Question

An Individual consumes commodity bundles including two goods: P and Q. The satisfaction/value/utility obtained for the individual is provided by the following equation – 3*(units of P)^2 + 2*(units of Q). Now answer the following questions

a. Compute the utility got the following bundles – (12, 5), (2,7) and (8,5)
b. Considering a utility level of 75 units; identify one consumption bundle that provided that level of utility
c. Now draw an indifference curve for the individual with an utility of 75
d. Provided the unit cost of P is 12$ and unit cost of Q is 3$ - determine the cost of the consumption bundle you identified from part (b)
e. For 75 units of utility, given the units costs, find the optimal consumption bundle (find the bundle on the indifference curve that has the lowest cost)

Answer 1

Utility U(P,Q) = 3*P² + 2*Q

A)

U(12,5) = 3*12² + 2*5 = 3*144 + 10 = 442 units

U(2,7) = 3*2² + 2*7 = 3*4 + 14 = 26 units

U(8,5) = 3*8² + 2*5 = 3*64 + 10 = 202 units

B)

Utility level 75 units.

Thus we have to identify a consumption bundle(P,Q) such that U = 75

3*P² + 2*Q = 75

There are many possible values of P and Q that solves this equation. One such bundle is P=3 and Q = 24. In this case U = 3*9 +2*24 = 75

C)

Indifference curve for U = 75 shows all possible combinations of P and Q that give utility of 75.

Bundle A: P=0, Q=37.5, U = 75 [Put the values of P and Q in the utility function]

P,Q	3*P2	2*Q	U=75
A = 0,37.5	0	37.5*2	75
B = 1,36	3*1	2*36	75
C = 2,31.5	3*4	31.5*2	75
D = 3,24	3*9	2*24	75
E = 4,13.5	3*16	13.5	75
F = 5,0	3*25	0	75

Similarly, there can be many possible combinations between them. By tracing all such points the indifference curve for U=75 can be drawn.

MRS = MU_p/MU_q

MU_{p =}

MU_q =

MRS = 3P

We can see that the indifference curve is concave. Since the slope of IC equals MRS, which keeps on increasing as we move along the IC, thus IC is concave

D)

Unit cost P = $12

Unit Cost Q = $3

Consumption Bundle: P=3, Q=24

Cost of bundle = 3*12+24*3

= $108

E)

Since the indifference curve is concave, there will be corner solution i.e the consumer will consume only one good. Thus the solution will be either P=0 and Q =37.5 or P=5 and Q = 0.

Cost of P=0 and Q =37.5 = 0*12 + 37.5*3 = $112.5

Cost of P=5 and Q = 0. = 5*12 + 0*3 = $60

Thus the optimal consumption bundle P=5 and Q=0

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