Chapter 11: Saving, Capital Accumulation, and Output More detailed analysis of the last chapter

Saving rates in USA

Saving rates in EU vs. USA

Important question at this point is: How does saving and output correlate? Does higher saving lead to higher output?  Use a model to understand how they correlate

Effects of Capital on Output 𝑌 𝑁 =𝐹 𝐾 𝑁 , 𝑁 𝑁 = 𝐹 𝐾 𝑁 ,1 𝑓 𝐾 𝑁 = 𝐹 𝐾 𝑁 ,1 Assume that the following are constant to concentrate on the role of capital alone: Population Participation Rate Unemployment Rate Technology (for now)

Effects of Capital on Output Adding time subscripts and imposing our assumptions: 𝑌 𝑡 𝑁 =𝑓( 𝐾 𝑡 𝑁 ) Output per worker is a function of capital per worker Higher capital per worker leads to higher output per worker

Effects of output on capital accumulation We know from chapter 3 that the following holds: I = S + ( T – G ) What if there is no government sector? T = G = O (zero) S = I S = s*Y where s = savings rate  It = s*Yt The higher the savings rate, the higher Investment

Investment and capital accumulation What is the relation between investment and capital accumulation? Kt+1 = (1-δ) Kt + It Write it in per worker terms: 𝐾 𝑡+1 𝑁 = 1−𝛿 𝐾 𝑡 𝑁 + 𝑠 𝑌 𝑡 𝑁 Rewrite it a little to have a more convenient form: 𝐾 𝑡+1 𝑁 − 𝐾 𝑡 𝑁 =𝑠 𝑌 𝑡 𝑁 −𝛿 𝐾 𝑡 𝑁 In words: The change in capital stock per worker, is equal to savings per worker and depreciation of capital per worker.

Investment and capital accumulation 𝐾 𝑡+1 𝑁 − 𝐾 𝑡 𝑁 =𝑠 𝑌 𝑡 𝑁 −𝛿 𝐾 𝑡 𝑁 What can we say about the relationship between the savings rate, depreciation and growth of capital per worker? As long as s*Yt > δKt , capital per worker grows When s*Yt < δKt , capital decreases! How is this important for output? Since 𝑌 𝑡 𝑁 =𝑓( 𝐾 𝑡 𝑁 )  A higher savings rate implies a higher level of output!

Investment and capital accumulation Can you guess what the savings rate would be approximately for the graph above?

Investment and capital accumulation Proportion of the French Capital Stock Destroyed by the End of World War II Looking back at our graph, what happened to K/N and to Y/N?

Steady-state capital and output Suppose that we look at the very long run (e.g. 1000 years): According to the convergence hypothesis, growth of variables will be equal to zero Drop time subscripts 𝐾 ∗ 𝑁 − 𝐾 ∗ 𝑁 =𝑠 𝑌 𝑁 −𝛿 𝐾 ∗ 𝑁 𝑠𝑓 𝐾 ∗ 𝑁 =𝛿 𝐾 ∗ 𝑁 Steady state value of capital per worker: savings rate is just sufficient to cover depreciation of capital per worker

What about different savings rates for the following equation? 𝐾 𝑡+1 𝑁 − 𝐾 𝑡 𝑁 =𝑠 𝑌 𝑡 𝑁 −𝛿 𝐾 𝑡 𝑁

Steady-state capital and output What about different savings rates for this 𝑠𝑓 𝐾 ∗ 𝑁 =𝛿 𝐾 ∗ 𝑁 Steady state value of capital per worker: savings rate is just sufficient to cover depreciation of capital per worker

𝑌 ∗ 𝑁 = 𝑓 𝐾 ∗ 𝑁 Steady-state capital and output We know from the beginning of the chapter that: 𝑌 ∗ 𝑁 = 𝑓 𝐾 ∗ 𝑁 Does anyone have a guess on why there is a gradual increase of output?

Steady-state capital and output What if there is technological progress? The Effects of an Increase in the Saving Rate on Output per Worker in an Economy with Technological Progress

Steady-state capital and output Question: What is the “optimal” savings rate? Consider two extremes: There are no (and never have been savings). So, according to this: 𝑠𝑓 𝐾 ∗ 𝑁 =𝛿 𝐾 ∗ 𝑁  Output = 0 2. All income generated is saved. Consumption = 0 So, what number generates maximum consumption? A “golden” number between 0 and 1!

