Explain in words (and with a graph if that helps) why it is the case that if (ci

Question

Explain in words (and with a graph if that helps) why it is the case that if (ci,2) is the consumer's optimal solution to the problem max u(c,.c2) subject to p,G +Pc2 I, then (G,c2) is also the solution to the problem max hc,.c2) s.t. PG+P,c2 I where h(c,c2)u(c,c)-10. That is, demonstrate why this monotonic transformation of the utility function leaves the optimal consumption choice unchanged. Suppose the utility function u(c1,c2) is homogenous of degree k. What is the utility to the consumer from consuming (.G.. c) (Hint: i.e. find an expression for 1.C2 (r.cl , . c.) in terms of u(c14) ) Finally, if u is a homogenous function it must also be homothetic. Explain in words (possibly using a graph) what it means for the function to be homothetic and what knowledge of that fact tells us about the Marshallian demands (e.g. what can we say about the firm's income expansion path).

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Ans for 1

When we maximise the utility for given budget constraint as follows

L = u(c1,c2)- lambda*(m-p1c1-p2c2)

If we try to find optimum values of c1 and c2 we will differentiate with respect to c1 & c2

We get,

dL/dc1= u'(c1,c2)1-lambda(p1)

Similarly

dL/dc1= u'(c1,c2)2-lambda(p2)

Equalising bot equations we get

U'(c1,c2)1/U'(c1,c2)2= p1/p2 let's call it equation 1

Now if we use h(c1,c2)= u(c1,c2)^2-10

L= u(c1,c2)^2-10-lambda(m-p1c1-p2c2)

dL/dc1= 2u(c1,c2)*u'(c1,c2)1-lambda*p1

dL/dc1=2u(c1,c2)*u'(c1,c2)2-lambda*p2

On equalising we get the same equation for same budget line hence same slope of budget line and indifference curve in both the cases tells us the optimal bundle is same in both the cases .

Let's assume modified utility function u(c1,c1) then budget will be c1p1+c1p1=m1

Differentiating function L1

dL1/dc1=u'(c1,c1)-lambda(p1)=0

Similarly dL1/dc1=u'(c1,c1)-lambda(p2)

Let u(c1,c2)= sqrt(c1*c2)

Then u(c1,c1)= sqrt(c1*c2)=sqrt(^2*c1*c2)=sqrt(c1*c2)=*u(c1,c2)

Which is homogemous degree of 1 hence homothetic

As to be homothetic function of homogemous function it should have k>=1

If homogemous degree of zero then inspite of price change and budget change that is Propotional utility of consumer remains constant

If it non zero homogemous of degree greater than 1 then utility increases woh increasing sale.

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