Question
Spring 2014
Problem Set # 1 The due date is February 19 at the beginning of class.
1. Calculate the average annual growth rate of real GDP per capita (in 2011 dollars)
in the U.S. for 1960s, 1970s, 1980s, 1990s, 2000s, and during 1960-2010 by using the
data in the following table. Assume continuous compounding in computing the annual
growth rates. The average growth rate satis?es g = t2 1 t1 (ln y (t2 ) ln y (t1 )); where
y (t1 ) and y (t2 ) are the levels of GDP at time t1 and t2 :
Date
1960-1-1 1970-1-1 1980-1-1 1990-1-1 2000-1-1 2010-1-1
Real GDP/capita 17,747
23,585
29,041
36,378
45,026
47,772
2. Describe how, if at all, each of the following developments a?ects the break-even and
actual investment lines in our diagram for the Solow model with exogenous technology
growth, and the balanced-growth-path of the capital per e?ective worker:
(a) The rate of depreciation falls.
(b) The rate of technological progress rises.
(c) The production function is Cobb-Douglas, and the capital share rises. 3. Consider an economy with technological progress but without population growth that
is on its balanced growth path. Now suppose there is a one-time jump in the number
of workers. Note that there is a change to the labor stock at the time of the jump but
no change to the labor growth rate.
(a) At the time of the jump, does output per unit of e?ective worker rise, fall, or stay
the same? Why?
(b) After the initial change (if any) in output per unit of e?ective worker when the
new workers appear, is there any further change in output per unit of e?ective
worker? Is so, does it rise of fall? Why?
(c) Once the economy has again reached a balance growth path, is output per unit
of e?ective worker higher, lower, or the same as it was before the new workers
appeared? Why?
4. Consider the Solow model the technological progress. Assume that the production
function is Y = K (AL)1 . De?ne k = K=AL, y = Y =AL, and c = C=AL:
(a) Find expressions for the steady-state values of k , y , and c as functions of the
parameters of the model, s; n; ; g;and :
(b) What is the golden-rule value of k ?
(c) What saving rate is needed to yield the golden-rule capital stock? 1 5. The economy is described by the Solow model. There is no technological progress, so
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kt = st yt (n + )kt : Output is given by y = k : Suppose that the saving rate increases
with capital per capita according to st = skt ; where s is a baseline saving rate and
0 < < 1: Additionally, assume that + < 1:
(a) Derive the steady state value of capital per person for this economy.
(b) What is the golden rule level of s that maximizes consumption in the steady state?
(c) Under what conditions is the golden rule level of s greater than the golden rule
saving rate in the standard Solow model? 2 