Chapter 11: Saving, Capital Accumulation, and Output

Chapter 11: Saving, Capital Accumulation, and Output

Introduction to Chapter 11

• The chapter aims to delve into the relationship between saving, capital accumulation, and output in an economy by exploring several significant questions:

  • What determines the level of GDP and its growth rate in the long run?

  • What is the steady state or long-run equilibrium?

  • Does an increase in the saving rate lead to higher growth rates?

Dynamic vs Static Economics

• Dynamic Economics: Refers to the study of how economic variables evolve over time.

  • It focuses on the relationship of variable x over time, examining how it relates to its value at different times.

  • General relations include:

    • Relation of current value to future value: $xt$ to $x{t+1}$

    • Relation of current value to past value: $xt$ to $x{t-1}$

Interactions between Output and Capital

• The determination of output in the long run centers around two critical relations:

  • Relation 1: The amount of capital directly influences the amount of output produced.

  • Relation 2: The amount of output produced influences the amount of saving, and thus the amount of capital accumulated over time.

Capital, Output, and Saving/Investment

• Figure 11-1: Illustrates the interactions between output and capital.

The Effects of Capital on Output

• Under the concept of constant returns to scale, the following relation can be established regarding output ($Y$) and capital per worker ($K/N$):

  • Assumptions for analysis:

  • Population size, participation rate, and unemployment rate remain constant.

  • No technological progress is accounted for in this model.

• Relation: With these assumptions, it can be stated that higher capital per worker ($K/N$) leads to higher output per worker ($Y/N$).

Output and Capital Accumulation

• To derive the relationship between output and capital accumulation, the following two steps are taken:

  1. Establish the relation between output and investment.

  2. Establish the relation between investment and capital accumulation.

Formula for Capital Stock Evolution

• The evolution of capital stock can be expressed in differential form:

  • Rate of Capital Accumulation:

  • Let $d$ denote the rate of depreciation.

  • The change in capital stock per worker can be represented mathematically as:
    rac{dK}{dt} = I - dK

• Rearranging terms leads to the expression:

  • Change in capital per worker:

  • rac{K{t+1} - Kt}{N} = S - d

  • Where $S$ is savings per worker.

Main Relations between Output and Capital

• The two primary relations derived are:

  • First Relation: Capital determines output.

  • Second Relation: Output determines capital accumulation.

Dynamics of Capital and Output

• The change in capital from year $t$ to year $t + 1$ can be expressed in terms of depreciation and investment, leading to the following description:

  • ext{Change in capital} = ext{Investment} - ext{Depreciation}

• Conditions for capital stock change:

  • If investment per worker exceeds depreciation per worker, the change in capital per worker is positive (i.e., capital increases).

  • If investment per worker is less than depreciation per worker, the change in capital per worker is negative (i.e., capital decreases).

Steady State Concept

• Definition of Steady State: The state where output per worker ($Y/N$) and capital per worker ($K/N$) are no longer changing.

• At steady state, the left-hand side of the accumulation equation equals zero, signifying equilibrium.

• Steady-state capital per worker:

  • Given by $K^*/N$; and the steady-state output per worker is obtained using the production function.

The Saving Rate and Output

• Observations regarding the saving rate:

  1. The saving rate has a no effect on the long-run growth rate of output per worker (which equals zero).

  • This is linked to the concept of steady state being synonymous with long-run equilibrium.

  1. The saving rate does determine the level of output per worker in the long run.

  • Countries with higher saving rates tend to achieve higher levels of output per worker.

  1. An increase in the saving rate will lead to temporary higher growth of output per worker, but ultimately will not affect the long-run growth rate.

• Long-run outcome: Once the economy reaches the new steady-state after a higher saving rate, growth will cease.

Conclusion: Implications of Different Saving Rates

• Economies with a higher saving rate will achieve a higher steady-state level of output per worker.

• An increase in saving rate instigates a period of higher growth until the economy reaches its new steady-state.

• Figures illustrating these implications can provide a visual representation of capital and output dynamics in relation to the saving rate.