# A consumer is willing to trade 3 units of x for 1 unit of y when she has 6 units of x and 5 units of y. She is also willing to trade in 6 units of x for 2 units of y when she has 12 units of x and...

A consumer is willing to trade 3 units of x for 1 unit of y when she has 6 units of x and 5 units of y. She is also willing to trade in 6 units of x for 2 units of y when she has 12 units of x and 3 units of y. She is indifferent between bundle (6, 5) and bundle (12, 3). What is the utility function for goods x and y? Hint: What is the shape of the indifference curve?

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Hello!

**1**. Because a consumer is indifferent between bundles (6, 5) and (12, 3), these two points are on the same indifference curve.

**2**. How much units a consumer will to trade at a point (x, y) tell us the slope of this indifference curve at this point. And the slope of a curve at some point is the same as the slope of the tangent line to the graph of this function at this point.

Specifically, the slope at the point (6, 5) is -1/3 (3 units of x ~ 1 unit of y),

and the slope at the point (6, 5) is -1/3 also (6 units of x ~ 2 units of y).

**3**. It is known that an indifference curve is convex towards origin (0, 0) or at least weakly convex (i.e., is a straight line on some interval(s)). And note that "convex towards origin" means "concave upwards".

**4**. A concave upwards graph lies over a tangent line and under a secant line.

The equation of the tangent line at (6, 5) is `y = -1/3 x + 7.`

The equation of the tangent line at (12, 3) is `y = -1/3 x + 7.`

And the equation of the secant line between (6, 5) and (12, 3) is, surprise, `y = -1/3 x + 3` also.

So at least between x=6 and x=12 the indifference curve lies over and under the line 3y+x=21, thus coinciding with this line. So the utility function is

`U(x,y)=x+3y.`

Suppose that a person has initial amounts of the two goods that provide utility to him or her. These initial amounts are given by .

a. Graph these initial amounts on this person’s indifference curve map.

b. If this person can trade x for y (or vice versa) with other people, what kinds of trades would he or she voluntarily make? What kinds would not be made? How do these trades relate to this person’s MRS at the point

c. Suppose this person is relatively happy with the initial amounts in his or her possession and will only consider trades that increase utility by at least amount k. How would you illustrate this on the indifference curve map?