It would effect primarily the mass obtained (yield) and probably the quality, too, but that depends on how you measure this. Given that you use some procedure to purify your crude product that is equivalent in all cases independent on the mass of sodium hydroxide used, you simply will get less purified product.
Palm oil is a natural product and composed of a mixture of compounds. These compounds are know as fats, which are esters of propane-1,2,3-triol (glycerine) and fatty acids. Saponification is the hydrolysis of these esters. In that case sodium hydroxide is the strong base used, and the obtained soap is a mixture of glycerine, and the sodium salts of the fatty acids.
In principle for complete saponification you need one equivalent of base for each ester group.
$$\ce{RCOOR' + NaOH -> RCOONa + ROH}$$
Extending this scheme to palm oil, for the fat or triglyceride you need three equivalents of base:
$$
\ce{\underset{triglyceride}{H5C3-(OOCR^{1/2/3})3}
+ 3 \underset{base}{NaOH}
-> \underset{glycerine}{H5C3(OH)3}
+ 3 \underset{salts of fatty acids}{R^{1/2/3}COONa}}
\tag{1}\label{eq:fat}$$
Since it is a natural product you probably have a number of environmental factors influencing the composition of your specific sample of palm oil. Wikipedia cites the following approximate composition:
\begin{array}{lr}
\text{Type of fatty acid} & \text{Abundance}/\%\\\hline
\text{Myristic saturated C14} & 1.0 \\
\text{Palmitic saturated C16} & 43.5 \\
\text{Stearic saturated C18} & 4.3 \\
\text{Oleic monounsaturated C18} & 36.6 \\
\text{Linoleic polyunsaturated C18} & 9.1 \\\hline
\text{Other/Unknown} & 5.5 \\\hline
\end{array}
As you can imagine this composition of the palm oil has an uncertainty. You probably want to look up more accurate values and the errors of the employed techniques. However, the mass of base needed can therefore be only a crude recommendation. In one case be too much, resulting in leftover (unreacted) base, while in another case it might be too little, resulting in only partially or not at all hydrolysed fat.
A crudely (over)nsimplified calculation:
Assuming palm oil is made out of 100% triglyceride of palmitic acid, we know the molecular mass is $M(\text{palm oil})\approx\pu{807 g/mol}$, we also know $m(\text{palm oil})=\pu{1 kg}$, $m(\ce{NaOH})=\pu{142 g}$.
\begin{align}
M &= \frac{m}{n} &
\Leftrightarrow&&
n &= \frac{m}{M} &
\implies&&
n(\text{palm oil}) &= \frac{\pu{1kg}}{\pu{807 g/mol}} \approx \pu{1.24 mol}
\end{align}
From $\eqref{eq:fat}$ we know we need three equivalents of the base to produce three equivalents of soap. Ergo $n(\text{soap}) = n(\ce{NaOH}\text{ needed}) = \pu{3.72 mol}$. With $M(\ce{NaOH}) = \pu{40 g/mol}$ we know we need $m(\ce{NaOH},\text{ needed}) = \pu{149 g}$. We have $m(\ce{NaOH})=\pu{142 g}$, or $n(\ce{NaOH})=\pu{3.55 mol}$, ergo we only produce that much of soap. The molecular mass of our soap is $M(\text{soap})=\pu{278 g/mol}$. Therefore we will obtain $m(\text{soap}) = \pu{987 g}$.
I'll leave it to you to do the more precise calculations, since you are the mathematician of the both of us.
