Question

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

The utility function for each person, regardless of the real wage he/she receives, is identical: u(c,l)  ln c  ln l , 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 H (the Greeklowercase letter “tau”), and the labor income tax rate for individuals that earn low wagesis  L .

Low Wages, High Wages, and Taxes. There are two categories of people: those that receive high real wages and those that receive low real wages. Denote these two real wages as wH (high wages) and w (low wages), respectively, and wH> w The utility function for each person, regardless of the real wage he/she receives, is identical: u(c,l)- lnc +ln/, in which, exactly as in Chapter 2, c stands for consumption and / 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 r" (the Greek lowercase letter "tau"), and the labor income tax rate for individuals that earn low wages is τ a. What is the budget constraint for the low-wage individuals? (Display the budget constraint clearly by drawing a box around it.) b. What is the budget constraint for the high-wage individuals? (Display the budget constraint clearly by drawing a box around it.) c. Construct the Lagrange function for the low-wage individuals. (Note: use the utility functional form stated above.) d. Based on the Lagrange function constructed in part c, provide the first-order Display the two FOCs clearly by drawing a box conditions (FOCs) for c and l. around each. 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 Display the two FOCs clearly by drawing a box conditions (FOCs) for c and 1. around each.

Answer 1

Answer: u (c,l) In c+In l n+1=1 c:consumption, P: the nominal price of each unit ofc l: leisure n: labor, labor income tax rate higher wages (WH):tA H, the labor income tax rate for lower wages (WL): tAL Let c, I be the consumption and leisure units for WL Let c*, I* be the consumption and leisure units for WH Now time can either be spent on leisure or on labor considering n+ 1- 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 tAH>tAL and such that WH(1-tAH) WL(1-tAL). 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.