Consider the Solow growth model with government spending but without technological progress. Output $$ Y $$ is produced according to the production function $$ Y=F(K, N)=\sqrt{K N} $$ where $$ K $$ is the capital stock that depreciates at the rate $$ \delta $$. Employment $$ N $$ is equal to one half of the total population $$ L $$ and grows at the rate $$ n $$. The government levies a lump-sum tax $$ \tau0 $$ on each worker and collects $$ T \equiv \tau N $$ of the total fiscal revenues. Capital accumulation is financed solely from private saving, since the government spends all the tax revenues on public consumption and balances its budget, so $$ T=G $$. Private investment $$ l $$ and private consumption $$ C $$ absorb, correspondingly, constant fractions $$ s $$ and ( $$ 1-s $$ ) of disposable income. (a) [10 marks] What is the share of capital income in total income? (b) [10 marks] Derive analytically the balanced growth path conditions for capital, $$ K $$, and output per capita, $$ y $$. (c) [10 marks] Find the closed form algebraic solutions for all steady-state levels of capital per worker, $$ k $$. (d) [10 marks] Illustrate the steady-states graphically using the Solow diagram.

Question

Consider the Solow growth model with government spending but without technological progress. Output $$ Y $$ is produced according to the production function $$ Y=F(K, N)=\sqrt{K N} $$ where $$ K $$ is the capital stock that depreciates at the rate $$ \delta $$. Employment $$ N $$ is equal to one half of the total population $$ L $$ and grows at the rate $$ n $$. The government levies a lump-sum tax $$ \tau>0 $$ on each worker and collects $$ T \equiv \tau N $$ of the total fiscal revenues. Capital accumulation is financed solely from private saving, since the government spends all the tax revenues on public consumption and balances its budget, so $$ T=G $$. Private investment $$ l $$ and private consumption $$ C $$ absorb, correspondingly, constant fractions $$ s $$ and ( $$ 1-s $$ ) of disposable income. (a) [10 marks] What is the share of capital income in total income? (b) [10 marks] Derive analytically the balanced growth path conditions for capital, $$ K $$, and output per capita, $$ y $$. (c) [10 marks] Find the closed form algebraic solutions for all steady-state levels of capital per worker, $$ k $$. (d) [10 marks] Illustrate the steady-states graphically using the Solow diagram.

Mathis Li · Accepted Answer

**Share of Capital Income:** 50% of total income comes from capital. **Explanation:** In the production function $$ Y = \sqrt{K N} $$, the marginal product of capital is $$ \frac{1}{2} \frac{\sqrt{N}}{\sqrt{K}} $$. Multiplying this by the capital stock $$ K $$ gives capital income $$ rK = \frac{1}{2} Y $$. Since total income $$ Y $$ is split equally between capital and labor, the capital share is 50%.

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