Question

Suppose we have the Neoclassical Growth model that we have developed in class. Suppose At grows at the rate g. Assume a general period utility function, u(ct). a. Write down the Social Planners problem in terms of per capita consumption and capital stock. Set up the Lagrangian, solve, and give the equations that for every time period are necessary for a solution (Euler equation and resource constraint). b. Using the resource constraint, show why along a BGP per capita consumption and capital stock are growing at the rate of technological progress, g. c. Using the Euler Equation explain what restriction we must make on our period utility function in order for a BGP to exist. Write the Euler equation with this restriction, i.e. so it is independent of time. d. Solve for the initial capital stock per capita, k, and output per capita, y. Write down the expressions for the capital stock per capita at time t,kt, and output per capita at time t,yt. e. Suppose capital fully depreciates each period. What is the savings rate, yi along a BGP? Discuss how the savings rate changes with α and β, explain the intuition. How does the savings rate change with g ? Explain why the effect of g on s differs as σ changes.

Answer 1

The Social Planner's problem in the Neoclassical Growth model involves maximizing utility subject to the resource constraint. It can be solved using the Lagrangian, Euler equation, and resource constraint. Along a balanced growth path (BGP), per capita consumption and capital stock grow at the rate of technological progress. The period utility function must satisfy the constant relative risk aversion (CRRA) restriction for a BGP to exist. The savings rate along a BGP can be derived from the resource constraint.

Step-by-step explanation:

The Social Planner's problem in the Neoclassical Growth model is to maximize the utility of the representative agent subject to the resource constraint. The Lagrangian for this problem is:

1. L = ∑[β^t * u(ct)] + λ_t * [F(kt)-c_t-gkt]

where L is the Lagrangian, β is the discount factor, u(ct) is the period utility function, λ_t is the Lagrange multiplier for the resource constraint, F(kt) is the production function, c_t is per capita consumption, and g is the rate of technological progress.

The necessary equations for a solution are the Euler equation, which represents the intertemporal condition for consumption:

1. u'(ct) = β * [F'(kt+1)-g]

and the resource constraint, which represents the production function:

1. F(kt) = c_t + gkt + F(kt+1)

Along a balanced growth path (BGP), per capita consumption and capital stock are growing at the rate of technological progress, g, because the Euler equation requires that the marginal utility of consumption be equal to the expected marginal utility of future consumption. This implies that per capita consumption and capital stock must grow at the same rate as technological progress.

The restriction on the period utility function for a BGP to exist is that it must be constant relative risk aversion (CRRA), which means that it must have a constant elasticity of intertemporal substitution (EIS). The Euler equation with this restriction is:

1. u'(ct) = β * [F'(kt+1)-g] * ct^-σ

where σ is the coefficient of relative risk aversion.

The initial capital stock per capita, k, and output per capita, y, can be solved by setting the Euler equation and resource constraint equal to zero and solving for k and y. The expressions for the capital stock per capita at time t, kt, and output per capita at time t, yt, are:

1. kt = [(1/γ) * (g/β)]^(1/(α-1))
2. yt = [F(kt)/kt]^(1/(1-α))

Along a BGP, the savings rate, s, can be derived from the resource constraint and is given by:

1. s = g/(1+g)

The savings rate, s, increases with α and β because higher values of α and β imply a greater share of output being saved. The intuition is that a higher α represents a higher capital-output ratio, which leads to a higher savings rate. Similarly, a higher β represents a greater willingness to save for the future, which also leads to a higher savings rate. The effect of g on s differs as σ changes because the elasticity of intertemporal substitution (EIS) determines how consumption responds to changes in the interest rate. When σ is low, the EIS is high, and consumption is less responsive to changes in the interest rate. Therefore, the effect of g on s is small. When σ is high, the EIS is low, and consumption is more responsive to changes in the interest rate. Therefore, the effect of g on s is larger.