Exercise 4. Cost Minimization Short and Long Run (Minimum WageWorkers)

McDonald’s employs minimum wage workers operating registers totake orders. Let y be the output of orders taken in an hour, L bethe number of minimum wage workers working the registers, and let Kbe the number of registers. The production function isy=100∙L^½K^½.

1. Let the number of registers, K=4, be fixed. Are therediminishing marginal returns to labor? Explain why, and find∂y/∂L.

1. Let the number of counter employees, L=4, be fixed. Are therediminishing marginal returns to capital? Explain why, and find∂y/∂K.

1. If L=4 and K=4, how many customers are served in an hour? Ifboth inputs are doubled, what happens to output, y? Does it double,more than double or less than double?

1. Consider minimum wage at w=$10/hr. Each register costs over$3,600, however, the daily amortization cost is r=$10, whether theregister is used for one hour or several hours. What is the cost ofserving y=400 customers?

1. What are the conditional factor demands, L(w,r,y) and K(w,r,y)?What is the minimized cost function C=(w,r,y)? Given w=10, r=10,and y=400, what is the minimum cost?

1. The minimum wage is raised to w’=$20/hr. What happens in theshort run, with registers fixed at K=4? Find the new L(w,r,y) andthe short-run cost function.

1. What happens in the long run, when McDonald’s can change K aswell? McDonald’s automated kiosks allow for many registers to workdirectly with customers with minimum employee’s supervision. Do youthink minimum wage laws are affecting technological change, or whatother factor do you think might spur the introduction of automatedkiosks?