Exercise 1 Equilibrium in the Solow-Swan Model Is Characterized by the Market Clearing Condition S = I

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Exercise 1 Equilibrium in the Solow-Swan model is characterized by the market clearing condition S = I. With a ﬁxed savings rate of s, total savings are S = sY . Investment are given by the increase in the capital stock plus capital depreciation I = K˙ + δK. Thus market clearing implies

K˙ + δK = sY ⇒ K˙ = sF (K,L) − δK

Now transform this equation into per capita terms by diving through the total labour force L K˙ K K = sF , 1 − δ = sf(k) − δk L L L and note that

K˙ KL˙ − KL˙ K˙ K˙ k˙ = = = − kη ⇒ = k˙ + ηk L L2 L L from where it follows that

k˙ = sf(k) − (η + δ)k (1)

(a) In steady state k˙ = 0, thus

f(k∗) s = η + δ (2) k∗ Now we are given a Cobb-Douglas production function of the form

Y = F (K,L) = K1/3L2/3

The intensive form of this function is obtained by diving through L

K f(k) = F , 1 = k1/3 L

Plugging this functional form into the above deﬁnition of the steady state yields 1 s 1−α s(k∗)α−1 = η + δ ⇒ k∗ = η + δ

1 Per capita output in steady state is

α s 1−α y∗ = f(k∗) = (k∗)α = η + δ

(b) Profit maximisation by firms’ leads to the first order condition ( see p. 23 ) f 0(k) = r + δ so the real interest rate is given by

r = αkα−1 − δ

In steady state the real interest rate is

α(η + δ) r∗ = − δ s

C = Y − sY = (1 − s)Y.

So per capital consumption is c = (1 − s)y = (1 − s)f(k). In steady state

α s 1−α c∗ = (1 − s)y∗ = (1 − s) η + δ and steady state per capita savings are

α s 1−α sy∗ = s η + δ

Exercise 2 The golden rule savings rate, is the savings rate that maximises steady state per capita consumption over the steady state per capita capital input. Steady state consumption is given by

c∗(k∗) = (1 − s)f(k∗) = f(k∗) − sf(k∗) = f(k∗) − (η + δ)k∗

2 using equation (2) for the last step. If we maximise c∗ w.r.t. s we get the following ﬁrst order condition dc∗ = f 0(k∗) − (η + δ) = 0. dk∗ This gives the maximum as c∗ is globally concave in k∗: c∗00 (k∗) = f 00(k∗) < 0. Therefore, the golden-rule level of per capita capital kg is implicitly deﬁned by 0 f (kg) = η + δ (3)

(a) The golden-rule optimal level of per capita output is

α−1 αkg = η + δ The steady state level of per capita capital is

1 s 1−α k∗ = η + δ

∗ If s = α, k and kg coincide.

∗ (b) If s = 2α, k > kg and thus the economy is dynamically ineﬃcient. To see this note that as per capita consumption is maximised at the golden- rule optimal level of per capita capital, any level of k that exceed kg leads to a drop in steady state per capital consumption ( recall that c∗00 (k∗) < 0 and ∗0 c (kg) = 0 ). We could therefore decrease savings and increase consumption ∗ at the same time. This could for example be achieved by ”eating” k − kg of the capital stock.

(c) Recall the deﬁnition of steady state consumption as c∗ = f(k∗) − (η + δ)k∗ = (k∗)α − (η + δ)k∗ Now for the golden rule to hold η + δ αkα−1 = η + δ ⇒ kα = k g g α g So η + δ 1 − α c = − (η + δ) k = (η + δ)k g α g α g

3 Now 1 α 1−α k = g η + δ from where we get

1 2 1 − α α 1−α 1 1 1 c = (η + δ) = (η + δ) = g α η + δ 2(η + δ) 4 η + δ for α = 0.5. So ∂c 1 1 2 g = − < 0 ∂η 4 η + δ

Exercise 3 (a)and(b) ( see exercise 1(a) )

log k∗ s 10 = η + δ k∗

Exercise 4 (a) Again the steady state per capital capital stock is charac- terised by equation (2).

f(k∗) s = η + δ ⇒ s(k∗)−1/3 = η + δ k∗ so the steady state capital stock is

s 3/2 k∗ = . η + δ

(b) Firms’ proﬁts are given by

pr = [(1 − δ)k + f(k)] − (1 + r)k

Proﬁt maximisation implies to following ﬁrst order condition 2 f 0(k) = r + δ = π ⇒ π = k−1/3. 3

4 So in the steady state proﬁt rate is

2 2 η + δ 1/2 π∗ = (k∗)−1/3 = 3 3 s

The wage rate is the marginal product of labour ∂ ∂ w = F (K,L) = [Lf(k)] = f(k) − kf 0(k) ∂L ∂L So the steady state wage rate w∗ is

2 1 1 s w∗ = (k∗)2/3 − (k∗)2/3 = (k∗)2/3 = 3 3 3 η + δ

(d) Initially the economy is in steady state, with zero growth. Let’s call ∗ this steady state k1:

∗ 2/3 ∗ γk = s(k1) − (η + δ)(k1) = 0

For this capital stock, if the savings rate rises to s0 > s, the growth rate of the capital stock γk will become positive

0 ∗ 2/3 ∗ γk = s (k1) − (η + δ)(k1) > 0

The economy will start growing, both per capital capital and output go up. ∗ ∗ This will continue until the economy reaches its new steady state k2 > k1

0 ∗ 2/3 ∗ s (k2) − (η + δ)(k2) = 0 at which both per capita capital and output are higher than in the previous steady state. Per capita growth rates are however again zero.

