Use the Lagrangian method to answer the following questions: Each day Simon, who
ID: 1169660 • Letter: U
Question
Use the Lagrangian method to answer the following questions:
Each day Simon, who is a college student, eats lunch at campus. He only likes Twinkies (t) and iced Coffee (ic), and these provide him with a utility of U(t,ic)=t0.5ic0.5 If Twinkies cost $0.10 each and iced coffee costs $0.25per cup, how should Paul spend the $1 he has to maximize his utility? Depict this constrained maximization problem using a graph. If the school tries to discourage Twinkie consumption by increasing the price to $0.40, what would be his optimal bundle if he still has $1 to spend? Draw this increase in price and show how the optimal bundle changes for Simon.Explanation / Answer
(a) To make the solution easier, we should find the monotonic transformation of the given utility function by taking the natural logarithm of the function. Denoting the new utility function by v, we have
v(t, ic) = log(t0.5ic0.5)
v(t, ic) = log(t0.5) + log(ic0.5)
v(t, ic) = 0.5 log t + 0.5 log ic
which is as appropriate a utility function as given function U. Since, this function is in linear form, we shall use this to find the solutions to this problem.
Since the price of t is $0.10, that is ic is $0.25, and the amount to be spent is $1. The consumer’s budget constraint is of the following form
0.10 t + 0.25 ic = 1
The constrained maximization problem can be written as
maxt,icv(t,ic) subject to 0.10 t + 0.25 ic = 1
The Lagrangian for this problem is
L(t, ic) = v(t, ic) – (0.10 t + 0.25 ic – 1)
i.e. L(t, ic) = 0.5 log t + 0.5 log ic – (0.10)t – (0.25) ic +
The three necessary conditions for the solution are:
(a) L/t = 0
(b) L/ic = 0
(c) 0.10 t + 0.25 ic = 1
Solving (a), we have
L/t = 0
0.5/t + 0 – 0.10 + 0 + 0 = 0
0.5/t = 0.10
= 5/t …..(A)
Solving (b), we have
L/ic = 0
0.5/ic – 0.25 = 0
0.5/ic = 0.25
= 2/ic …..(B)
From (A) and (B), we have
5/t = 2/ic
5ic = 2t
t = 2.5ic
Inserting the above result in the condition (c) solving for ic, we have
0.10 (2.5ic) + 0.25 ic = 1
0.25ic + 0.25 ic = 1
0.5 ic = 1
ic = 2 ....(i)
which is the optimal consumption of iced coffee
and,
t = 2.5(2)
= 5
which is the optimal consumption of Twinkies.
(b)
(c) Since the price of t is $0.40, that is ic is $0.25, and the amount to be spent is $1. The consumer’s budget constraint is of the following form
0.40 t + 0.25 ic = 1
The constrained maximization problem can be written as
maxt,icv(t,ic) subject to 0.40 t + 0.25 ic = 1
The Lagrangian for this problem is
L(t, ic) = v(t, ic) – (0.40 t + 0.25 ic – 1)
i.e. L(t, ic) = 0.5 log t + 0.5 log ic – (0.40)t – (0.25) ic +
The three necessary conditions for the solution are:
(a) L/t = 0
(b) L/ic = 0
(c) 0.40 t + 0.25 ic = 1
Solving (a), we have
L/t = 0
0.5/t + 0 – 0.40 + 0 + 0 = 0
0.5/t = 0.40
= 5/4t …..(A)
Solving (b), we have
L/ic = 0
0.5/ic – 0.25 = 0
0.5/ic = 0.25
= 2/ic …..(B)
From (A) and (B), we have
5/4t = 2/ic
5ic = 8t
t = 0.625ic
Inserting the above result in the condition (c) solving for ic, we have
0.40 (0.625ic) + 0.25 ic = 1
0.25ic + 0.25 ic = 1
0.5 ic = 1
ic = 2 ....(i)
which is the optimal consumption of iced coffee
and,
t = 0.625(2)
= 1.25
which is the optimal consumption of Twinkies.
(d)
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