$lGU(l,G)=\sqrt[\phantom{2}]{l}+G$

and the $to$ keep things simple, $we$ will normalize the price $of$ the composite Good $p_G=1.$

This way, when $we$ consider the labor market that pays a constant neoclassical price for

leisure denoted $as$ the wage rate $w\frac{w}{p_G}=wTto$ denote the time endowment for

the consumer.

$To$ link $up$ with the UBI setting, let $us$ say that $in$ addition $to$ the wage labor market

$to$ which the consumer can supply what $we$ will label $as{h}^{M}w$ discussed $above{h}^H$

hours $of$ work, $to$ denote Home hours $of$ labor supply. $To$ link $up$ with our Rural Labor

Supply notes, let this 'home production' $of$ other Goods have a production function given $by$

$G=f(h^H)=A\sqrt[\phantom{2}]{h^H}$

$So$ that with these two different ways $(orsectors)in$ which $to$ supply labor, the consumer's

time accounting equation constraint $is$

$T-l=h^M+h^H-=h^{\text{T}\text{o}\text{t}\text{a}\text{l}}orif$ you prefer $l=T-h^M-h^H$

The exogenous variable A here are the asset$(s)$ used for the home business, and $wedo$

not specify what they are, other than $to$ say you should take $itas$ exogenous, and not $a$

choice variable. I will say more about how the A variable links $to$ the UBI transfers $in$

parts $of$ the problem below, and you should again take all $of$ that $as$ exogenous. The choice

variables will thus $be(G,h^M,h^H),$ because given the choices $of{h}^M$ and $h^H,$ then given the

time accounting equation $ofl=T-h^M-h^H,$ those choices will thus determine the choice

$ofl.$ The exogenous variables are $(w,T,A).$ Owing $to$ the nature $of$ the home production

function, for a finite $w,$ the consumer will always choose $at$ least some $h^H>0f^{\text{'}}(h^H=0)=h^M>0as$ well, $so$ that the consumer also always works some

hours $in$ the Labor Market, and again, $we$ also only concern ourselves with 'interior' $l>0$

and $G>0.$

$(a)To$ get started, first write down the Budget Constraint linking the total quantity $of$

$Gto$ the two ways $of$ procuring $G,$ meaning through home production and via the purchase

$in$ the market, which $in$ both cases requires giving $up$ leisure either via $h^{H}or{h}^{M}G,h{h}^{H}is$ determined. Also add $in$ the direction $of$ increasing

Utility $on$ this diagram, and how for a consumer with $an$ optimal interior $h^M>0$ the

optimal 'tangency condition' for optimal $G^{*}$ and $(h^{\text{T}\text{o}\text{t}\text{a}\text{l}}{)}^{*}$ shows $upon$ this graph. Without

worrying about the specific form $of$ the utility function $at$ this point, draw $in$ a canonical,

smooth indifferen