Derive The Steady State

Derive The Steady State

Derive the steady-state condition in terms of the equilibrium capital stock per person, k* , starting from the law of motion for total capital, in a Solow model with positive population growth.

Consider a Solow model with population growth but no technology growth, where output is given by Y = F(K, L). Explain what happens to the steady-state and the transition path for income per worker when a war destroys both a large chunk of the population and a large chunk of the capital stock.

Take two countries with different values of all parameters in the Solow model. Contrast and compare the long-run convergence of these two countries with and without technological progress.

Explain, in words, the golden rule condition for capital in the Solow model with and without population growth (no technology growth).

In the Solow model, is there any link between saving and technological progress? Does this make sense?

Numerically, solve for equilibrium in a Solow model with a production function Y = K^{0.3}L^{0.7} and parameters s = 0.2, δ = 0.08, η = 0.02. Should public policy try to encourage more or less saving here? Why?

(Ch.5 Review) Consider some small open economy with fixed factors of production, given as follows: Y = C + I + G + NX; Y = F(K, L); G = G, T = T are all fixed. I = I(r), C = C(r), NX = NX( ), where consumption/savings depend on the interest rate. Why might savings increase or decrease with the interest rate? Show whether or not savings equals investment in this economy. Explain what happens when the government makes regulatory changes to increase household saving.

Paper For Above instruction

The Solow growth model is a foundational framework in economic growth theory, primarily used to analyze the determinants of long-term economic growth and the factors influencing capital accumulation, population growth, and technological progress. A critical aspect of the model involves understanding the steady-state condition, which describes a balanced situation where the capital stock per worker remains constant over time, assuming population growth and other parameters are held steady. This paper will derive the steady-state condition in the context of a Solow model with positive population growth, analyze the effects of shocks such as wars, compare convergence behaviors across countries with different parameters, and explore policy implications, particularly related to savings and technological progress.

Derivation of the Steady-State Condition with Population Growth

The law of motion for total capital stock in the Solow model is given by:

 \(\dot{K} = s Y - (δ + η) K \)

where \(s\) is the savings rate, \(\delta\) is the depreciation rate, and \(\eta\) is the population growth rate. Population growth affects the per-worker capital since, as the population increases, the existing capital must be spread over a larger workforce, diluting capital per worker. To analyze this, it is standard to express the model in per-worker terms, defining \(k = \frac{K}{L}\), the capital stock per worker.

Rearranging the law of motion in per-worker terms yields:

 \(\dot{k} = s f(k) - (\delta + \eta) k \)

where \(f(k) = \frac{F(K, L)}{L}\) is the per-worker production function. The steady state occurs where \(\dot{k} = 0\), leading to the steady-state condition:

 \( s f(k^) = (\delta + \eta) k^ \)

This equation defines the equilibrium capital per worker, \(k^*\), in terms of the model parameters and the per-worker production function. It illustrates that in steady state, savings per worker exactly offset the depreciation and dilution of capital caused by population growth.

Effects of War on Steady State and Transition Path

In a model with no technological progress, a war that destroys both a significant portion of the population and capital stock will initially cause a sharp decline in income per worker. The economy will then follow a transition path towards a new steady state. The new steady state will generally involve a lower level of capital per worker and income per worker, given the reduced capital and population base. Over time, by saving and capital accumulation, the economy will gradually rebuild to the lower steady state, assuming no change in other parameters.

The transition dynamics depend on the savings rate and the destruction magnitude. The economy's path involves a period of growth as it adjusts to the new, lower steady state, during which investment exceeds depreciation and population growth. This adjustment underscores the resilience of the economy but also highlights the importance of policies to facilitate recovery post-shock.

Convergence of Countries with Different Parameters

In the Solow model, countries with different initial conditions, savings rates, population growth rates, or productivity parameters will tend to converge towards their respective steady states over time. Without technological progress, the convergence is conditional—dependent on structural similarities. When technological progress is included, the convergence path involves growth at the rate of technological advancement, and differences in steady states are primarily in levels, not growth rates.

