Mankiw, Romer and Weil (1992)

# Mankiw, Romer and Weil (1992)
## Macroeconomics
### Kenji Sato
### Day 06

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## MRW

In today's class and the next, we study the theory proposed by [Mankiw, Romer and Weil (1992, QJE)](https://doi.org/10.2307/2118477) (henceforth, MRW)

This paper is extremely influential. Please read it.

---

## Cobb-Douglas production function

We consider the following function form:

`$$F(K, L) = K^\alpha L^{1 - \alpha}$$`

The intensive-form:

`$$f(k) = k^\alpha$$`

---

## Steady state

Verify that the steady state is given by

`$$k^* = \left(\frac{
  s
}{
  \delta + g + n
}\right)^{\frac{1}{1 - \alpha}}$$`

and that output per worker is given by

`$$\frac{Y}{L}
=
A \left(\frac{
  s
}{
  \delta + g + n
}\right)^{\frac{\alpha}{1 - \alpha}}$$`

---

## Elasticity of output per capita wrt saving rate

`$$\ln \left( \frac{Y}{L} \right) =
\log A(0) + gt + \frac{\alpha}{1 - \alpha} \ln (s)
-\frac{\alpha}{1 - \alpha} \ln (\delta + g + n)$$`

Data suggests that

`$$\alpha \simeq \frac{1}{3} \Longrightarrow
\frac{\alpha}{1 - \alpha} \simeq \frac{1}{2}$$`

The Solow model, therefore, suggests that 1% increase of `$s$` implies 
0.5% increase of `$Y/L$`.

Increment of `$\delta + g +n$` by 1% decreases `$Y/L$` by 0.5%.

---

## Question

Do data support these predictions of the Solow model
concerning the determinant of standard of living?

---

## Assumptions

MRW assumes the following:

* `$g$` and `$\delta$` are constant across countries.
    * advancement of knowledge, which `$g$` reflects, is not country-specific
    * no strong reason to expect `$\delta$` to vary across countries
* A(0) is decomposed into `$A(0) = a + \epsilon_j$`, where `$a$` is common to 
  all countries and `$\epsilon_j$` is country-specific shock.

---

## Empirical specification

Consider `$t = 0$`.

MRW's first empirical specification:

`$$\ln \left( \frac{Y_j}{L_j} \right) =
a + \frac{\alpha}{1 - \alpha} \ln (s_j)
-\frac{\alpha}{1 - \alpha} \ln (\delta + g + n_j) + \epsilon_j$$`

Under the assumption that `$s$` and `$n$` are independent of `$\epsilon$`
we can estimate the above with the ordinary least squares.

`$$\ln \left( \frac{Y_j}{L_j} \right) =
\beta_0 + \beta_s \ln (s_j) +
\beta_n \ln (\delta + g + n_j) + \epsilon_j$$`

---

## Empirical specification (cont'd)

`$$\ln \left( \frac{Y_j}{L_j} \right) =
\beta_0 + \beta_s \ln (s_j) +
\beta_n \ln (\delta + g + n_j) + \epsilon_j$$`

If the Solow model is a good model of the economy, then the 
OLS would predict

`$$\beta_s \simeq 0.5$$`

`$$\beta_n \simeq - 0.5$$`

To perform the empirical exercise, we need to determine
`$Y_j/L_j$`, `$s_j$`, and `$\delta + g + n_j$` from data.

---

## Empirical specification (restricted regression)

Since the model predicts that `$\beta_s$` and `$\beta_n$` are 
the same in maginitude and opposite in sign, the 
following specification should work.

`$$\ln \left( \frac{Y_j}{L_j} \right) =
\beta_0 + \beta_1 \left( 
  \ln (s_j) - \ln (\delta + g + n_j)
\right) + \epsilon_j$$`

---

## Data

MRW uses data set constructed by Summers and Heston (1988), an earlier version of 
the Penn World Table that you are already familiar with.

Their dataset convers the period of 1960 to 1985.

---

## Data (cont'd)

They measure

* `$n_j$` as the **average rate of growth of the working-age population**.
  (Unlike PWT v9.0, the earlier version does not contain total number of the 
  employed.)
* `$s_j$` as the average share of real private and public investment. (**average 
  share of gross capital formation**)
* `$Y_j/L_j$` as **real GDP devided by the working-age population** in 1985.
  (1985 is the latest sample in their dataset.)

They simply assume that `$\delta + g = 0.05$`.

---

## Non-oil, Intermediate, OECD

They used three sample for the empirical study.

* Non-oil countries exclude countries that are heavily reliant on oil production.
* Intermediate countries exclude countries whose population in 1960 is less than 1 million.
* OECD countries consist of OECD countries with populations greater than 1 million.

.small[
According to a recent data by World Bank, Venezuela and Chad are more heavily reliant 
on oil than Iran but MRW includes those two countries while excluding Iran. 
]

---

Table 1 on p. 414 (emphasis added)

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## Implied `$\alpha$`

The restricted regression predicts higher `$\alpha \simeq 0.59$` from 
the coefficient on ln(I/GDP);

`$$\frac{\alpha}{1- \alpha} \simeq 1.43$$`

implies

`$$\alpha \simeq 0.588..$$`

which is much higher than the common wisdom `$\alpha \simeq 1/3$`.

---

## Exercise (A part of the homework series)

Do the above two regression analyses

* non-restricted, bivariate OLS
* restricted, simple OLS

with PWT v9.0 dataset with

* countries the populations of which in 1960 are more than 1 million,

Report on what you observe.

