Question

In: Economics

A consumer’s utility is given by ?(?, ?) = ? ^1/2? ^1/4 where Z is the...

A consumer’s utility is given by ?(?, ?) = ? ^1/2? ^1/4 where Z is the consumption of pizza and C is the consumption of coffee. Let pZ and pC denote the prices of pizza and coffee, and let m be the consumer’s income.

(a) State the “optimality condition” characterizing the best affordable bundle.

(b) Derive the consumer’s demands for pizza and coffee. You must show your work.

(c) What is the consumer’s optimal consumption if ?? = ?? = 1 and m=120? ? ∗ =_________ ? ∗ =_________

(d) Suppose that the price of coffee rises to ?? = 2 and at the same time income increases to m=160? Is the bundle (? ∗ , ? ∗ ) in (c) still affordable? Is the consumer off or worse off as a result of the change in price and income?

Expert Solution

Consumer's utility function;

U(z,c) = z^1/2 c^1/4

where, z = pizza
c = coffee

Prices of pizza = P_z
Prices of coffee = P_c

Consumer's income = M

(a) The optimal level is achieved when the slope of indifference curve (MRS) is equal to the slope of budget line (P_z/P_c);

MRS = P_z/P_c

Marginal rate of substitution (MRS) : It is the rate at which a consumer is willing to give up some units of one good in order to gain one more unit of another good. It is calculated as;

MRS = MUz / MUc

MUz = d/dz (z^1/2 c^1/4)
MUz = c^1/4 / 2z^1/2

MUc = d/dc(z^1/2 c^1/4)
MUc = z^1/2 / 4c^3/4

MRS = (c^1/4 / 2z^1/2) / (z^1/2 / 4c^3/4)
= 4c / 2z
MRS = 2c/z

At optimal level;

MRS = P_z/P_c
2c/z = P_z/P_c
2c = zP_z/P_c
c = zP_z/2P_c

(b) Consumer's demand will be;

c = zP_z/2P_c

Putting in the budget constraint;

P_zz + P_cc = M
P_zz + P_c (zP_z/2P_c) = M
P_zz + P_zz/2 = M
2P_zz + P_zz = 2M
3P_zz = 2M
z = 2M/3P_z

c = (2M/P_z)P_z / 2P_c
c = 2M / 2P_c
c = M/ P_c

The demand function for pizza will be; z = 2M/3P_z

The demand function for coffee will be; c = M/ P_c

(c) Given;

Prices of pizza; P_z = 1
Prices of coffee; P_c = 1

Consumer's income; M = 120

So, optimal bundle will be;

c = M/ P_c
c = 120/1
c = 120

z = 2M/3P_z
z = 2*120 / 3
z = 80

(d) Now, price of coffee rises to; P'_c = 2
Consumer's income increases to; M' = 160

Now, the optimal consumption bundle will be;

c = M'/ P'_c
c = 160/2
c' = 80

z = 2M'/3P_z
= 2(160) / 3(1)
= 320/3
z' = 106.67

Utility before change in price and income;

U(z,c) = z^1/2 c^1/4
U(120,80) = 120^1/2 80^1/4
U(120,80) = 13.94

Utility after change in price and income;

U(z,c) = z^1/2 c^1/4
U(80,106.67) = 80^1/2 106.67^1/4
U(80,106.67) = 12.16

As we can see, after the changes in price and income consumer's utility decreases from 13.94 to 12.16. Thus, we can say that consumer is worse off.

Rahul Sunny answered 1 year ago

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