Question

Low Wages, High Wages, and Taxes. There are two categories of people: those that receive high nominal wages and those that receive low nominal wages. Denote these two nominal wages as WH (high wages) and WL (low wages), respectively, and WH > WL .

The utility function for each person, regardless of the nominal wage he/she receives, is identical: u (c,l) =lnc +lnl , in which, exactly as in Chapter 2, c stands for consumption and l stands for leisure. Furthermore, after defining n as labor, keep in mind that n + l = 1 (which is also identical to the framework considered in Chapter 2).

The labor income tax rate for individuals that earn high wages is t^H (the Greek lowercase letter “tau”), and the labor income tax rate for individuals that earn low wages is t^L. To complete the notation, P is the nominal price for each unit of c (which, once again, is identical to the notation in Chapter 2).

The rest of this question focuses on the tax rates t^L and  t^H . For the sake of clarity, suppose that WH , WL , and P are all unaffected regardless of the particular policy setting of the two tax rates.

e. Construct the Lagrange function for the high-wage individuals. (Note: use the utility functional form stated above.)

f. Based on the Lagrange function constructed in part e, provide the first-order conditions (FOCs) for c and l. Display the two FOCs clearly by drawing a box around each.

Answer 1

u (c,l) = ln c +ln l

n + l = 1

c: consumption, P: the nominal price of each unit of c

l: leisure

n: labor, labor income tax rate higher wages (WH): t^ H, the labor income tax rate for lower wages (WL): t^L

Let c, l be the consumption and leisure units for WL

Let c *, l* be the consumption and leisure units for WH

Now time can either be spent on leisure or on labor considering n + l = 1, its rationale that an individual who’s getting high wages would spend more time on labor than on leisure considering that each unit of c costs the same to both the individuals and as WH can get more units if c it would prefer to spend more time on labor. Similarly, someone who’s getting low wages would spend more time on leisure than on wages because WL would allow only a few units of c to be bought while the if the time is allotted to leisure the individual gets more utility.

When taxes come into the picture it’s possible that the numerical values of c, I and c*, I* be same because the after-tax wage for WH can be made equal to the after-tax wage of WL. In such a scenario the t^H > t^L and such that WH(1-t^H) = WL(1-t^L). As after-tax wages for both the high wages and low wages individuals would be the same and hence the optimal choices for both the individuals would be the same to achieve the same highest utility possible.