Savings and Economic Growth
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1 Savings and Economic Growth
2 Savings and Economic Growth Question: How does the savings rate affect the long-run average growth rate of a country?
3 Savings and Economic Growth Question: How does the savings rate affect the long-run average growth rate of a country? We will answer this question using a very simple aggregate (or economywide) model of economic growth.
4 Savings and Economic Growth Question: How does the savings rate affect the long-run average growth rate of a country? We will answer this question using a very simple aggregate (or economywide) model of economic growth. The model we will study is called the Solow model (after the Nobel Prize-winning economist Robert Solow at M.I.T.).
5 The Aggregate Production Function Revisited
6 The Aggregate Production Function Revisited Recall the aggregate production function:
7 The Aggregate Production Function Revisited Recall the aggregate production function: Y = AK α L 1 α.
8 The Aggregate Production Function Revisited Recall the aggregate production function: Y = AK α L 1 α. For now, fix A =1andL = 1, so that neither the level of technology (A) nor the aggregate amount of labor supply(l) is changing.
9 The Aggregate Production Function Revisited Recall the aggregate production function: Y = AK α L 1 α. For now, fix A =1andL = 1, so that neither the level of technology (A) nor the aggregate amount of labor supply(l) is changing. (Later, we will allow A to grow over time.)
10 The Aggregate Production Function Revisited Recall the aggregate production function: Y = AK α L 1 α. For now, fix A =1andL = 1, so that neither the level of technology (A) nor the aggregate amount of labor supply(l) is changing. (Later, we will allow A to grow over time.) Physical capital K, however,will change over time.
11 The Aggregate Production Function Revisited Recall the aggregate production function: Y = AK α L 1 α. For now, fix A =1andL = 1, so that neither the level of technology (A) northeaggregateamountoflaborsupply(l) is changing. (Later, we will allow A to grow over time.) Physical capital K, however,will change over time. Let s study the shape of the aggregate production function (again, holding technology and employment constant).
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14 Diminishing Marginal Product of Capital
15 Diminishing Marginal Product of Capital The slope of the production function is called the marginal product of capital.
16 Diminishing Marginal Product of Capital The slope of the production function is called the marginal product of capital. The marginal product of capital is the amount by which output increases when capital increases by a (very) small amount.
17 Diminishing Marginal Product of Capital The slope of the production function is called the marginal product of capital. The marginal product of capital is the amount by which output increases when capital increases by a (very) small amount. The declining marginal product of capital suggests that it will be difficult to generate sustained growth simply by increasing capital over time.
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19 The Solow Growth Model
20 The Solow Growth Model At the beginning of every year t, the economy has a stock of (physical) capital K t.
21 The Solow Growth Model At the beginning of every year t, the economy has a stock of (physical) capital K t. In year t, the economy produces output (or GDP) Y t according to the aggregate production function:
22 The Solow Growth Model At the beginning of every year t, the economy has a stock of (physical) capital K t. In year t, the economy produces output (or GDP) Y t according to the aggregate production function: Y t = K α t.
23 The Solow Growth Model At the beginning of every year t, the economy has a stock of (physical) capital K t. In year t, the economy produces output (or GDP) Y t according to the aggregate production function: Y t = K α t. Some of this output is consumed today and the rest is invested (here, investment means the formation of physical capital).
24 The Solow Growth Model At the beginning of every year t, the economy has a stock of (physical) capital K t. In year t, the economy produces output (or GDP) Y t according to the aggregate production function: Y t = K α t. Some of this output is consumed today and the rest is invested (here, investment means the formation of physical capital). To keep things simple, suppose that the entire current stock of capital is depleted (or used up) through depreciation during the course of production.
25 The Solow Growth Model At the beginning of every year t, the economy has a stock of (physical) capital K t. In year t, the economy produces output (or GDP) Y t according to the aggregate production function: Y t = K α t. Some of this output is consumed today and the rest is invested (here, investment means the formation of physical capital). To keep things simple, suppose that the entire current stock of capital is depleted (or used up) through depreciation during the course of production. In other words, if the economy does not invest today, there will be no capital with which to produce tomorrow.
26 The Key Equations of the Solow Model
27 The Key Equations of the Solow Model Let C t be (aggregate) consumption in year t.
28 The Key Equations of the Solow Model Let C t be (aggregate) consumption in year t. Let I t be (aggregate) investment in year t.
