# Kuhn Tucker optimization problem and game theory [duplicate]

Consider a game with two players, where each player i= 1 ,2 has preferences $u_i$= $s_i^a$$c_i^{1-a}, where c_i is the consumption and s_i is social interaction. s_i is given by : s_i = t_i + t_{ij}*t_{ji}, where t_i is time spend by player i alone, and t_{ij} is the time player i spends with player j. Player i has to decide how much of his or her time T to allocate between work, having time alone, {t_i} and social interaction t_{ij}. Assume that for each hour, player i works, he or she earns the wage w and assume that the price of the consumption good c_i is normalized to p=1. i. Define the optimization problem for player 1. Write down the Kuhn-Tucker conditions and discuss these conditions. Explain why player 1 faces a strategic situation. Find the best response functions for player 1 and 2. Graph these functions. ii. There are some cultures that emphasize social interaction more than others. In this exercise how can I determine the role of '' culture'' if exists? My solution is :$$max s_i^a c_i^{a-1}$$subject to c_i+ws_i=Tw and s_i\le T Lagrangian$$L(\lambda, \mu, s_i, c_i) = max s_i^a c_i^{a-1} -\lambda [c_i+ws_i-Tw]+\mu [s_i-T]$$FOCs$L_{\lambda}=0$so,$c_i+ws_i=Tw$(Eqn-1)$L_{s_i}=0$so,$as_i^{a-1}c_i^{1-a}$-$\lambda w$-$\mu=0$(Eqn-2)$L_{c_i}=0$so,$(1-a)s_i^{a}c_i^{-a}$-$\lambda =0$(Eqn-3) Slackness condition :$\mu [s_i-T]=0$If$s_i-T<0$implies$s_i < T$so$\mu =0$Thus by equation (2),$as_i^{a-1}c_i^{1-a}$=$\lambda w$By equation (3),$(1-a)s_i^{a}c_i^{-a}$=$\lambda $By equation (2)/(3),$(a/1-a)(c_i/s_i)=w$implies$c_i={w(1-a)\over a} s_i$(Eqn-4) Equation 4 and 1,${1-a\over a}ws_i +ws_i=Tw$implies$s^*_i=aT$Thus,$c_i^*=(1-a)wT$In order to find the best response function$s_i^*=aTt_i^*+t_{ij}*t_{ji}=aT$By equation 5,$t_{12}(t_{21})={aT-t^*_1\over t_{21}}$and$t_{21}(t_{12})={aT-t^*_2\over t_{12}}$equation (6) When I plug (5) into (6)$t_{21}(t_{12})={aT-t^*_2\over (aT-t^*_1)/t_{21}}$So,$t_{21}={(t_{21}aT-t_{21}t^*_2)\over aT-t^*_1}t_{21}aT-t_{21}t^*_1= t_{21}aT-t_{21}t^*_2(t^*_2-t^*_1) t_{21}=0$Each player faces a strategic situation because his best response depends on the others action. The term$t_{ji}\$ determine by other players. Thus you have to choose your best response conditional on the others’ action.I really try to find the best response functions but i stuck at this point.

Thank you.

• Yes it was again me but then I upload my solution more analytically – Stefanos Makridis May 10 '18 at 21:30