Economic Growth I Economics 331 J. F. O’Connor

"A world where some live in comfort and plenty, while half of the human race lives on less than $2 a day, is neither just, nor stable." Quote of the Day, NYT of Wednesday, July 18, 2001

Model of Capital Accumulation Y = F(K, L) where F displays constant returns to scale and diminishing marginal products Y = C + I C = (1-s)Y where s is the saving rate  is the change in the capital stock and  is the depreciation rate of capital

Per Unit of Labor Version Y/L = F(K/L,1) = f(k) Y/L = C/L + I/L C/L = (1-s)Y/L  L  L  L Note: y - c = i y - c = sy =sf(k) y = f(k) y = c + i c = (1-s)y  k  i -  k sf(k) = i  k  sf(k) -  k

Key Relationship  k  sf(k) -  k The change in capital per worker, k=K/L, is equal to saving (investment) per worker less depreciation of capital per worker. Capital per worker increases if saving per worker is greater than the depreciation in capital per worker, and vice versa.

Growth and the Steady State From above, the equation governing the growth in capital per worker is:  k  i -  k = sf(k) -  k In the steady state equilibrium, the change in the capital per worker is zero,  k =0. Investment = depreciation.

The Steady State Equilibrium Given the equilibrium capital-labor ratio, k*, we can find output, f(k*) and consumption per capita, (1-s)f(k*). Note that the economy converges in the long run to a steady state, where there is no growth! It arrives at a stationary state and stays there.

Saving and Growth Note experience of Germany and Japan after WWII See how a change in the saving rate affects growth. It causes the capital to grow towards a new steady state. But once that is reached, there is no more growth. So, what does a high rate of saving and investment do for you? It gives you a higher per capita income in the steady state.

Prediction: Higher s  higher k *. And since y = f(k), higher k *  higher y *. Thus, the Solow model predicts that countries with higher rates of saving and investment will have higher levels of capital and income per worker in the long run.

International Evidence on Investment Rates and Income per Person

A numerical example Production function (aggregate): To derive the per-worker production function, divide through by L: Then substitute y = Y/L and k = K/L to get

A numerical example, cont. Assume: s = 0.3  = 0.1 initial value of k = 4.0

Approaching the Steady State: A Numerical Example Year k y c i  k  k Year k y c i  k  k

Exercise: solve for the steady state Continue to assume s = 0.3,  = 0.1, and y = k 1/2 Use the equation of motion  k = s f(k)   k to solve for the steady-state values of k, y, and c.

Solution to exercise:

Population Growth Assume that the population--and labor force-- grow at rate n. (n is exogenous) EX: Suppose L = 1000 in year 1 and the population is growing at 2%/year (n = 0.02). Then  L = n L = 0.02  1000 = 20, so L = 1020 in year 2.

Introducing Population Growth Recall that k = K/L and therefore, %  k = %  K - %  L  k/k =  K/K -  L/L = ( I -  K )/K - n where n is the rate of population growth Multiply both sides by k = K/L to get  k = I/L -  k - nk where I/L = i

Growth and the Steady State From above, the equation governing the growth in capital per worker is now :  k  i -  k - nk = sf(k) - (  n)k In the steady state equilibrium, the change in the capital stock is zero,  k =0. Investment per capita = the depreciation rate + population growth rate.

Break-even investment (  + n)k = break-even investment, the amount of investment necessary to keep k constant. Break-even investment includes:  k to replace capital as it wears out n k to equip new workers with capital (otherwise, k would fall as the existing capital stock would be spread more thinly over a larger population of workers)

The equation of motion for k With population growth, the equation of motion for k is  k = s f(k)  (  + n) k break-even investment actual investment

The Solow Model diagram Investment, break-even investment Capital per worker, k sf(k) ( + n ) k( + n ) k k*k*  k = s f(k)  (  +n)k

The impact of population growth Investment, break-even investment Capital per worker, k sf(k) ( +n1) k( +n1) k k1*k1* ( +n2) k( +n2) k k2*k2* An increase in n causes an increase in break- even investment, leading to a lower steady-state level of k.

Prediction: Higher n  lower k *. And since y = f(k), lower k *  lower y *. Thus, the Solow model predicts that countries with higher population growth rates will have lower levels of capital and income per worker in the long run.

International Evidence on Population Growth and Income per Person

Chapter Summary 1.The Solow growth model shows that, in the long run, a country’s standard of living depends  positively on its saving rate.  negatively on its population growth rate. 2.An increase in the saving rate leads to  higher output in the long run  faster growth temporarily  but not faster steady state growth.