3 Task 3. Model with endogenous population and technologies Consider a model with endogenous technology and population . Population growth is an increasing function of the difference between current income per capita, y, and the minimum level of income bar (y) required for subsistence: n=beta (y-bar (y)), where beta is a positive constant . Total output is a function of labor (L) and land (R) Y=(AR)^alpha L^1-alpha Assume that available land does not grow over time. Growth of technology depends on output per capita: dot (A)/A=gamma y where 0lt gamma lt beta . 1. Analyze the dynamics and steady state(s) of the model. Solve for any steady state values of output per capita y^ast . Draw a grath of the growth rate equation with dot (y)/y on the vertical axes and y on the horisontal axes 2. Suppose that y is rising to a new level. What is the change in the steady state level of output per capita? population growth? Illustrate your answer on a diagram 3. Suppose that R is rising to a new level.. What is the change in the steady state level of output per capita? population growth? Illustrate your answer on a diagram.

1. To analyze the dynamics and steady state(s) of the model, we need to first derive the growth rate equation for output per capita, y. The growth rate of output per capita is given by:<br /><br />$\frac{\dot{y}}{y} = \frac{d}{dt}(\frac{Y}{R}) = \frac{d}{dt}((AR)^{\alpha}L^{1-\alpha}) = \alpha \frac{d}{dt}(AR) + (1-\alpha) \frac{d}{dt}(L)$<br /><br />Since land (R) does not grow over time, the growth rate of output per capita simplifies to:<br /><br />$\frac{\dot{y}}{y} = \alpha \dot{A} + (1-\alpha) \frac{\dot{L}}{L}$<br /><br />Given that the growth of technology depends on output per capita, we have:<br /><br />$\dot{A}/A = \gamma y$<br /><br />Substituting this into the growth rate equation, we get:<br /><br />$\frac{\dot{y}}{y} = \alpha \gamma y + (1-\alpha) \frac{\dot{L}}{L}$<br /><br />To find the steady state(s), we set $\frac{\dot{y}}{y} = 0$ and solve for y:<br /><br />$0 = \alpha \gamma y + (1-\alpha) \frac{\dot{L}}{L}$<br /><br />Assuming that the population growth rate $\frac{\dot{L}}{L}$ is constant, we can solve for the steady state level of output per capita $y^*$:<br /><br />$y^* = \frac{(1-\alpha) \frac{\dot{L}}{L}}{\alpha \gamma}$<br /><br />2. Suppose that y is rising to a new level. This means that the output per capita is increasing. In this case, the growth rate of output per capita $\frac{\dot{y}}{y}$ will be positive. As a result, the population growth rate $\frac{\dot{L}}{L}$ will also increase to maintain the steady state level of output per capita. This can be illustrated on a diagram where the growth rate equation $\frac{\dot{y}}{y} = \alpha \gamma y + (1-\alpha) \frac{\dot{L}}{L}$ is plotted with $\frac{\dot{y}}{y}$ on the vertical axis and y on the horizontal axis. As y increases, the growth rate $\frac{\dot{y}}{y}$ will shift upwards, indicating an increase in population growth rate $\frac{\dot{L}}{L}$.<br /><br />3. Suppose that R is rising to a new level. This means that the available land is increasing. In this case, the growth rate of output per capita $\frac{\dot{y}}{y}$ will decrease as the denominator in the equation $\frac{\dot{y}}{y} = \alpha \gamma y + (1-\alpha) \frac{\dot{L}}{L}$ increases. As a result, the population growth rate $\frac{\dot{L}}{L}$ will decrease to maintain the steady state level of output per capita. This can be illustrated on a diagram where the growth rate equation $\frac{\dot{y}}{y} = \alpha \gamma y + (1-\alpha) \frac{\dot{L}}{L}$ is plotted with $\frac{\dot{y}}{y}$ on the vertical axis and y on the horizontal axis. As R increases, the growth rate $\frac{\dot{y}}{y}$ will shift downwards, indicating a decrease in population growth rate $\frac{\dot{L}}{L}$.