29 The Key Equations of the Solow Model Let C t be (aggregate) consumption in year t. Let I t be (aggregate) investment in year t. All output in year t is either consumed or invested:
30 The Key Equations of the Solow Model Let C t be (aggregate) consumption in year t. Let I t be (aggregate) investment in year t. All output in year t is either consumed or invested: Y t = C t + I t.
31 The Key Equations of the Solow Model Let C t be (aggregate) consumption in year t. Let I t be (aggregate) investment in year t. All output in year t is either consumed or invested: Y t = C t + I t. The usual national income accounting identity is
32 The Key Equations of the Solow Model Let C t be (aggregate) consumption in year t. Let I t be (aggregate) investment in year t. All output in year t is either consumed or invested: Y t = C t + I t. The usual national income accounting identity is Y t = C t + I t + G t + NX t,
33 The Key Equations of the Solow Model Let C t be (aggregate) consumption in year t. Let I t be (aggregate) investment in year t. All output in year t is either consumed or invested: Y t = C t + I t. The usual national income accounting identity is Y t = C t + I t + G t + NX t, where G t is government spending in year t and NX t is net exports in year t.
34 The Key Equations of the Solow Model Let C t be (aggregate) consumption in year t. Let I t be (aggregate) investment in year t. All output in year t is either consumed or invested: Y t = C t + I t. The usual national income accounting identity is Y t = C t + I t + G t + NX t, where G t is government spending in year t and NX t is net exports in year t. But in this very simple model, we are ignoring government spending and we are imagining that the economy is closed (so that it does not trade with the rest of the world).
35 The Savings Decision
36 The Savings Decision Key decision facing any economy: how to split today s output between today (consumption) and tomorrow (savings, or investment).
37 The Savings Decision Key decision facing any economy: how to split today s output between today (consumption) and tomorrow (savings, or investment). Let s assume that the economy has a constant savings rate:
38 The Savings Decision Key decision facing any economy: how to split today s output between today (consumption) and tomorrow (savings, or investment). Let s assume that the economy has a constant savings rate: S t = sy t,
39 The Savings Decision Key decision facing any economy: how to split today s output between today (consumption) and tomorrow (savings, or investment). Let s assume that the economy has a constant savings rate: S t = sy t, where the savings rate s is a number between 0 and 1.
40 The Savings Decision Key decision facing any economy: how to split today s output between today (consumption) and tomorrow (savings, or investment). Let s assume that the economy has a constant savings rate: S t = sy t, where the savings rate s is a number between 0 and 1. In a closed economy, S t = I t,soi t = sy t.
41 The Savings Decision Key decision facing any economy: how to split today s output between today (consumption) and tomorrow (savings, or investment). Let s assume that the economy has a constant savings rate: S t = sy t, where the savings rate s is a number between 0 and 1. In a closed economy, S t = I t,soi t = sy t. Because capital depreciates completely during production, investment (I t ) is the only source of capital goods in the future:
42 The Savings Decision Key decision facing any economy: how to split today s output between today (consumption) and tomorrow (savings, or investment). Let s assume that the economy has a constant savings rate: S t = sy t, where the savings rate s is a number between 0 and 1. In a closed economy, S t = I t,soi t = sy t. Because capital depreciates completely during production, investment (I t ) is the only source of capital goods in the future: K t+1 = I t.
43 The Savings Decision Key decision facing any economy: how to split today s output between today (consumption) and tomorrow (savings, or investment). Let s assume that the economy has a constant savings rate: S t = sy t, where the savings rate s is a number between 0 and 1. In a closed economy, S t = I t,soi t = sy t. Because capital depreciates completely during production, investment (I t ) is the only source of capital goods in the future: K t+1 = I t. (Note: We are assuming that it takes one year to build and install new capital goods.)
44 The Economy s Law of Motion
45 The Economy s Law of Motion In a typical year t:
46 The Economy s Law of Motion In a typical year t: Y t = K α t (production)
47 The Economy s Law of Motion In a typical year t: Y t = Kt α (production) S t = sy t (savings)
48 The Economy s Law of Motion In a typical year t: Y t = Kt α (production) S t = sy t (savings) I t = S t (investment)
49 The Economy s Law of Motion In a typical year t: Y t = Kt α (production) S t = sy t (savings) I t = S t (investment) K t+1 = I t (new capital goods)
50 The Economy s Law of Motion In a typical year t: Y t = Kt α (production) S t = sy t (savings) I t = S t (investment) K t+1 = I t (new capital goods) Putting it all together: K t+1 = skt α.
