth rate from capital accumulation alone.

Worked Policy Comparisons

Worked Example: A Saving Incentive in the AK Model

Suppose an economy has $A=0.50$, a depreciation rate of $\delta_K=0.05$ per year, and an initial capital stock per worker of $k_0=\$100$ thousand. Its saving rate is initially $s=0.20$. A permanent policy raises the saving rate to $s=0.24$, with all other assumptions unchanged. Compare output after five years under the original and new saving rates.

Method. Calculate the AK growth factor under each saving rate, then apply that factor for five annual periods. In this example, $A$ is constant, so $y_t=Ak_t$.

Under the original policy, the annual growth rate is $sA-\delta_K=(0.20)(0.50)-0.05=0.10-0.05=0.05$, or 5%. The annual capital factor is $1.05$. Therefore,

k_5=100(1.05)^5=100(1.2762815625)=127.62815625 \quad\text{thousand dollars per worker.}

Output per worker after five years is $y_5=(0.50)(127.62815625)=63.814078125$ thousand dollars. Under the new saving rate, the growth rate is $(0.24)(0.50)-0.05=0.12-0.05=0.07$, or 7%, and the annual factor is $1.07$. Thus,

k_5=100(1.07)^5=100(1.4025517307)=140.25517307 \quad\text{thousand dollars per worker},

and $y_5=(0.50)(140.25517307)=70.127586535$ thousand dollars per worker. Rounded to the nearest thousandth of a thousand dollars, the outputs are $\$63.814$ thousand and $\$70.128$ thousand per worker, respectively. The policy raises the model’s growth rate by 2 percentage points per year, not just its level. The check is that $70.127586535/63.814078125=(1.07/1.05)^5$, as expected when both paths start at the same capital stock. The common mistake is to call $sA=12\%$ the new growth rate; that omits 5% depreciation. The correct rate is $sA-\delta_K=7\%$.

Worked Example: Research Effort and Current Output

A hypothetical economy has knowledge stock $A_0=100$, research productivity $\eta=0.10$, and research share $r=0.20$. Use the Romer-style equations above to compare output now and after four years with a policy that permanently raises $r$ to $0.30$. Hold $A_0$ and $\eta$ fixed.

Method. First calculate the knowledge growth rate, then calculate current output using the labor share $1-r$. Apply the knowledge growth factor for four years.

Initially, $g_A=\eta r=(0.10)(0.20)=0.02$, or 2% per year. Current output is $y_0=(1-0.20)(100)=0.80(100)=80$. After four years, $A_4=100(1.02)^4=100(1.08243216)=108.243216$, so $y_4=0.80(108.243216)=86.5945728$.

With $r=0.30$, knowledge grows at $g_A=(0.10)(0.30)=0.03$, or 3% per year. Current output becomes $y_0=0.70(100)=70$. After four years, $A_4=100(1.03)^4=100(1.12550881)=112.550881$, and $y_4=0.70(112.550881)=78.7856167$. Rounded to three decimal places, output is 86.595 without the policy and 78.786 with it, in the model’s output units. The policy raises the growth rate of knowledge but reduces current goods-producing labor. It has not yet overcome the lower labor share after four years. The common mistake is to infer that faster knowledge growth must mean higher output at every date; the production-labor effect prevents that conclusion.

Worked Example: Education Time and Human-Capital Growth

In a hypothetical economy, $h_0=10$, $\phi=0.10$, $A=2$, and output per worker is $y_t=A(1-e)h_t$. Education time is initially $e=0.40$. A permanent education policy raises it to $e=0.60$. Compare output now and after four years, assuming the education share is constant at its new value after the policy takes effect.

Method. Calculate each education share’s human-capital growth factor and its production-time share separately. This is important because education affects both future human capital and the time available for current production.

At $e=0.40$, human capital grows at $\phi e=(0.10)(0.40)=0.04$, or 4% per year. Current output is $y_0=2(1-0.40)(10)=12$. After four years, $h_4=10(1.04)^4=10(1.16985856)=11.6985856$, so $y_4=2(0.60)(11.6985856)=14.03830272$.

At $e=0.60$, human capital grows at $(0.10)(0.60)=0.06$, or 6% per year. Current output is $y_0=2(1-0.60)(10)=8$. After four years, $h_4=10(1.06)^4=10(1.26247696)=12.6247696$, giving $y_4=2(0.40)(12.6247696)=10.09981568$. Rounded to two decimal places, the original-policy outputs are 12.00 initially and 14.04 after four years; the new-policy outputs are 8.00 initially and 10.10 after four years. The policy raises long-run growth in this model but lowers output over this four-year comparison. A common error is to use $h_t$’s 6% growth rate as if it were the output level or to ignore the production-time factor $1-e$. The calculation checks both effects separately.

Reading the Paths and Comparing Policy Effects

A useful graphical test is to plot the logarithm of output against time. If output grows at a constant proportional rate $g$, its log rises by the same amount each year: $\log(y_{t+1})-\log(y_t)=\log(1+g)$. The path is a straight line in a log-output graph. A one-time level increase shifts the line upward without changing its slope. A permanent change in a growth parameter changes the slope from the policy date onward; if the policy also changes current production, the path can jump to a different starting point as well.

