The cost of internal equity (using the dividend discount model) is
ke = (D1/P0) + g
The cost of external equity is just like the formula for internal equity (retained earnings) except that you base it on the net proceeds after flotation costs rather than the market value of the stock.
ke' = (D1/Pnet) + g
Because Pnet will be somewhat lower than P0 (because of the flotation costs), ke' will be higher than ke.