4. Worth 25 points total. Consider an economic agent who has instantaneous utili

Question

4. Worth 25 points total. Consider an economic agent who has instantaneous utility function where: C and Gt are consumption goods (variables). Let the price of these goods be given respectively by Po and pGt, and assume that the agent's income is It. This agent treats the prices of goods as well as income as given (i.e., they are exogenous variables from the agent's perspective). The agent's utility maximization problem can be stated formally as max CG Ct, G such that (a) Set up the Lagrangian associated with this problem. (Hint: because instantaneous utility is increasing and concave in each of its arguments, then the constraint will bind that is, it will hold with equality.) each good. instantaneous utility. (b) Use the Lagrangian to derive mathematically the agent's optimal consumption of (c) Use your answer to (b) to derive mathematically the agent's optimal level of

Explanation / Answer

Let Lagrangian function to be set is Q

We have to maximise utility at given time t

Ut=Ct^0.5×Gt^0.5

Such that Pc×Ct+Pg×Gt<It

There fore Lagrangian can be written as

Q= (Ct×Gt)^0.5-(It-Pc×Ct+Pg×Gt)

To find the optimization for each good we have to find First order condition

dQ/dC=0.5(GT/Ct)^0.5-(-Pc) & dQ/dG=0.5(Ct/GT)^0.5-(-Pg)

0.5(Gt/Ct)^0.5/Pc==0.5(Ct/Gt)^0.5/Pg

Gt/Ct=Pc/Pg

It=Pc×Ct+Pc×(Ct/Gt)×Gt=2Pc×Ct

It/(2Pc)=Ct and It/(2Pg)=Gt

Ut=(Ct×Gt)^0.5=(It/2Pc×It/2Pg)^0.5=It/2×(1/Pc×Pg)

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