Jim spends most of his time in Starbucks. He has $12 to spend on coffee and muffins. Starbucks sells muffins for $2 each and coffee for $1.20 per cup. Jim consumes qc cups of coffee and qm muffins per week. His utility function is U(qc,qm) = qc^(1/2) qm^(1/2)

a) Use the Lagrange technique to find Jim’s optimal bundle. What utility does he attain?

b) Now Starbucks has introduced a frequent buyer card. It costs $2, but it gives him a discount of 20 cents per cup of coffee. If Jim buys this card, how will his consumption of coffee and muffins change? What utility will he attain?

c) Should Jim buy this card?

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I know the budget constraint is 2qm + 1.20qc = $12, but not sure where to go from there.