Steady-state capital and output Figure 11-6 The Effects of the Saving Rate on Steady-State Consumption per Worker

Small detour: Social Security Social security: Pension to retired workers. - Cost: As of 2013, about 8% of GDP ≈ $1.2 trillion - Keeps 20% of all Americans, age 65 or older, above the Federally defined poverty level Two ways to fund: Fully funded system: Workers pay today for future benefits, these funds will be invested and paid back as “pensions”. Pay-as-you-go system: Workers pay today for current retirees’ pensions and next generation will pay for today’s workers’ pensions.  Current system

Small detour: Social Security Fully funded system: Workers pay today for future benefits, these funds will be invested and paid back as “pensions”. Private savings↓ Why? Ask yourself: if you knew that the state will pay you a pension once you retire, will you save less or more today? (On average, people save less, hence, private savings go down) What about a switch to this system? Current workers would have to for themselves and for the current retirees.  Double cost on current workers which is (especially now) not easy to implement

Small detour: Social Security 2. Pay-as-you-go system: Workers pay today for current retirees’ pensions and next generation will pay for today’s workers’ pensions. Private savings↓ Why? Same argument as last slide. But more problematic: Population ages quickly but population does not increase to catch up Either future retirees have to take less pensions (ask yourself, would you like to have that?) Or current workers have to pay more today (ask yourself again, would you like to pay more today?) Problem of social security……..

Figure 11-7B The Dynamic Effects of an Increase in the Saving Rate from 10% to 20% on the Level and the Growth Rate of Output per Worker (cont.) Assignment: Replicate these two figures using Excel. Due date is next Friday  Gain five points on your highest midterm!

Steady-state consumption and savings But can we get this relationship between savings and consumption in a formal way? Consider this: We learned that in the steady-state the following holds: 𝑠𝑓 𝐾 ∗ 𝑁 =𝛿 𝐾 ∗ 𝑁 There is no consumption! Any idea on how to get consumption?

Steady-state consumption and savings Remember that (without a government): Y = C + S (all income is either consumed or saved) And 𝑌 𝑁 – 𝐶 𝑁 = 𝑆 𝑁 = 𝑠𝑓 𝐾 ∗ 𝑁 Hence, 𝑌 𝑁 – 𝐶 𝑁 = 𝛿 𝐾 ∗ 𝑁 So, 𝐶 𝑁 = 𝑌 𝑁 − 𝛿 𝐾 ∗ 𝑁

Steady-state consumption and savings Remember that (given production function Y=(K*N)^0.5: Y N = K 0.5 N 0.5 = s 2∗0.5 δ 2∗0.5 = s δ C N = Y N − δ K ∗ N C N = s δ − δ s 2 δ 2 = s(1−s) δ Another way of writing consumption: C N = Y N - s 𝑌 N = (1-s)* Y N = (1-s)* s δ = s(1−s) δ Question: Which savings rate maximizes consumption per worker? same

Steady-state consumption and savings Max consumption at s = 0.5; after that, consumption goes down!

Understanding the bigger picture You might be lost amid all the equations in regards to the “big picture”. What we should get out of this chapter is: We now (hopefully) understand that: Savings lead to capital accumulation Capital accumulation leads to higher output Higher output leads (ceteris paribus) to higher standard of living

Understanding the bigger picture Sources of Mechanical Drive in American Manufacturing Thousand Horsepower Year Steam Engines Steam Turbines Internal Combustion Engines Water Wheels and Turbines Electric Motors Total  Horsepower per Manufacturing Worker 1869 1216 1130 2346  1.04 1879 2186 1225 3411  1.10 1889 4581 9 1242 16 5848  1.25 1899 8022 120 1236 475 9853  1.57 1909 12026 90 592 1273 4582 18563  2.25 1919 11491 465 856 970 15612 29394  2.68 1929 6857 1112 722 623 33844 43158  3.90 1939 4216 1736 866 394 44827 52039  4.35 1948 86095  6.60 1953 105007  7.18 Source: Brad DeLong “Slouching Towards Utopia”