Question

1.Consider the following production function: Y = KθL1-θ, where the labor (L) is growing at the rate n = 0.03 every year. The capital stock (K) is depreciating at a rate δ = 0.05 annually. The value of θ is 0.62).The saving rate is equal to 40%. The government imposes a proportional tax rate τ on y. Thus, the disposable income is equal to (1 - τ)y only. Let y = Y/L and k = Y/L.

a) Please transform the production function into per capita format and find out the steady state values of k, y and c in terms of τ. Show your answers in a diagram with k as the x-axis.

b) Find out the values of k, y, c at the Golden Rule steady state. (Note: when you derive golden rule steady state, assume no tax policy is used. That is, τ = 0.)

c) What tax rate can be used to give y* (steady state y) = y** (golden rule y) ? (Note: the tax rate  you find can be a positive or a negative number.)

Answer 1

Solution-

Here the disposable income is equal to yd= (1-t)k^$\Theta$

At the steady state change in capital stock per labor is 0.

(a)

Y= kolle 7 = KOLLO y = ko @jy = ky where y us output per Labor and R is ki, capital per labe Also, Disposable Income will be ( (1-?)y = (1-2) Rol 1 yd = (1-2) ko) At the steady state, TAR=0 ie. i-SRor=0 i = SRINR sy = SRIOR 9 where s is otomer saving rate and S vis depreciations

000) syd = m+ 8) R. Here, m=0.03 S = 0.05 O= 0.62 s=0.4 > (s.1-77*° = (1+6)R → STEADY STATE . Putting the values we get 0.4 (147) RO62 (0.03 + 0.05 TRA PRO: 62 = 0.4 (1-2) poo: 62 = 0.08 R 0.38 = 5(1-2). R* = (5 (1-1)] 0:38 R* = 50.38 (1-2) 0.38 R* = 69.09 (1-732.687

Also, y = RO y* (163.09) (1-772-037**** y* = 69.09 )0:62 (1-2) 2.63X0 62 y* - 13-82 (-2)!.63 2 2.63X0.62 (* = (1-0.4) g* c* = 0.6513-82 (1-1) Tex = 8.3 (1-2) 1.63 / g. 1 - 41c7983 (n+8)R *88-3 (1-2)'63 13.82(11) -- R² R (69.09 (1-2) 2.63)

At the golden enele, Consumption is maximized This at Steady state statt sy=8R+ sn ey-sy-y-SR-NR MPix = 5th , 2= d.c=4 - BR-AP de – dy - S.tn 0 ie, ar J (RO) = 8.tn AR de dues HER) 6) JO=mPk - S ORO-I = Stn 0.62 RO 62-1 = 0.05 +0.03 / MPK = 8th 0.62 38 = 0.05-0.03 R 0.38 = 0.62 0.08

F = 0.62) 038 (0.08) ۹۷ (415) 5 2* 218.۹۱/ * * | * * 8. = (218.9) 262 - 2825 ** = (S)** = ()- 0.9) 23.2 2 = 25-6728-0 16 . 95 ر

© y* - y** 13.82(1-2) 1.63 = 28.25 (1-2) 163 = 28.25 1-2 = (2.044) 1.63 1-2= 1055 2=-0.55 (12) 73.82