(e) The golden-rule optimal per capita capital stock is given by

3 2 2 2 f 0(k ) = η + δ ⇒ k−1/3 = η + δ ⇒ k = g 3 g g 3(η + δ)

5 The steady state capital stock is given by

s 3/2 k∗ = η + δ

∗ The economy is dynamically ineﬃcient if k > kg, i.e.

s 3/2 2 3/2 2 > ⇒ s > η + δ 3(η + δ) 3 irrespective of the value of δ and η.

Exercise 5 (a) ( see exercise 4(a) )

s 2 k∗ = η + δ

(b) ( see exercises 4(b) and (c) ) 1 π∗ = (k∗)−1/2 2 1 w∗ = (k∗)1/2 2 sy∗ = s(k∗)1/2 c∗ = (1 − s)(k∗)1/2

As can easily be seen, the steady state capital stock declines as η increases. Thus wages, pc savings, and pc consumption fall when η increases. The proﬁt rate however rises, as capital becomes relatively scarce as compared to labour.

Exercise 6 We will consider the ﬁxed proportion ( Leontieﬀ ) production function K L Q = F (K,L) = min , a b with a > 0 and b > 0.

6 a. 1) Calculate the marginal product:

Let us ﬁrst calculate the marginal product of capital. We will have to consider 2 cases i) K/a < L/b and ii) K/a ≥ L/b i) for K/a < L/b the production function will be K Q = a and thus the marginal product of capital will be ∂Q 1 = > 0 ∂K a ii) for K/a ≥ L/b the production function is L Q = b and the marginal product ∂Q = 0 ∂K Same procedure for the marginal product of labour i) for K/a < L/b ∂Q 1 K = = 0 ∂L ∂L a ii) for K/a ≥ L/b ∂Q 1 L 1 = = > 0 ∂L ∂L b b 2) Average products: The average product of capital is deﬁned as Q/K, thus i) for K/a < L/b, Q/K = 1/a ii) for K/a ≥ L/b, Q/K = L/(bK)

The average product of labour is Q/L, thus i) for K/a < L/b, Q/L = K/(bL) ii) for for K/a ≥ L/b, Q/L = 1/b. graph for unit isoquant here

7 b. 1) Deﬁnition: A product function has constant returns to scale (CRS), if for all λ > 0 F (λK, λL) = λF (K,L) Let’s verify this for the ﬁxed proportions production function

λK λL K L F (λK, λL) = min , = λ min , = λF (K,L) a b a b

So this production function indeed has constant returns to scale.

2) Deﬁnition: A function f : X → R is quasi-concave if for x, y ∈ X we have f(λx + (1 − λ)y) ≥ min {f(x), f(y)} for all 0 < λ < 1. ( Note: Any concave function is also quasi-concave. The reverse is not necessarily true. )

So let’s check this for our ﬁxed proportions production function. Take two admissible values (K1,L1) and (K2,L2). Then

λK + (1 − λ)K λL + (1 − λ)L λK + (1 − λ)K λL + (1 − λ)L F 1 2 , 1 2 = min 1 2 , 1 2 a b a b K L K L ≥ min min 1 , 1 , min 2 , 2 = min {F (K ,L ),F (K ,L )} a b a b 1 1 2 2

So this function is indeed quasi-concave.

3) To see that both factors are essential, simply note that F (0,L) = 0, and F (K, 0) = 0. c. To solve for Solow’s basic diﬀerential equation, let us ﬁrst transform the production function into intensive form. As we have shown that the production function has constant returns to scale, we can simply divide it by L to get k 1 f(k) = min , a b For all k/a ≤ 1/b, the production function will be f(k) = k/a, for all k/a > 1/b it will be f(k) = 1/b.

8 So we get two basic diﬀerential equations depending on these two cases: i) for k/a ≤ 1/b: k˙ s = − (η + δ) k a ii) for k/a > 1/b: k˙ s = − (η + δ) k bk

Let us assume that s/a > η + δ, for otherwise k would always decrease, and the only possible steady state would be k∗ = 0. With this assumption, the capital stock k will eventually exceed the value of 1/b and thus the steady state is deﬁned by s s = (η + δ) ⇒ k∗ = bk∗ b(η + δ)

The real rental rate π is given by

π = f 0(k)

Thus, for k/a < 1/b, π = 1/a, and for k/a > 1/b, π = 0. ( Note that there is a kink in the production function at k = a/b, so the derivative at this point is not well deﬁned. ) Real wages w are given by

w = f(k) − kf 0(k)

Thus for k/a < 1/b k 1 w = − k = 0 a a and for k/a > 1/b, w = 1/b.

So suppose we start with an initial level of per capita capital below a/b. Then initially capital is relatively rare and labour is relatively abundant. The wage rate will be zero, the rental rate will be positive. The capital stock will start growing until it reaches its steady state value of k∗ > a/b. Once k exceeds a/b labour will now be relatively scarce and the factor capital abundant, thus

9 the rental rate will drop to zero, the wage rate will suddenly become positive. At k = a/b, the exact rental and wage rate cannot be pinned down. Any values for π and w such that f(a/b) = w + π(a/b) are possible. ( Remem- ber that the isoquants of this production function have a kink at K/a = L/b.)

The output - capital ratio is given by Q/K. We can divide both numer- ator and denominator by L to see that this is equivalent to f(k)/k. So the capital - output ratio is i) for k/a ≤ 1/b, f(k)/k = 1/a ii) for k/a > 1/b, f(k)/k = 1/(bk).