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ECON 711 Macroeconomic Theory and Policy

Problem Set 1

Module 1: The Solow Growth Model

Suggested Solutions

1. Solow model. Consider a standard Solow-Swan model in continuous time with Cobb-

Douglas production function y = kα, constant savings rate s, depreciation rate δ, productivity growth g and employment growth n.

(a) Derive expressions for the steady state values k* , y* , c* in terms of the model parameters s, δ, g, n and α .

(b) Use a diagram to explain how an increase in s aﬀects k* , y* , c* . Does this change in s increase or decrease long run output and consumption per worker? Explain.

(c) Use a diagram to explain how an increase in α aﬀects k* , y* , c* . Does this change in α increase or decrease long run output and consumption per worker? Explain.

Answer:

(a) Steady state capital k* solves

sf )k* ) = )δ ＋g ＋n)k*

Hence with y = f )k) = kα we have the solution (ignoring the trivial k* = 0 case)

k* = ╱ 、1 一α

and therefore α

y* = ╱ 、1 一α

c* = )1 一 s) ╱ 、1 一α

(b) An increase in s unambiguously increases k* and hence increases y* . Whether an increase in s increases c* depends on where the level of s is relative to the ‘golden rule’ level (which with this Cobb-Douglas production function, is equal to α). If s < α then a marginal increase in s increases c* but if s > α then a marginal increase in s decreases c* .

is greater than 1.

● If > 1 then a higher α increases k* and hence increases y* and c* . In this situation, the parameters are conducive to capital accumulation (the savings rate is high relative to eﬀective depreciation), so less curvature in the production function () higher) makes for a larger steady state level of capital per eﬀective worker.

● If < 1 then a higher α decreases k* and hence decreases y* and c* . In this situation, the parameters are not conducive to capital accumulation (the savings rate is low relative to eﬀective depreciation), so less curvature in the production function makes for a lower steady state level of capital per eﬀective worker.

2. Linear production function. Suppose the production function is linear y = k and for simplicity suppose no productivity or employment growth, g = n = 0. Does the Solow-Swan model have a steady state capital stock in this setting? Why or why not? Explain the dynamics of kt in this economy. How do these dynamics depend on the values of s and δ? Explain. What standard assumptions about the production function does this example violate?

Answer:

With y = f)k) = k, capital accumulation is given by

kat = sf )kt) 一 )δ ＋g ＋n)kt = )s 一 δ)kt

(since g = n = 0). Steady states k* are given by points such that kat = 0. In this case there is generally no steady state except the trivial one at k* = 0. In the ‘knife-edge’ case where s 一 δ , any k is a steady state. More generally, for s δ, either (i) the savings rate is greater than depreciation so that

kat = )s 一 δ)kt > 0

for all t and the capital stock grows without bound, or (ii) the savings rate is less than depreciation so that

kat = )s 一 δ)kt < 0

for all t and the capital stock shrinks towards 0.

This linear production y = f)k) = k function has positive marginal product, f\ )k) = 1, but does not exhibit diminishing returns f\\ )k) = 0. Moreover, since f\ )k) = 1, for all k, this linear

production function does not satisfy the usual Inada conditions [f\ )0) = o and f\ )o) = 0]. In particular, the failure of the ﬁrst Inada condition f\ )0) = o exposes the economy to kt → 0 when s < 6 while the failure of the second Inada condition f\ )o) = 0 allows the economy to experience unbounded growth kt → o when s > 6 . In this latter case, the basic conclusion of the Solow model — i.e., that capital accumulation alone cannot sustain long run growth — is overturned.

3. Inada conditions. Consider a production function in intensive form y = f )k) . Brieﬂy explain the role played by the Inada conditions f\ )0) = o and f\ )o) = 0 in analyzing the Solow-Swan model. In particular, suppose f\ )k) > 0 and f\\ )k) < 0 but that the Inada conditions are not satisﬁed. What possibilities does this lead to?

Answer:

As alluded to above, the Inada conditions guarantee the existence of an interior steady state k* > 0. For simplicity, again suppose that g = n = 0. Consider two cases:

. If sf\ )0) < 6 :

– Investment never exceeds depreciation.

– The only steady state is the trivial k* = 0.

– Capital shrinks toward this steady state kt → 0.

. If sf\ )o) > 6 :

– Investment is never less than depreciation.

– Again, the only steady state is the trivial k* = 0.

– Capital grows without bound, kt → o , away from this steady state.

Notice that the usual Inada conditions f\ )0) = o and f\ )o) = 0 are suﬃcient to ensure these cases do not arise, but are not necessary. The necessary conditions are that