Consider two consumer, LaTeX: h=1,2h = 1 , 2, with preferences LaTeX: U=\ln\left(x^h ight)-\frac{z^h}{s^h}U = ln ? ( x h ) ? z h s h, where LaTeX: x^hx h is consumption, LaTeX: z^hz h is income, and LaTeX: s^hs h is skill level.Assume that LaTeX: s^1=1s 1 = 1 and LaTeX: s^2=2s 2 = 2.

(i) Consumer 1 is allocated consumption LaTeX: x^1=1x 1 = 1 and income LaTeX: z^1=1z 1 = 1. Consumer 2 is allocated consumptionLaTeX: x^2=2.718x 2 = 2.718. What is the maximum income 2 can earn before incentive compatibility is violated?

(ii) Combine the incentive compatibility constraint and the aggregate resource constraint (LaTeX: z^1+z^2=x^1+x^2z 1 + z 2 = x 1 + x 2) to solve for LaTeX: z^1z 1 and LaTeX: z^2z 2. If LaTeX: x^1=2.7183x 1 = 2.7183 and LaTeX: x^2=7.3891x 2 = 7.3891, what is the implied value of LaTeX: z^1z 1?

(iii) Assume social welfare is given by LaTeX: W=U^1+U^2W = U 1 + U 2. Using the answers to (ii) write social welfare as LaTeX: W=\beta^1\ln\left(x^1 ight)+\beta^2\ln\left(x^2 ight)-\gamma\left[x^1+x^2 ight]W = ? 1 ln ? ( x 1 ) + ? 2 ln ? ( x 2 ) ? ? [ x 1 + x 2 ]. What is the value of LaTeX: \beta^2? 2?

(iv) What is the optimal value of LaTeX: x^1x 1?

Question 19 4 pts (iii) Assume social welfare is given by W = U1 + U2. Using the answers to (ii) write social welfare as W = {31 1110.31) + 2 h1(;'c2) "y [m1 + $2] . What is the value of g? Question 20 4 pts {iv} What is the optimal value of 2:1? Question 17 4 pts 2" Consider two consumer, h = 1, 2, with preferences U : 111(32") T' where 33'- is consumption, 5 21% is income, and 313 is skill level. Assume that 31 = 1 and 32 = 2. (i) Consumer 1 is allocated consumption 31 : 1 and income zl : 1. Consumer2 is allocated consumptionm2 : 2.713. What is the maximum income 2 can earn before incentive compatibility is violated? Question 18 6 pts {ii} Combine the incentive compatibility constraint and the aggregate resource constraint ( 31 + 22 = 5.5-1 + :32) to solve for 2:1 and 22. If 31 = 2,7183 and 5:3 = 7,3391, what is the implied value of 21