(Solved): 3. Consider the standard Solow growth model with population growing at rate \( n \) and CobbDougla ...

3. Consider the standard Solow growth model with population growing at rate \( n \) and CobbDouglas technology with capital share parameter \( \alpha \). Suppose that the government decides to subsidize investments at rate \( \theta \), i.e. for each unit invested in the economy the government adds \( \theta \) units. You can ignore the issue of how to finance this subsidy. In sum, the economy can be described by the following equations: [Capital law of motion] \( k_{t+1}-k_{t}=(1+\theta) i_{t}-(\delta+n) k_{t} \), [Production function] \( y_{t}=A k_{t}^{\alpha} \), [Saving rule \( ] \quad s_{t}=s y_{t} \), where \( \delta \) is the depreciation rate, \( A \) is the technology parameter, and \( s \) denotes the saving rate. \( k_{t}, i_{t} \), and \( y_{t} \) are all per capita terms. a. Calculate the steady-state capital per worker \( k^{*} \). b. An interesting question in the Solow model is: what is the level of capital that maximizes consumption in steady state, which is often referred to as the golden rule capital stock. Notice that the steady state consumption is given as \[ c^{*}=y^{*}-(\delta+n) k^{*}=A\left(k^{*}\right)^{\alpha}-(\delta+n) k^{*} . \] Using this equation, calculate the golden rule level of capital stock \( k_{g} \). c. Calculate the level of investment subsidy \( (\theta) \) that achieves the golden rule.