51 The Economy s Law of Motion In a typical year t: Y t = Kt α (production) S t = sy t (savings) I t = S t (investment) K t+1 = I t (new capital goods) Putting it all together: K t+1 = skt α. This is the law of motion for the economy s capital stock.
52 An Alternative Expression for the Law of Motion
53 An Alternative Expression for the Law of Motion Subtract K t from both sides to get:
54 An Alternative Expression for the Law of Motion Subtract K t from both sides to get: K t+1 K t = sk α t K t.
55 An Alternative Expression for the Law of Motion Subtract K t from both sides to get: K t+1 K t = sk α t K t. ΔK t+1 K t+1 K t is the change in the capital stock from year t to year t +1.
56 An Alternative Expression for the Law of Motion Subtract K t from both sides to get: K t+1 K t = sk α t K t. ΔK t+1 K t+1 K t is the change in the capital stock from year t to year t +1. ΔK t+1 is positive if sk α t > K t.
57 An Alternative Expression for the Law of Motion Subtract K t from both sides to get: K t+1 K t = sk α t K t. ΔK t+1 K t+1 K t is the change in the capital stock from year t to year t +1. ΔK t+1 is positive if sk α t > K t. ΔK t+1 is negative if sk α t < K t.
58 An Alternative Expression for the Law of Motion Subtract K t from both sides to get: K t+1 K t = sk α t K t. ΔK t+1 K t+1 K t is the change in the capital stock from year t to year t +1. ΔK t+1 is positive if sk α t > K t. ΔK t+1 is negative if sk α t < K t. ΔK t+1 is zero if sk α t = K t.
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61 Solving for the Steady State
62 Solving for the Steady State K = sk α K = s 1/(1 α).
63 Solving for the Steady State K = sk α K = s 1/(1 α). The steady-state value of the capital stock depends on the savings rate s and the exponent α in the production function.
64 Solving for the Steady State K = sk α K = s 1/(1 α). The steady-state value of the capital stock depends on the savings rate s and the exponent α in the production function. The higher is the savings rate, the higher is the steady-state capital stock.
65 Solving for the Steady State K = sk α K = s 1/(1 α). The steady-state value of the capital stock depends on the savings rate s and the exponent α in the production function. The higher is the savings rate, the higher is the steady-state capital stock. Steady-state output (GDP) is: Y = K α.
66 Solving for the Steady State K = sk α K = s 1/(1 α). The steady-state value of the capital stock depends on the savings rate s and the exponent α in the production function. The higher is the savings rate, the higher is the steady-state capital stock. Steady-state output (GDP) is: Y = K α. Steady-state consumption is: C =(1 s)y.
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68 Dynamics
69 Dynamics Question: If the economy doesn t start at the the steady-state capital stock K, does it ever get there?
70 Dynamics Question: If the economy doesn t start at the the steady-state capital stock K, does it ever get there? Short answer: The economy always converges to K (as long as the initial capital stock is positive).
71 Dynamics Question: If the economy doesn t start at the the steady-state capital stock K, does it ever get there? Short answer: The economy always converges to K (as long as the initial capital stock is positive). However, it takes an infinite amount of time to get to the steady state.
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74 An Important Theoretical Discovery
75 An Important Theoretical Discovery Growth in the long run is ZERO!
76 An Important Theoretical Discovery Growth in the long run is ZERO! The savings rate does NOT affect growth in the long run (that is, after the economy converges to its steady state).
77 An Important Theoretical Discovery Growth in the long run is ZERO! The savings rate does NOT affect growth in the long run (that is, after the economy converges to its steady state). Increases in the savings rate DO affectgrowthintheshortrun but NOT in the long run.
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80 Another Important Theoretical Discovery
81 Another Important Theoretical Discovery Sustained increases in technology lead to sustained increases in output, consumption, and the capital stock.
82 Another Important Theoretical Discovery Sustained increases in technology lead to sustained increases in output, consumption, and the capital stock. Improvements in technology overcome the problem of diminishing returns to capital.
83 Another Important Theoretical Discovery Sustained increases in technology lead to sustained increases in output, consumption, and the capital stock. Improvements in technology overcome the problem of diminishing returns to capital. This is what Sergey Brin means by building ladders to larger, higher-hanging fruit.
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