Countries with higher savings rates or lower population growth tend to attain higher steady-state income levels, but the convergence speed varies depending on the parameters. If technological progress exists, the long-run growth rates of per capita income are similar across countries and driven by technological change, making convergence in levels more prominent than in rates.

The Golden Rule Condition

The golden rule level of capital accumulation maximizes steady-state consumption. In the absence of population growth and technological progress, this condition is achieved when the marginal product of capital equals the depreciation rate, i.e., \(f'(k_{GR}) = δ\). When population growth is present, the golden rule adjusts to account for the dilution effect, leading to the condition \(f'(k_{GR}) = δ + η\). This ensures that saving and investment are directed such that consumption is maximized in the steady state.

Saving and Technological Progress

In the basic Solow model without technological progress, saving facilitates capital accumulation and growth until the steady state. When technological progress is included, saving can indirectly influence the growth rate by accumulating capital that fosters innovation and productivity. However, in growth models with technological progress, the growth rate of output per worker is primarily driven by the rate of technological change, which is considered exogenous. Thus, while saving is crucial for accumulating capital, it does not directly influence the growth rate determined by technological progress, making the link somewhat limited but still significant in fostering higher levels of income.

Equilibrium in the Model with Given Parameters

Given the production function \(Y= K^{0.3}L^{0.7}\) and parameters \(s=0.2\), \(\delta=0.08\), and \(\eta=0.02\), the steady-state per-worker capital \(k^*\) can be derived by solving:

 \( s f(k) = (\delta + \eta) k \)

which translates to:

 \( 0.2 \times k^{0.3} = (0.08 + 0.02) \times k \)

Leading to:

 \( 0.2 \times k^{0.3} = 0.10 \times k \)

Dividing both sides by \(k^{0.3}\) (assuming \(k > 0\)):

 \( 0.2 = 0.10 \times k^{0.7} \)

So:

 \( k^{0.7} = 2 \Rightarrow k^* = 2^{1/0.7} \approx 2^{1.4286} \approx 2.7 \)

This value indicates the steady state capital per worker. To determine whether policy should encourage saving, note that higher saving raises the steady-state capital and income, but at the cost of reduced consumption in the transition. Since higher savings lead to a higher steady state, policy focus depends on long-term growth and welfare considerations. If the goal is sustainable growth and higher consumption levels, encouraging saving may be beneficial, provided it does not suppress current consumption excessively.

Savings, Interest Rates, and External Shocks in Open Economies

In a small open economy, the relationship between savings and interest rates is governed by the interaction of domestic and international capital markets. An increase in the interest rate tends to attract foreign capital inflows, increasing domestic savings, and vice versa. Conversely, higher interest rates can also reduce domestic consumption, leading to increased savings. When government policies aim to augment household savings through tax incentives or mandatory contributions, these measures can shift the overall savings rate, influencing investment and current account balances.

In the model where savings and investment depend on the interest rate, equilibrium occurs where savings equal investment. Policy interventions that encourage household saving, such as tax incentives or stricter economic regulations, tend to increase the savings rate, thereby promoting higher investment and potentially economic growth. However, such policies should balance short-term consumption reductions with long-term benefits to ensure overall welfare.

This interconnectedness underscores the importance of carefully calibrated policies that consider the effects on consumption, investment, and external balances.

Conclusion

The Solow model remains a powerful tool for understanding economic growth dynamics, especially how savings, population growth, and technological progress interact. Deriving the steady-state condition with population growth underscores the importance of savings and investment in maintaining long-term income levels. External shocks like wars have significant impacts, but economies exhibit resilience through capital accumulation pathways. Differences among countries can be explained by structural parameters, with convergence depending on these factors. Policy implications, particularly concerning savings and technological progress, highlight the delicate balance necessary for fostering sustainable growth and maximizing welfare.

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