Around 150 million people are employed in the United States, each working around 2,000 hours per year, for a total of 300 billion hours of labor per year. Suppose that number of hours is fixed, with full employment, but some of those workers are in the underground economy where they get paid in cash not reported to the IRS. Many earn a salary that does get reported, and they pay income tax. But the underground sector of the economy includes many who evade tax, including landscapers and some migrant farm workers, but also a lot of small contractors such as painters, plumbers, carpenters, housecleaners, and other helpers.

Your firm, Policy Analysts Inc. (PAI), is asked to evaluate the effects of the difference in tax between the two sectors on economic efficiency (deadweight loss, DWL) and on economic incidence (on the sources side and uses side). To find the effects on the allocation of labor, you hire an econometrician who gets data and runs regression to estimate the downward-sloping demand for labor in each sector. For the sector using taxable labor (), the height of the demand curve is the gross-oftax wage that the sector is willing to pay: () = 60 0.3

Here, the amount of labor is counted in billions of hours, and the height is in $/hour (or, in $billions paid per 1 billion hours). Thus, starting from zero labor, the sector is willing to pay $60/hour ($60 billion for the first one-billion hours). If the gross wage were $18, then you can solve for desired labor by writing 18 = 60 0.3 and solve for as 140 (billion hours). To calculate dollar amounts in this case, $18 times 140 billion hours is $2,520 billion ($2.52 trillion reported to the IRS). Next, your econometrician estimates that the underground labor () sector is willing to pay: () = 40 0.2 so this sector is willing to pay = $40/hr for the first hour of untaxed labor . Finally, you note that + = 300.

With these three equations, you are now able to solve for the three unknowns in the equilibrium with no tax: , , and the wage in both sectors. With no tax, the gross of tax wage is equal to the untaxed wage because of arbitrage: if the two sectors' wage rates differed, then workers would move to the sector with the higher wage and only stop moving when the two wages are equal. Solve for the equilibrium two ways: use algebra to solve three equations for three unknowns, and then draw the allocation diagram, label every curve, label each axis, and label every point on each axis. Explain why this is an equilibrium where each sector is on their demand curve and the two demanded quantities add up to the total hours of labor.