---

## What's missing?

MRW's strategy to fix the bad prediction:

**Add human capital to the Solow model**

---

## Production function

Let `$0 < \alpha + \beta < 1$`, `$\alpha, \beta > 1$`. The output is given by

`$$Y = K^\alpha H^\beta (AL)^{1 - \alpha - \beta}$$`

* `$H =$` stock of human capital
* `$K =$` stock of physical capital
* `$A =$` knoledge
* `$L =$` labor

`$$\begin{aligned}
  k = \frac{K}{AL},\quad
  h = \frac{H}{AL},\quad
  y = \frac{Y}{AL} = k^\alpha h^\beta
\end{aligned}$$`

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## Investment

They assume that the investment rates are constant for both capitals and that the depreciation rate is common.

* `$s_k =$` saving rate for physical capital `$K$`
* `$s_h =$` saving rate for human capital `$H$`
* `$\delta =$` depreciation for both `$K$` and `$H$`

Verify that

`$$\begin{aligned}
  \dot k &= s_k k^\alpha h^\beta - (\delta + g + n) k\\
  \dot h &= s_h k^\alpha h^\beta - (\delta + g + n) h
\end{aligned}$$`

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## Your turn

Compute the steady state values

`$$k^* \quad \text{and} \quad h^*$$`

and verify that

`$$\begin{multline}
  \ln \left( \frac{Y}{L} \right) =
  \log A(0) + gt + 
  \frac{\alpha}{1 - \alpha - \beta} \ln (s_k) \\ + 
  \frac{\beta}{1 - \alpha - \beta} \ln (s_h) - 
  \frac{\alpha + \beta}{1 - \alpha - \beta} \ln (\delta + g + n)
\end{multline}$$`

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## Convergence to the steady state

When we studied the Solow model, we justfied our strategy to focus on the steady state because the economy eventually converges to the steady state regardless of the initial capital.

The MRW model too has this property. Please verify this by using the phase diagram. See the Problem Set for today for details.

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## Model 1: Decomposition of Y/L, Eq. (11)

* The coefficient, `$\alpha / (1-\alpha-\beta)$` on `$\ln (s_k)$` is greater 
  than `$\alpha / (1 - \alpha)$` because `$1 - \alpha > \beta > 0$`
* The coefficient on `$\ln (\delta + g + n)$` is larger in absolute value 
  than the coefficient on `$\ln (s_k)$` i.e.,
  `$$\frac{\alpha}{1 - \alpha - \beta} < \frac{\alpha + \beta}{1 - \alpha - \beta}$$`

---

## Model 2: Another decomposition, Eq. (12)

`$$\begin{multline}
  \ln\left(\frac{Y}{L}\right) = 
  \ln A(0)+gt+\frac{\alpha}{1-\alpha}\ln(s_{k}) \\ -
  \frac{\alpha}{1-\alpha}\ln(\delta+g+n)+\frac{\beta}{1-\alpha}\ln(h^{*})
\end{multline}$$`

This specification is almost identical to the Solow model.

The bias in the linear regression for the baseline model may well be due to the omitted variable. By omitting `$\ln(h^{*})$` term, 
the previous linear regression is subject to omitted-variable bias because higher `$h^*$` results in

* higher `$s_k$` (e.g., by improving sense of economy)
* lower `$n$` (e.g., by increasing cost of marriage)

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## Model Choice made by MRW

Which of `$s_h$` and `$h^*$` is easier to measure?

As a proxy for `$s_h$`, MRW measured the percentage of the working-age population 
that is in secondary school.

`$$\begin{multline}
\texttt{SCHOOL} =
\frac{\text{# of people in secondary school}}{\text{# of people aged 12-17}}\\
\times
\frac{\text{school age population (15-19)}}{\text{working age population}}
\end{multline}$$`

Then they used this variable to test Model 1.

---

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## Coefficients

`$$\begin{aligned}
  \frac{\alpha}{1 - \alpha - \beta} 
    &\simeq 0.70\\
  \frac{\beta}{1 - \alpha - \beta}
    &\simeq 0.73\\
  \frac{\alpha + \beta}{1 - \alpha - \beta}
    &\simeq -1.50
\end{aligned}$$`

These estimates suggest

`$$\alpha \simeq 0.3 \quad \text{and}\quad \beta \simeq 0.3$$`

Consistent with the common wisdom.

---

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## Question

Suppose that you have just heard that your tax you had already paid 
(say, 10,000 Yen) will be reimbursed next month.

What would you do with the expected windfall gains?

1. Nothing. Wait until you have the cash.
2. Go shopping or go for a drink this weekend, expecting the refund.

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## Why Solow not enough?

In the Solow model, **the saving rate is exogenous**. Take a tax reduction 
for example. In the Solow model, consumers raises consumption at the time 
of enforcement of the new tax system.

But usually governments would inform the change of tax well before the enforcement and we might drink a toast to the tax reduction.

Similarly, in case of tax hike, it is possible that consumers consume
more immediately after knowing that they will have to pay more to the
government.

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## Why Solow not enough? (Cont'd)

So, the Solow model is missing an important aspect of our economic activity: **Expectation formation**.

Although rigorously dealing with the problem of expectations is beyond the scope of this 
course, the **Ramsey model** will serve as the skeleton of more elaborate models.