Growth

Growth Theory

Mark Huggett1

1Georgetown

January 26, 2018

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Growth Theory: The Agenda

1. Facts motivating theory

2. Basic Solow model

3. Model properties

4. How to use the model

5. Full Solow model

6. Use the model to interpret data

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Kaldor’s Growth “Facts”

1. Output per capita grows over time2. Capital per capita grows over time3. Capital-Output ratio is approx constant over time4. Capital and Labor’s share is approx constant over time5. Return to Capital has no trend6. Output per capita varies widely across countries at apoint in time

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Solow Growth Theory

Key Assumptions:1. Agg. Production Function w/ Diminishing MPK2. Can Accumulate Physical Capital3. Technology Grows Exogenously4. Constant Saving/Investment Rate.

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Basic Solow Model

1. Ct + It = Yt = F (Kt, Lt)

2. It = sF (Kt, Lt) - investment

3. Kt+1 = Kt(1− δ) + It - Capital Accumulation

4. Lt = L

Implication: Kt+1 = Kt(1− δ) + sF (Kt, Lt)Steady State: δK = sF (K,L)

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Restate Model in Terms of k ≡ K/L

ct + it = yt = F (kt, 1) - CRSit = sF (kt, 1) - investmentkt+1 = kt(1− δ) + it - Capital Accumulation

Implication: kt+1 = kt(1− δ) + sF (kt, 1)Steady State: δk = sF (k, 1)

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Solow Model—Steady State

0 k

dk

y =F(k ,1)

i=s F(k ,1)

k *

i*

y *

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Main Properties of the Basic Solow Model

Define k = KL

1. One positive capital steady state capital-labor ratio k∗.2. Higher savings rate s implies a higher steady state valuek∗.3. The economy converges over time to the steady state k∗.4. There is a max feasible steady state capital-labor ratiok∗∗ where the production function intersects the”depreciation line”.

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How to Use the Model

1. Use the graph to get qualitative insights to two typesof experiments: (1) exogenous one-time changes incapital or labor (e.g. war or disease) and (2)permanent changes in model parameters (e.g. changethe savings rate s).

2. One can get insight into how factor prices move fromthe production function if one adopts competitivetheory of factor prices.

3. Make assumptions on the parametric form of theproduction function and choose all parameter values.Use the model for quantitative insights (e.g. how muchdoes increasing the saving rate increase steady stateoutput?).

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Example: Recovery from WWII

1. War begins and ends in period 3.

2. Capital-labor ratio falls as, we assume, more capital isdestroyed than people.

3. After the war ends, the model mechanism againoperate.

4. After you determine what happens to the capital-laborratio over time, then time variation in all othervariables are easy to determine.

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2

2.5

3

3.5

4

Solow Model: War Recovery

0

0.5

1

1.5

2

0 5 10 15 20 25

Time

Capital‐labor Output‐labor Wage

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Example: Plague

1. The Black Death killed a large fraction of the Englishpopulation in the 1300’s.

2. This increased the capital-labor ratio suddenly anddramatically for exogenous reasons.

3. Assume that the plague in the model permanentlydecreases the total labor in the model economy.

4. Determine implications of the model. Compare toClark’s data on real wages.

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012345678

0 5 10 15 20 25

Time

Solow Model: Black Death

Capital‐labor Output‐labor wage Gross interest rate

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0

0.1

0.2

0.3

0.4

0.5

1340 1345 1350 1355 1360 1365 1370

Time

England: Black Death

Farm Wage Nonfarm Wage Population (millions/10)

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Does the Basic Model Produce Kaldor’s Facts?

1. In steady state y and k do not grow!

2. Thus, the only possibility to explain sustained growth iny = Y/L and k = K/L (Kaldor’s Facts 1-2) is to have allcountries be BELOW steady state.

3. This is unsatisfactory as then growth should be slowing downover time with a fixed saving rate. The data say that growthrates of Y/L are increasing over the last two hundred years inadvanced countries!

4. This motivates the full Solow model.

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“Full” Solow Model: (Use kt ≡ Kt

LtAt)

Ct + It = Yt = F (Kt, LtAt)It = sF (Kt, LtAt)Kt+1 = Kt(1− δ) + ItLt+1 = Lt(1 + n)At+1 = At(1 + g)

Implication:Kt+1 = Kt(1− δ) + sF (Kt, LtAt)kt+1(1 + n)(1 + g) = kt(1− δ) + sF (kt, 1)

Steady State: k[(1 + n)(1 + g)− (1− δ)] = sF (k, 1)

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Figure 1: Solow Model—Steady State

0

k

k (d+ n+ g+ ng)

y=F(k ,1)

i= sF(k ,1)

k*

i*

y*

Figure 2: Solow Model—Golden Steady State

0 k*

k(d+n+g+ng)

y=F(k,1)

k*max k* g

i* g

y* g

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Interpretation of Steady State: kt ≡ Kt

LtAt

1. (kt, yt, it, ct) constant BUT (Kt/Lt, Yt/Lt, It/Lt, ...) grow2. (Yt/Lt, Kt/Lt) grow at rate g and (Yt, Kt) grow (approx)at rate n+ g.3. Yt = WtLt +RtKt implies (in steady state) Wt grows atrate g and Rt is constant.4. In steady state, labor and capital’s share of output areEXACTLY constant.5. Note: Points 1-4 are Kaldor’s Facts 1-5!

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What Are the Effects of Increasing the Savings Rate?

Distinguish between the “steady state” or long-run effects andthe effects in “transition”.Steady State: increasing savings rate s does NOT change thelong-run growth rate of Y/LTransition: increasing savings rate s increases the growth rateof output for any finite time period

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Assessing the Solow Model: Cross-Country Differences

1. Are countries with high capital-labor ratios rich?2. Are countries with high savings rates rich?3. Are countries with high population growth rates poor?4. Do observed differences in savings rates across countriesimply large GDP per worker differences within the Solowmodel?5. Is the technology level the same across countries?

We will give a first-pass answer to all of these questions.

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Assessing the Importance of Savings RateDifferencesTwo countries:same technology: y = F (k, 1) = Akβ

savings rate differs: sH = .30, sL = .05yHyL

= F (kH ,1)F (kL,1)

=kβHkβL

= (kHkL

)β

In steady state: sAkβ = k(n+ g + ng + δ) implies

k = (sA

n+ g + ng + δ)1/(1−β)

yHyL

= (kHkL

)β = ( sHsL

)β

1−β = ( .30.05

)β

1−β and yHyL

.= 2.15 if β = .3!

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Assessing the Importance of TechnologyDifferencesAssume the production function is Cobb-DouglasYi = AiF (Ki, Li) = AiK

βi L

1−βi

Yi = AiF (Ki, Li)⇒ Ai = Yi/F (Ki, Li) = Yi/(Ki)β(Li)

1−β

Assume (Yi, Ki, Li) can be measured (with some error)across countries and set β to the US value.

Standard Finding: low Yi/Li countries have very lowimplied Ai levels. Labor quality differences across countriesare perhaps poorly measured.

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