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...

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...

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.`