Question

I. Suppose that you satisfy the hypotheses of the discounted utility model, your yearly discount factor is dand your instantaneous utility function for money is given by u(m)-Vm (the same at every date t). You have been offered a job for three years, starting immediately. You have three options: (1) seniority compensation: you are paid Sw this year, S(w+s,) next year and S(w+s2) two years from now, (2) hiring bonus: you will be paid Sw all three years but you get an extra Sb this year as a sign- up bonus, (3) reverse seniority: you are paid Sz this year, S(e-s,) next year and Se -s2) two years from now For parts (a) and (b) assume that w S40,000 and$49,000 (a) If 60.9, b- $6,400, s 4,100, and s2-8,400 how do you rank the three options? (b) Suppose that you don't have a choice: seniority compensation is what the firm does and it implements it in such a way that every year your salary goes up by Ss (thus you get Sw the first year, S(w+s the second year and S(w+2s) the third year). You suggest to the manager the following: “I need more money this year than the next two years, because I am relocating; could we agree on my giving up the seniority increases (so that I get a flat salary of Sw each year), in exchange for a relocation bonus of Sb this year?" The amount b that you suggested is that amount that makes you indifferent between seniority compensation (as just described) and the following: wb this year and w in each of the next two years. (b.1) Write an equation whose solution tells us the amount that you suggested if -0.75 and s -$8,000. No need to simplify or solve the equation.] (b.2) Write an equation whose solution tells us your discount factor if s-$8,000 and you suggested b- 12,000. No need to simplify or solve the equation.] (c) Assume that δ-05. Suppose now that you don't have a choice and the firm gives you the following deal: you will not be paid in the first two years and will be given $140,000 in the third year. Write down an equation whose solution gives an annual salary w that would make you indifferent between getting Sw every year and accepting the deal proposed by the firm. (d) Assume that δ-08. You are given a choice between a contract that pays a fixed salary of S40,000 every year and a contract that pays $46,000 the first year and $34,000 in the second and third year (d.1) [... points] Calculate your utility from the two contracts. (d.2) [... points] Calculate your utility from the two contracts for the case where you are a hyperbolic discounter with parameters 0.8 and B 0.5

Answer 1

a. The discounted utility function is given as U=U(1) + U(2)/(1+$\delta$) + U(3)/(1+$\delta$)²

Total earning from option 1 during the three years is

w+(w+s₁)/(1+$\delta$) + (w+s₂)(1+$\delta$)² = $40000+ $(40000+4100)/(1+0.9) + $(40000+8400)/(1+0.9)² = $76617.73.

Total earning from option 2 during the periods

= (w+b) + (w/(1+$\delta$)) + (w/(1+$\delta$)² ) = $(40000+6400) + $40000/(1+0.9) + $40000/(1+0.9)² = $78532.96

Similarly from Option 3

= z + (z- s₁ )/(1+$\delta$)+(z-s₂)/(1+$\delta$)² = $49000 =$(49000 - 4100) +$ (49000 - 8400) = $83878.12

since the utility function for money is same for every date of time. Therefore, the ranking of three option will be as

(option 3 :Reverse seniority) P (Option 2 :hiring hours) P (Option 1 :Seniority compensation)

(Note:Here P denotes prefer to) .

b.1. The required equation will be

w + ((w+s)/(1+$\delta$)) +((w+2s)/(1+$\delta$)²)

i.e, $40000 + $ (40000+8000)/(1+0.75) +$(40000+16000)/(1+0.75)².

c. If I get w in each year, then the earning will be

w+w/(1+$\delta$) + w/(1+$\delta$)² = $40000 + $ (40000/(1+0.5) + $ (40000/(1+0.5)² = $84444.44

which is much less than $140000.

Therefore, I will accept the deal proposed by the firm.

d. 1. The total earning from contract 1 is

= $ 40000 +$ 40000/(1+0.8) + $ 40000/(1+0.8)² = $ 74567.90

Since Utility of money = U(m) = $\sqrt{}$m =$\sqrt{}$74567.90=273.01

the total earning from contract 2 is = $46000 + $34000/(1+0.8) +$34000 (1+0.8)² = $ 69382.72

The utility of that earning will be $\sqrt{}$69382.72=263.41