Model and policy change	Immediate or level effect	Long-run growth effect under the stated assumptions
AK: one-time increase in $k_t$	Higher capital and output levels	None; $s$, $A$, and $\delta_K$ are unchanged
AK: permanent increase in $s$	Changes investment and the path of capital	Yes; $g_k=sA-\delta_K$ rises if $A$ is fixed
Romer-style: one-time increase in $A_t$	Higher productivity, with $r$ unchanged	None; the knowledge growth factor is unchanged
Romer-style: permanent increase in $r$	Less labor produces current goods	Yes; $g_A=\eta r$ rises if $\eta$ is fixed
Human capital: permanent increase in $e$	Less time is available for current production	Yes; $g_h=\phi e$ rises if $\phi$ is fixed

These are conditional model results, not general claims that a particular policy always raises welfare or output. In particular, the research and education examples make a resource trade-off explicit: more effort devoted to creating future productivity means less labor time devoted to current output. The time at which a faster-growing path overtakes a lower-investment path depends on the initial level difference and the growth-rate difference. A higher growth rate alone does not establish that output is already higher over a short horizon.

Common Mistakes to Avoid

Confusing an initial jump with faster growth. A one-time increase in $A_t$ changes the knowledge level. If $\eta$ and $r$ do not change, it does not change the factor $1+\eta r$.
Omitting depreciation in the AK model. The rate is $sA-\delta_K$, not $sA$. Saving finances new capital, but depreciation removes some existing capital.
Ignoring the date and persistence of a policy. A temporary change to a growth parameter affects growth only while it is in force. A permanent change affects each subsequent period under the model’s assumptions.
Equating a higher growth rate with a higher current level. The Romer-style and human-capital examples show how diverting labor time can lower current production even when the knowledge or human-capital stock grows faster.
Comparing slopes on the wrong graph. On an ordinary output-versus-time graph, a growing series curves upward. On a log-output graph, a constant proportional growth rate gives a straight line whose slope represents $\log(1+g)$.

Key takeaway: Identify what the policy changes: an initial stock, a persistent growth parameter, or both. Then calculate the initial production effect separately from the subsequent growth factor. The model determines which of these changes the policy can produce.

Check Your Understanding

For each problem, distinguish the immediate level effect from the continuing growth effect. Unless stated otherwise, use the simplified model equations defined above.

In an AK model, $A=0.40$, $s=0.25$, and $\delta_K=0.06$. A one-time transfer raises $k_t$ by 10% but does not change any parameter. What is the growth rate before and after the transfer, and what happens to the capital level? (Answer: The growth rate is $0.25(0.40)-0.06=4\%$ both before and after; the capital level rises by 10%.)
An AK economy has $A=0.60$ and $\delta_K=0.04$. A permanent policy raises $s$ from 0.15 to 0.20. Calculate both annual growth rates. (Answer: Initially $0.15(0.60)-0.04=5\%$; afterward $0.20(0.60)-0.04=8\%$, an increase of 3 percentage points.)
In a Romer-style model, $\eta=0.08$, and $r$ increases permanently from 0.25 to 0.375. Calculate knowledge growth before and after the policy. If $A_t=200$ on the policy date, calculate $A_{t+1}$ under the new policy. (Answer: Growth rises from $0.08(0.25)=2\%$ to $0.08(0.375)=3\%$; $A_{t+1}=200(1.03)=206$.)
A human-capital model has $\phi=0.05$ and education share $e=0.40$. The education share rises permanently to $0.60$, with $A=1.5$ and $h_t=20$. Calculate the output level before and after the change, and each human-capital growth rate. (Answer: Before, $y_t=1.5(0.60)(20)=18$ and growth is $2\%$; after, $y_t=1.5(0.40)(20)=12$ and growth is $3\%$.)
A policy permanently raises a Romer-style model’s research productivity from $\eta=0.04$ to $\eta=0.06$, while $r=0.25$ and $A_t=400$ remain fixed at the policy date. Find next period’s knowledge under each value of $\eta$, and state whether this is a level or growth-rate policy effect. (Answer: At $\eta=0.04$, $A_{t+1}=400[1+(0.04)(0.25)]=404$; at $\eta=0.06$, $A_{t+1}=400[1+(0.06)(0.25)]=406$. The permanent parameter change raises the growth rate from 1% to 1.5%.)

Model and policy change	Immediate or level effect	Long-run growth effect under the stated assumptions
AK: one-time increase in \(k_t\)	Higher capital and output levels	None; \(s\), \(A\), and \(\delta_K\) are unchanged
AK: permanent increase in \(s\)	Changes investment and the path of capital	Yes; \(g_k=sA-\delta_K\) rises if \(A\) is fixed
Romer-style: one-time increase in \(A_t\)	Higher productivity, with \(r\) unchanged	None; the knowledge growth factor is unchanged
Romer-style: permanent increase in \(r\)	Less labor produces current goods	Yes; \(g_A=\eta r\) rises if \(\eta\) is fixed
Human capital: permanent increase in \(e\)	Less time is available for current production	Yes; \(g_h=\phi e\) rises if \(\phi\) is fixed

What You'll Learn