CLASS #16: THE GOLDEN RULE OF CAPITAL ACCUMULATION AND THE SOLOW GROWTH MODEL WITH EXOGENOUS TECHNOLOGICAL PROGRESS

• The golden rule of capital accumulation
  • The Solow growth model predicted convergence toward a steady state.
  • The steady state depends positively on the saving rate.
  • What is the optimal saving rate?
  • Remember: we want to maximize happiness (utility), not output
    Why should we poor people make sacrifices for those who will in any case live in luxury in the future?
    Robert Solow (Nobel prize winner)
• How can we reconcile the Solow growth model with ongoing growth: the Solow growth model with exogenous technological progress

The golden rule of capital accumulation

Assume people's utility depend on the amount of consumption (one good only in this economy): if we want to maximize consumption in the long run we want to find the saving rate s that maximizes the steady state level of per capita consumption. Depreciation + Decreasing marginal returns to capital imply that there is a level of k beyond which further capital accumulation is suboptimal: the increase in capital depreciation outweighs the increase in output. Remember the formula for the steady state:

$$k^*=(\frac{sA}{\delta+\lambda})^{\frac{1}{1-\alpha}}$$

Depreciation ($\delta$) and population growth ($\lambda$) are - to some extent - out of our control, and - in the neoclassical framework, to some extent - so is A (productivity).
The only variable that we can control is the saving rate s: what is the optimal saving rate? Intuitive approach. Since we are in a closed economy: y_t=c_t+i_t. What is investment in the steady state? from its definition:

$$K_{t+1}=(1-\delta)K_t+I_t$$

Divide both sides by N_t and obtain (remember $\frac{N_{t+1}}{N_t}=1+\lambda$):

$$(1+\lambda)k_{t+1}=(1-\delta)k_t+i_t$$

but by the definition of steady state k_t+1=k_t=k^*, so:

$$i^*=(\lambda+\delta)k^*$$

$$c^*=y^*-i^*=Ak^{*\alpha}-(\lambda+\delta)k^*$$

What is the level of k^* that maximizes per capita consumption? take derivatives!

$$\alpha Ak^{*\alpha-1}-\delta=\lambda$$

or

$$MPK-\delta=\lambda$$

You want to save up until the level of k such that the MPK is equal to depreciation plus population growth: a further increase in capital would trigger gains in terms of output that are lower that the sum of depreciation and population growth (graphical analysis). Alternative approach.
Remember that c=(1-s)y. This implies:

c^*=(1-s)y^*=

$$=(1-s)A(\frac{sA}{\delta+\lambda})^{\frac{\alpha}{1-\alpha}}$$

The Solow growth model with technological progress

Exogenous technological progress reconciles the Solow (neoclassical) growth model with sustained growth. Add technological progress to the model in a special way, as labor augmenting.
Call the variable E_t efficiency of labor
for a given number of hours spent working, the higher E the higher the labor input in terms of ``efficiency units".
Example: two workers can work the same number of hours, but depending on how efficiently they perform their task, their effective input can be quite different. We will see later why we did not assume growth in A. The production function becomes:

$$Y_t=AK_t^\alpha(E_tN_t)^{1-\alpha}$$

Efficiency of labor grows with improvements in education, health, skills. Assume that it grows at a constant rate $\gamma$ per year:

$$E_{t+1}=(1+\gamma)E_t$$

How do we measure the growth rate in E?

$$\ln{Y_t}-\ln{Y_{t-1}}-\alpha(\ln{K_t}-\ln{K_{t-1}})$$

$$-(1-\alpha)(\ln{N_t}-\ln{N_{t-1}})=(1-\alpha)(\ln{E_t}-\ln{E_{t-1}})$$

or

$$\ln{E_t}-\ln{E_{t-1}}=\frac{TFPgrowth}{1-\alpha}$$

How do we measure the level of E today? A and E are indistinguishable. But the level of E does not matter, as we will see: so assume E₀=1. Use the definition of investment:

$$K_{t+1}=(1-\delta)K_t+I_t =$$

$$=(1-\delta)K_t+sY_t$$

and divide both sides by E_t+1N_t+1 to obtain:

$$\frac{K_{t+1}}{E_{t+1}N_{t+1}}=\frac{(1-\delta)K_t+ sAK_t^\alpha(E_tN_t)^{1-\alpha}}{E_{t+1}N_{t+1}}$$

Call $\bar{k}_t=\frac{K_t}{E_tN_t}$, the "capital per effective labor" ratio, and recall that

$$E_{t+1}=(1+\gamma)E_t$$

and

$$N_{t+1}=(1+\lambda)N_t,$$

and obtain:

$$\bar{k}_{t+1}=\frac{(1-\delta)\bar{k}_t+sA\bar{k}_t^\alpha} {(1+\lambda)(1+\gamma)}$$

This is -almost- the same non-linear difference equation we studied last class. Hence it has the same properties. In particular:

• it has a steady state in terms of $\bar{k}$: $\bar{k}^*$.
• it displays convergence toward the steady state.

No matter where the economy starts, at some point it will reach a stationary state where $\bar{k}=\bar{k}^*$. The steady state is in terms of $\bar{k}$, not k!
Both per capita capital and per capita output are growing at the rate $\gamma$.
By definition $\bar{k}_t=\frac{K_t}{E_tN_t}$, or

$$k_t=\frac{K_t}{N_t}=\bar{k}^*_tE_t$$

This implies that the growth rate of per capita capital is:

$$\ln{k_{t+1}}-\ln{k_t}=\ln{E_{t+1}}-\ln{E_t} \simeq \gamma$$

and the growth rate in per capita output is:

$$\ln{y_{t+1}}-\ln{y_t}=\alpha(\ln{k_{t+1}}-\ln{k_t}) \simeq \gamma$$

What does the steady state in the "capital per effective labor ratio" $\bar{k}$ depend upon?
solve:

$$\bar{k}=\frac{(1-\delta)\bar{k}+sA\bar{k}^\alpha} {(1+\lambda)(1+\gamma)}$$

and obtain:

$$\bar{k}^*=(\frac{sA}{(1+\lambda)(1+\gamma)-(1-\delta)})^ {\frac{1}{1-\alpha}}$$

Conclusions

• The model displays persistent growth in output per capita.
• The growth in per capita output depends only on the exogenously given rate of technological progress.
• A key assumption of the neoclassical growth model is that technological progress is -at least eventually- the same for all countries (technology diffusion): therefore the model still predicts convergence in the levels of output across countries.
• The steady state of the "capital per effective labor ratio" depends, among other things, on the savings rate.
• However, governments can do very little to affect growth: by increasing the savings rate they can affect the level of output, but not its growth, which is exogenously determined.

the Asian miracle

• "perspiration" (Krugman): growth in k
• "inspiration": growth in technology

Alwyn Young: for most countries, perspiration
does the neoclassical growth model then explain the recent crash?
perspiration suggests a gradual slowdown, not a crash!