# Impact of technology shock on hours worked in basic RBC model - short and long run with logarithmic preferences

First off, I apologise if, in the following, I do not present the model adequately - I'm at an intermediate level of the study of economics. My question is at the bottom.

Anyway, consider a basic RBC model, where there are a large number of representative firms and infinitely-lived agents, who behave according to the rational expectations and permanent income hypotheses.There is perfect competition. We also assume that agents know exactly what will happen in the future.

The model presented mathematically:

The consumer's problem:

$max\sum_{t=0}^{∞}β^t[u(c_t)+v(1-h_t)]$

s.t. $\sum_{t=0}^{∞}βR^{-(t+1)}c_t=A_0+\sum_{t=0}^{∞}βR^{-(t+1)}w_th_t$

where $β=$ the discount rate; $R^{-1}=\frac{1}{1+r}$, $u=$ a function representing the value of consumption, $v=$ a function representing the value of leisure, $h_t=$ hours worked

The firm's problem:

$max{π=A_t}f(K_t,h_t)-w_th_t-r_tK_t$

We assume all savings are invested in capital, so we have:

$K_{t+1}=(1+r+δ)K_t+w_th_t-c_t$

where $δ=$ depreciation rate of capital.

Then, using the fact that the three markets of the model clear (goods, labour, capital), and assuming constant returns to scale, we have three equilibrium conditions (after some maths):

$u'(c_t)=β(1+A_{t+1}f_1(K_{t+1},h_{t+1}-δ)u'(c_{t+1})$ - intertemporal savings/consumption choice

$\frac{v'(1-h_t)}{u'(c_t)}=A_tf_2$ - consumption/leisure tradeoff

$K_{t+1}=(1-δ)K_t+A_tf-c_t$ - law of motion for capital

What I'm interested in is what happens to hours worked when there is a positive technology shock with logarithmic preferences. I have already shown that the income and substitution effects cancel each other out in the steady state of the model - but supposedly, in the short run, the substitution effect dominates the income effect with logarithmic preferences. Can someone explain this to me why and show this to me mathematically?