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2. Consider an economy with the following production Y = K? L? (a) With aid of a sketch for the Solow-Swan growth model developed in class, show the [3 Golden Rule steady state value for capital per worker k? that maximises the level of consumption per worker Ct. (b) The Golden Rule capital per worker kär is the level of k* that maximises the consumption per worker. Policy makers can influence the saving rate and thus the k* with the aim of having a level of k* that maximises the consumption per worker i.e. kor- Consumption is maximised where the gap between output per worker Yt and (n+8) kt is at is largest. Using your answer for (a) and that at = y 14 = f(k!)- 8 kt and the with a depreciation rate 8 = 0.025 and that the population rate is n = 0.025, complete the The table below to find where consumption per worker Ct is maximised. [4r 1 kt Yt = k? (+n) k c = f(t)- (+n)kt St = syt = sk (8 +n) = 0.05 0 4 16 25 36 49 64 100 121 144 (c) From part (b), calculate what the savings rate will have to be to establish the Golden Rule level of capital per worker kar [1r
a) GIven Y = K1/2L1/2
In per capita terms: Y/L = K1/2L1/2/L = K1/2 / L1/2 => y = k1/2 where y= per capita output, Y/L and k = per capita capital, K/L
Per capita investment, i = per capita saving,S = sy where, s: saving rate
kt+1 = per capita investment + (1-δ) kt = syt + (1-δ-n)kt, where, n: population growth rate, δ: rate of depreciation
=> change in capital stock = kt+1 -kt = s(kt1/2) - (n+δ)kt
Condition for golden rule steady state level of capital per worker that maximizes consumption per worker is when the slope of production function,MPk is equal to the slope of breakeven investment line, i.e.:
Marginal Productivity of capital , MPk= δ + n (slope of breakeven investment line)
MPk = dy/dk = dk1/2/dk = 1/2k-1/2
golden rule steady state level of capital per worker is where: 1/2k*GR-1/2 = δ + n
squaring both sides, we get: 1/4k*GR-1 = (δ + n)2 = >
=> k*GR = 1/(4(δ + n)2)
b)
| kt | yt | (δ+n)kt | ct = yt - (δ+n)kt | st = syt = yt -ct | saving rate, s = st/yt |
| 0 | 0 | 0 | 0 | 0 | 0 |
| 4 | 2 | 0.2 | 1.8 | 0.2 | 0.10 |
| 16 | 4 | 0.8 | 3.2 | 0.8 | 0.20 |
| 25 | 5 | 1.25 | 3.75 | 1.25 | 0.25 |
| 36 | 6 | 1.8 | 4.2 | 1.8 | 0.30 |
| 49 | 7 | 2.45 | 4.55 | 2.45 | 0.35 |
| 64 | 8 | 3.2 | 4.8 | 3.2 | 0.40 |
| 100 | 10 | 5 | 5 | 5 | 0.50 |
| 121 | 11 | 6.05 | 4.95 | 6.05 | 0.55 |
| 144 | 12 | 7.2 | 4.8 | 7.2 | 0.60 |
Consumption is maximized at k = 100
c) saving rate needed to maintain k*GR = 100 is 0.5 as shown in the table above. Rate of saving is calculated by dividing savings with output.
