Answer to Question #273337 in Microeconomics for cookie

Question

Robinson's preferences between apples (a) and bananas (b) are expressed by the following:

 

U = (a+2)^0.5(b+1)^0.5

 

(a) Show that Robinson's indifference curves are negatively sloped.

(b) Are they convex to the origin? Explain.

Answer 1

Robinson's interference curves

a. From $U = U = (a + 2)^{0.5} \times(b+ 1)^{0.5}$

Where U Is the utility, a the quantity of apples consumed and b the quantity of bananas consumed.

Considering three different utilities I.e U ={20,30,40}

 $U = (a + 2)^{0.5} \times(b+ 1)^{0.5}$

a = U²/(b+ 1)

U = 20

a = 400/(b+ 1)

U = 30

a = 900/(b+ 1)

U = 40

a = 1600/(b+ 1)

Plotting all the three results gives the utility curve below.

From these curves, as Robinson increases on the consumption of bananas, he gives up the number of apples he consumes in order to achieve the same levels of utility. Thus making the indifference curves negatively sloped with higher curves representing Higher levels of satisfaction.

b. The curves are convex in origin, because as Robinson begins to substitute bananas for apples, the marginal rate of substitution diminishes as a for b along the indifference curve. The slopes of the indifference curves are referred to as the marginal rate of substitution, which is the rate at which Robinson sacrifices one commodity in order to consume more units of another commodity.

As Robinson moves from left to right along the indifference curves the willingness to substitute bananas for apples diminishes, the indifference curve is steeper closer to the y axis and is gradual towards the x axis thus making it convex in origin.