# Economics for Business Lecture 17

In the last few lectures, we've looked at growth theory and the stakes involved in getting the basics wrong. We saw the Solow model's predictions about sustained capital accumulation ($\Delta k/k$): keep population growth low, keep savings ($s$) and therefore investment($I$) rather high, and try to curb depreciation on assets. Technical progress and human capital move the economy forward, potentially stimulating convergence of growth rates.

Now we'll move onto a micro-founded macro model developed in the later chapters of the Barro book. The basic idea is to specify four markets inside the economy: the products market, the bond market, the money market, and the labour market. We'll build our macroeconomic equilibria from behavioural assumptions about the actors in the model: households and firms. Households are assumed to want to maximise their incomes from all of these markets, subject to a budget constraint. Firms want to maximise profits. Their interactions, along with the usual macroeconomic accounting identities, such as $Y=C+I+G$, give us the macroeconomic equilibrium, called a general equilibrium.

# Economics for Business Lecture 16: Growth and Convergence

Why are we so rich and they so poor? This is the fundamental question in macroeconomics. Economists agree that the main way to enrich a country and its people is to create the conditions which allow it to grow its way out of poverty. By growth, I mean GDP growth of course.

We have seen in the last few lectures that the determinants of GDP growth are increases in the rate of capital accumulation through saving and investment, increasing rates of technological change, and a steady population growth rate.

Now we'll look at the predictions of the Solow growth model and it's extensions while asking how much the theory actually explains when we look at the data.

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The Solow model predicts that growth rates tend to diminish over time as the economy approaches a steady state level of output per worker. The steady state level of output per worker is shown to increase as savings rates or technology increase. The steady state level of output per worker falls as the population or grows.

Changes to the labour force can affect the growth rate because they change the capital labour ratio, but they do not affect the ultimate steady state level of output per worker. In all of these cases, the curves in the basic steady state diagram are shifted to illustrate the effects of changing parameter values on the steady state level of capital.

Following a quick recap (for the five or six of you that didn't make it), we'll examine the concept of absolute convergence next. Since the rate of capital accumulation per worker is essentially determined by the current stock of capital per worker, lesser developed countries are predicted by the model to grow more quickly than developed countries. However, the capital per worker will only generate faster growth rates if the values of the other parameters (savings, technology, population growth, etc.) are somewhat comparable. This implies that there is only conditional convergence.

We'll look at the data, talk about growth and convergence across the world.

Oh, and there will be a quiz in class.

Click the link below for slides, a handout version of the slides, and links to interesting articles.

# Economics for Business Lecture 15: Growth Models

The Solow Growth model attempts to explain the features of growth we encountered in the last lecture. We need to be able to explain movements in capital accumulation, labour force growth, and technology.

Beginning from the growth accounting equation

$\frac{\Delta Y}{Y} = \frac{\Delta A}{A} + \alpha \cdot \frac{\Delta K}{K} + (1-\alpha)\frac{\Delta L}{L}$

and simplifying the notation to look at accumulation in per worker  ($y = Y/L$, etc) terms with no technological change, we should see that the growth rate of real GDP per worker will be the difference between the growth rate of real GDP and the growth rate of labour, because of diminishing marginal productivity of labour. In our notation,

$\frac{\Delta y}{y} = \frac{\Delta Y}{Y} - \frac{\Delta L}{L}$

And using the same idea, we can show

$\frac{\Delta k}{k} = \frac{\Delta K}{K} - \frac{\Delta L}{L}$.

The equation above just says the growth rate of capital per worker is equal to the growth rate of capital minus the growth rate of labour.

Rearranging our first equation above using the new identities, we get

$\frac{\Delta Y}{Y} - \frac{\Delta L}{L} = \alpha \cdot \left( \frac{\Delta K}{K} - \frac{\Delta L}{L} \right)$

But we know that in per worker terms, we can reduce this equation to

$\frac{\delta y}{y} = \alpha \cdot \frac{\Delta k}{k}$

The growth of real GDP per worker depends only on the growth rate of capital per worker.

So, Solow says we should spend time thinking about policies to increase the growth rate of capital per worker in an economy in order to develop.

Solow Lesson 1: Focus on growth rate of capital per worker, $\Delta k/k$.

How does the growth rate of capital change?

The growth rate of the capital stock depends on how much the economy saves. This is because, in the medium term, everything saved gets invested. Real income in the macroeconomy must equal the Net Domestic Product, which is GDP taking depreciation ($\delta$) of the capital stock, $K$, into account. We can define real saving as the saving rate times the level of real income, or

Real Saving = $s \cdot (Y - \delta K)$

We know that household income equals the sum of what gets consumed and what gets saved, so the following equation must be true:

$Y - \delta K = C + s \cdot (Y - \delta K)$

And, because in macroeconomic equilibrium savings will always equal net investment, we can say

$Y - \delta K = C + (I + \delta K)$.

The change in capital stock will equal gross investment (that $I$ in the equation above), so we can write

$s \cdot (Y - \delta K) = I - \delta K$, and then because the change in the capital stock will equal gross investment minus depreciation of the capital stock, we have something like

$\Delta K = s \cdot (Y - \delta K)$.

Which in per worker terms, after some rearranging, which we'll do in class, is

$\Delta k/k = \Delta K / K - \Delta L / L$.

Combine this result with the requirement that the growth rate of labour should be constant, or

$\frac{\Delta L}{L} = n$

to get the result that the growth rate of capital per worker is dependent on the amount saved out of output per worker minus the cost of replacing depreciated capital per year minus the labour force growth rate, and we have

$\frac{\Delta k}{k} - s \cdot (\frac{y}{k}) - s\delta - n$.

Reversing this equation and plugging in the value for the growth rate of capital per worker, we have

$\frac{\Delta y}{y} = \alpha \cdot [s \cdot (\frac{y}{k})-s \delta - n]$.

Phew! That was a bit of a struggle, but don't worry, we'll go through some numerical examples tomorrow. You can also download the slides and a podcast after the lecture.

You really should read Barro, chapters 3 and 4, to understand this material deeply.