Chapter+6+-+part+I

Part I: Consumer choice, welfare, elasticity and state-contingent commodities model
1. Perfect substitutes: U(x,y)=x+y We want to max U(x,y)=x+y s.t. x.px + y.py = m From where we have that y=(m - x.px)/py Therefore U(x,y) = x + (m - x px)/py U'=1-px/py = 0 => px=py=p Back to the budget constraint p (x+y) = m => x=m/p - y Being substitutes, we know that for px>py, x=0 and for px x = m/px Holding py constant (e.g. at 1) we have the following demand curve:

2. If d.x/d.p = d.xh/d.p - d.x/d.m x0 Then d.xh/d.p = d.x/d.p + d.x/d.m x0 Given that d.x/d.m > 0 then d.xh/d.p > d.x/d.p => Hicks demand curve is steeper than ordinary demand curve

3. U(x1,x2)=x1.x2 We want to max U(x1,x2) s.t. x1p1+x2p2=M => x2= (M - x1.p1)/p2 So U(x1,x2) = x1.(M- x1.p1)/p2 U'=(M - 2.x1.p1)/p2 = 0 => x1 = M/2.p1 and, back to the budget constraint, x2=M/2.p2

We can now obtain a pseudo utility function V = (M/2.p1)(M/2.p2) = M^2/(4.p1.p2) For the initial situation, p1=4 and p2=1, V=M^2/16 For the final situation, p1=1 and p2=1, V=M^2/4

CV: amount of money that has to be removed so that the initial utility is maintained after the price reduction: V(CV) = (M-CV)^2/4 = M^2/16 => CV=M/2

EV: amount of money that has to be given prior to price reduction that gives the same utility as the price reduction: V(EV) = (M+EV)^2/16 = M^2/4 => EV = M

CS is the integral of x1 in relation to p1 between the initial and the final price: CS = 0.7 M

4. If Elasticity = 1.5, 10% increase in price leads to a 15% reduction in quantity, so that the revenue will be: R = 1.1p*0.85q = 0.935*pq => Revenue is 6.5% lower If elasticity = 0.75, 10% increase in price leads to 7.5% reduction in quantity, so the revenue will be: R = 1.1p*0.925q = 1.1075*pq => Revenue is 1.75% higher

5. Show graphically that a risk averse will buy insurance for a premium higher than the expected loss: