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2. Lin is interested in starting her own business, a cafe in town. Her research shows that the local demographic enjoys coffee and mini muffins according to the utility function:

\[ U(c,m) = \ln(m) + 2\ln(c) \]

a. What would customer demand be for coffee and muffins given each customer's breakfast budget \( B \), and prices of coffee and muffins \( p_c \) and \( p_m \)?

b. What is the indirect utility function of a representative customer? If the customers had twice the budget to spend at the cafe, would they be twice as well off? Why or why not?

c. There has been a supply chain disruption and the price of coffee has doubled. What are the income effect and substitution effect for coffee and muffins demand? Draw and label both the income effect (IE) and substitution effect (SE) for both coffee and muffins. Clearly label the new and old budget constraints. Put coffee on the x-axis and muffins on the y-axis.

d. What is the compensated demand for \( c \)? (Hint: Note that \( U(c,m) = \ln(m) + 2\ln(c) \equiv \ln(mc^2) \))

Answer :

a) Customer demand for coffee and muffins given each customer's breakfast budget of B and prices of coffee and muffins pc and pm is given by the indirect utility function of a representative customer, which is defined as U(c,m)=ln(m)+2ln(c).

Let, B denote the breakfast budget, pc and pm are the prices of coffee and muffins respectively.Then, the customer demand for coffee and muffins can be expressed as follows:-

Max U(c,m)Subject to pc c + pm m ≤ BBy using the Lagrangian function, we haveL(c,m,λ) = ln(m) + 2ln(c) - λ(pc c + pm m - B)FOC:Lc: 2c/λp c = 0Lm: 1/λp m = 0Lλ: pc c + pm m = B From the FOCs, we havepc c = 0 and pm m = 0 => c = 0 and m = 0Orpc c ≠ 0 and pm m ≠ 0 => 2c/λp c = 1/λp m, λ = 2c/p c = m/p mSubstituting this value of λ in the budget constraint, we getpc c + pm m = B => p c c + p m m = BWe can solve the above two equations to get the customer demand for coffee and muffins.

Then the customer demand for coffee and muffins would be:c = (Bp c )/3 and m = (Bp m )/3b) The indirect utility function of a representative customer is defined as U(c,m)=ln(m)+2ln(c).If the customers had twice the budget to spend at the cafe, then their new budget would be 2B. By using the indirect utility function, the utility function with the new budget constraint is given asV(c,m)=ln(m)+2ln(c)subject to pc c + pm m ≤ 2BThus, we can write the indirect utility function asV(c,m)=ln(m)+2ln(c) = U(c,m) + λ(2B - pc c - pm m)Now, we can maximize the above function to get the indirect utility function of a representative customer for the new budget constraint. Max V(c,m) subject to pc c + pm m ≤ 2BFOC:Lc: 2c/λp c = 0Lm: 1/λp m = 0Lλ: pc c + pm m = 2BFrom the FOCs, we getpc c = 0 and pm m = 0 => c = 0 and m = 0Orpc c ≠ 0 and pm m ≠ 0 => 2c/λp c = 1/λp m, λ = 2c/p c = m/p mSubstituting the above value of λ in the budget constraint, we getpc c + pm m = 2B => p c c + p m m = 2B

Then the indirect utility function of a representative customer is given asV(c,m)=ln(m)+2ln(c) + λ(2B - pc c - pm m) = ln(m) + 2ln(c) + λ(2B - p c c - p m m)Substituting the values of c and m from the customer demand function (part a), we getV(c,m) = ln(B/3) + 2ln(2B/3) + λ(2B - pc c - pm m)Therefore, if the customers had twice the budget to spend at the cafe, they would not be twice as well off because their new utility function is not twice the original function.

However, their welfare would increase.c) When the price of coffee doubles due to supply change disruption, the new price of coffee would be 2p c and the new budget constraint is given as 2pc c + pm m = B. We can write the customer demand function with the new price of coffee as:c = B/3pc and m = B/3pmSubstituting the values of c and m in the new budget constraint, we get 2pc (B/3pc) + pm (B/3pm) = B => B/3 = pm /(2pc)The income effect and substitution effect for coffee and muffins demand is given as follows:Income Effect (IE):The IE of coffee measures the change in the demand for coffee resulting from the increase in consumer income.The IE is negative for normal goods and positive for inferior goods.The IE of muffins measures the change in the demand for muffins resulting from the increase in consumer income.The IE is negative for normal goods and negative for inferior goods.Substitution Effect (SE):The SE of coffee measures the change in the demand for coffee resulting from the relative price change between coffee and muffins.The SE is positive for normal goods and negative for inferior goods.The SE of muffins measures the change in the demand for muffins resulting from the relative price change between coffee and muffins.The SE is positive for normal goods and positive for inferior goods.The compensated demand for c is given as:ln(m) + 2ln(c) ≡ ln(mc2)Then, by using the chain rule of differentiation, we getdln(mc2)/dpc = (dln(m)/dpc) + 2(dln(c)/dpc) => -c/m + 2(1/c) => (2-cm)/(mc)Thus, the compensated demand for coffee (c) is (2 - cm)/(mc) units.

To learn more about "Customer's Demand" visit: https://brainly.com/question/1552746

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