Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Chapter 3 PHYSICAL CAPITAL

The five features of capital 1. Capital is productive 2. Capital is itself produced through the process of savings and investment Not a natural resource Produced privately or publicly: buildings or computers vs. Messina bridge 3. Capital is rival in its use Only limited number of people can use one piece of capital at a time Are ideas a form of capital? 4. If capital privately produced, rate of return (ror) motivates its production; if publicly produced, other criteria matter. But “social” ror important as well 5. Capital wears out Capital gradually depreciates. Depreciation for buildings is slower than for computers Depreciation may be physical, economic, fiscal – different types of depreciation need not coincide 3-2

Capital’s role in production Given that capital is productive, it enables workers to produce more output Study relation between K and output (GDP) through production function Focus on capital accumulation is crucial in the Solow model 3-3 Robert Solow Nobel Prize, 1978

The main properties of the production function Y=F(K,L) Constant returns to scale –F(zK,zL) = z F(K,L) –If z=1/L: (1/L)Y = (1/L) F(K,L) = F(K/L,L/L)= F(K/L,1) Y/L = F(K/L) or y=F(k,1) or, ignoring 1, y=f(k) –So we convert output into output per worker and obtain a function that relates output per worker with capital per worker only (L disappears) Positive but diminishing returns on each factor K, L Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 3-4

3-5 Figure 3.1 GDP per worker and Capital per Worker, 2005

The main properties of the production function Second property of the production function Positive but diminishing returns on each factor K, L Or Diminishing marginal product of capital –Marginal product of capital = extra output from using one additional unit of K, holding L fixed –MPK = f(k+1) – f(k) –Aka (also known as) the partial derivative of y with respect to k –∂F/∂K = f’(k) 3-6

3-7 Figure 3.2 A Production Function with Diminishing Marginal Product of Capital

Decreasing Marginal Product of Capital

Example: the Cobb Douglas production function Y=AK α L 1-α –A, a measure of productivity (Why?) –α says how exactly K and L combine to produce output –Empirically, α not too far from 1/3 (see box in next slide) In per worker terms: y = Ak α –where y = Y/L and k = K/L 3-9

3-10 Capital’s Share of Income in a Cross- Section of Countries: average not too far from 1/3

The Solow model Constant population (hence L is constant) Capital accumulation ΔK = I – D –I investment, D depreciation In per worker terms: Δk = i – d Investment is constant fraction of output –i = γy = γf(k) Depreciation constant fraction of capital –D = δk 3-11

3-12 Combining the three equations here is the equation of capital accumulation

3-13 Figure 3.4 The Steady State of the Solow Model

3-14 Figure 3.6 Effect of Increasing the Investment Rate on the Steady State

3-15 The Solow model in a Cobb-Douglas world – capital accumulation first

The steady state level of capital per worker If Δk = 0 (investment net of depreciation = 0) k ss = (γA/δ) 1/(1-α) The steady state capital stock is higher The higher the propensity to save and invest The lower the depreciation rate The higher the efficiency of the economy The same holds for the level of output (Gdp) per worker 3-16

3-17 The Solow model in a Cobb-Douglas world – output per worker in the steady state equilibrium second

The Solow model as a theory of income differences Using the Solow model, we have obtained the steady state equilibrium levels of Gdp per worker Hence we have a theory to explain income differences across countries This was our purpose to start with! Question: is the Solow model a good theory of income differences? Figure 3.7 contrasts the predicted values of cross-country income gaps with respect to the US with the actual values of these gaps 3-18

Question: what would be the picture like if the Solow model were an exact predictor of the data we see in the real world? 3-19 Predicted Versus Actual GDP per Worker Actual Gdp: real data Predicted Gdp: level of Gdp obtained using the equation from the Solow model and the actual data on γ, K and A

The Solow model as a theory of relative growth rates Can the Solow model explain cross-country differences in growth rates as well? First of all, so far we have a model of income differences in steay states where by definition growth is zero. So as such the model cannot say anything about growth BUT The model may still have something to say as to relative growth rates, hence why some countries grow faster than others This is the case when we think of countries AWAY from their steady states. In steady state growth is zero, but away from the steady state some transitional growth will occur (exactly to take that country to the steady state!). Hence the Solow model predicts that countries converge - by growing fast or slow - to their steady state 3-20

Transitional growth in the Solow model (Cobb-Douglas version) Dividing by k we obtain the growth rate of k: K hat =Δk/k = γAk α-1 – δ This equation can be nicely graphed as follows: 3-21

3-22 Growth in the Solow model This picture is very useful to learn immediately about the growth rate of k (k with the hat in the picture) and the speed of convergence to the steady state (diminishing as the steady state equilibrium is approached)

Three predictions on transitional growth rates If two countries have the same rate of investment (as well as depreciation and productivity) but different levels of income, the country with lower income will experience faster growth –Same investment rate means same steady state level of income. So the initially rich country will grow less fast than the initially poor country 3-23

Three predictions on transitional growth rates If two countries have the same level of income (as well as depreciation and productivity) but different rates of investment, the country with a higher rate of investment will experience faster growth –Higher investment means higher steady state level of output, and thus faster growth along the way 3-24

Three predictions on transitional growth rates A country that raises its investment rate will experience faster growth –Higher investment raises the steady state level of income. So even if a country starts from a steady state equilibrium, it will start growing to reach the new, higher, steady state equilibrium implied by the higher investment rate 3-25