# income effect and substitution effect

Question:Assume no non-labor income, Alice's labor supply function is $$h(w) =\frac{w}{16}$$. Her preferences are given by $$u(c, l) =\sqrt{c}+3l$$. When the wage rate increases, Alice's labor supply increases because of? (Substitution effect or income effect only, or both? If both, are they moving along the same direction?)

This is our homework question and when I solve this, I apply lagrange to the utility function and get $$h(w) =\frac{w}{36}$$,which contradicts with the equation given by $$h(w) =\frac{w}{16}$$. Does this mean that it is not appropriate to apply lagrange's method in this question? I found the utility function to be quasilinear so I selected that there's no income effect, but I got it wrong. What am I supposed to do to analyse this equation? Any help will be appreciated.

• How did you get $h(w) = w/36$?
– Art
Oct 17 '19 at 6:29
• I use lagrange's method, take derivatives for c and l respectively, and plug it in the budget constraint c+wl=wL, which means wh=c, h=c/w and replace c with an expression related to w.
– Alex
Oct 17 '19 at 14:21
• If you’re asking us to help you should include all the information. Without @Art inquiring we had no budget constraint to work with. You should update your question and include your work so that we can help see where you might have gone wrong with it. Another tip is to use LaTeX to format equations as it is much clearer (and a great skill to have) Oct 17 '19 at 16:28