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) Consumption is equal total production minus savings 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∗ (c) ( see exercise 4(d) ) 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 (c) Per capita savings are sy∗ = sf(k∗) = s(k∗)2/3 and per capital consump- tion is c∗ = (1 − s)(k∗)2/3. (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. (c) ( see exercise 4(e)and 2(a) ) The economy is dynamically inefficient as long as s > 1/2 irrespective of η. 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 con- sider 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.