Question

Suppose the utility function is given by U(x₁, x₂) = 14 min{2x, 3y}. Calculate the optimal consumption bundle if income is m, and prices are p₁, and p₂.

Answer 1

U(x,y) = 14 min{2x, 3y}

Since the utility fucntion remains unchanged under monotonic increasing transformation, so we can write the utility function as U(x,y) = min{2x, 3y}

The price of x is p1 and y is p2. The consumer has a income m. Thus the budget constraint is p1*x+p2*y=m

Since the utility function is the min function, for the consumer to be in equilibrium he/she should derive the same amount of utility from x and y. If does not get the same utility from x and y then the consumer will get lower amount that is 2x or 3y which ever is less.

Thus for equilibrium 2x=3y which implied x=1.5*y.

Substituting the above x in the budget constraint of the consumer p1*x+p2*y=m, we get p1*(1.5*y.)+p2*y=m. Solving for y, we get the optimal consumption of y as y=m/(1.5*p1+p2). Thus the optimal consumption of x is x=1.5*y=1.5*m/(1.5*p1+p2)