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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 $c_i$+w$s_i$=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

$s_i^*=aT$

$t_i^*+t_{ij}*t_{ji}=aT$

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.

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marked as duplicate by Giskard, Theoretical Economist, Amit, Herr K., Maarten Punt May 17 '18 at 19:55

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