1. This question is about using our model of rational choice "backwards": to make inferences about the parameters in the model from observed choices.Alice always maximizes her utility subject to her budget constraint. Suppose we observe her choices at two different points in time, before a price change and after the price change. That is, we observed her old choice bundle when we saw old prices and her new choice bundle after seeing the new prices.Suppose Alice tells us that as a result of the price change she ended up better off (that is,the new bundle gave higher utility). Based on this, what can we infer about the cost of her old bundle compared with what the new bundle would have cost at the old prices? Give a mathematical argument. Note that the question does not specify how many goods are in a bundle

2. Karl and Clara maximize their respective utilities subject to budget constraints; they shop at the same grocery store that sells only apples and bananas. Karl's utility function is given by UK (xA, xB) = xA + 2xB, where xA denotes his consumption of apples and xB his consumption of bananas. Clara's utility function is UC (xA, xB) = 3xA + 2xB.(a) When the store observes that Karl leaves with some apples, can they infer that Clara also buys some apples? How?(b) When the store observes that Karl leaves with some bananas, can they infer that Clara also buys some bananas? How?(c) The store is interested in setting the prices of apples and bananas in such a way that both consumers buy strictly positive amounts of both goods. Can you advise the store on what those prices might be?

3. Suppose the total demand from all current consumers for a given good is estimated by econometricians to be p = 60 - 2q (this is called "inverse" demand function, since it's written as if prices are functions of quantity demanded). Now suppose that the number of consumers doubles - for each consumer in the market, another identical consumer appears.Model this with a graph to answer the following: does the demand curve shift to the right in a parallel way, doubling demand at every price, to model this? What is the equation of the new demand line, after the doubling?

Could you give a complete answer?