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SIPA INAF U8145 Spring 2024 Problem Set 2: Growth Models
发布时间：2024-05-27
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SIPA INAF U8145
Spring 2024
Problem Set 2: Growth Models
1 The Harrod-Domar Model
Assume that population growth is given by:
dt/dP = nP
(as we assumed when discussing the Solow model in lecture). Using the notation for the Harrod- Domar model used in lecture, find the growth rate of per capita income, y = P/Y.
2 The Solow Model
1. Properties of Cobb-Douglas production function.
Rather than the general function discussed in lecture, suppose that aggregate production is governed by a Cobb-Douglas production function:
Y = AKα P1−α
where 0 < α < 1. Show that this function satisfies the following assumptions that we imposed on the general function discussed in lecture:
(a) Constant returns to scale.
(b) Positive marginal products of capital and labor.
(c) Diminishing marginal products of capital and labor.
2. The steady-state with a Cobb-Douglas production function.
Assuming that the aggregate production function is as given in part 1 of this question and that there is constant population growth (∂t/∂P = nP) but no technical change (that is, A is constant), find the following:
(a) An expression for ∂t/∂k, the rate of change over time of the per capita capital stock.
(b) An explicit solution for k* , the steady-state value of per capita capital stock. (By “explicit” I mean a solution is which k* is by itself on the left-hand side of an equation and the expression on the right-hand side does not involve k*.)
Use the notation that we used in lecture and that is also used in the book (s=savings rate, δ =depreciation, etc.)
3. Comparative statics.
Show both algebraically (i.e. using a partial derivative) and graphically what happens to k* , the steady-state value of the per-capita capital stock from question 2, when the following changes occur:
(a) The savings rate, s, is increased.
(b) The rate of depreciation, δ, is decreased.
(c) The rate of population growth, n, is increased.
Consider each change in isolation from the others. That is, do not worry about an economy that experiences a change in more than one of the parameters at the same time.
4. Introducing technical change into the Solow model.
Suppose now that there is technical change in our economy that tends to increase the efficiency of labor. Population growth is as above. Let the efficiency of labor be E(t) and suppose that the aggregate production function is given by:
Y = F(K(t), E(t)P(t))
Assume that the rate of growth of efficiency is given by:
π = E/∂t/∂E
Let L(t) be the number of “efficiency units” of labor in the economy: L(t) = E(t)P(t). Define the capital stock per efficiency unit and the income per efficiency unit, respectively, as follows:
k = L/K
y = L/Y
Note that you will also want to define a function f such that y = f(k(-)).
(a) Derive an expression for ∂t/∂k, the rate of change of capital per efficiency unit with respect to time.
(b) Write down an equation that implicitly defines k(-)* , the steady state value of capital per
efficiency unit. (An implicit solution is a solution that pins down the value of k* but that cannot be simplified to where k* is by itself on the left-hand side. Here I have not told you exactly what the production function is, and so you will not be able to arrive at an explicit solution. Your implicit solution will have both a k(-)* and an f(k(-)* ) in it.)
(c) Find the growth rate of per-capita income, y, at the steady state, that is, when k = k* . Hint: note that y = E/y.
(d) Find the growth rate of total income, Y , at the steady state.
