We need to show that when the marginal rates of substitution for the two individuals are equal (MRS=MRS)the allocation lies on the contract curve For example,consider the utility function =yThen If MRS'equals MRS,then - Is this point on the contract curve?Yes,because =X.x andy2=Y-y1 四-) This means that 专 X -%=r-%义=gc男-（x With this utility funetion we findMRS=MRSand the ontract curve isa straight line.However,if the two traders have identical preferences but different incomes,the contract curve is not a straight line when one good is inferior. 7.Give an example of conditions when the production possibilities frontier might not be concave. The production possibilities frontier is concave if at least one of the production functions exhibits decreasing turn. If both production functions exhibit co nstant retu ,then the productio possibilities frontier is a straight line.If both production functions exhibit increasing returns to scale,then the production function is convex.The following numerical examples can be used to illustrate this concept. Assume that L is the labor input,and X and Y are the two goods.The fir example is the g returns to scale ca the second e ample is th constant returns to scale case,and the third example is the increasing returns to scale case.Note further that it is not necessary that both products have identical production functions. We need to show that when the marginal rates of substitution for the two individuals are equal (MRS1 = MRS2 ), the allocation lies on the contract curve. For example, consider the utility function U = xi yi 2 . Then MRSi = MUx i MUy i = 2xi yi xi 2 = 2yi xi . If MRS1 equals MRS2 , then 2 1 y x1       = 2 2 y x2       . Is this point on the contract curve? Yes, because x2 = X - x1 and y2 = Y - y1 , 2 y1 x1       = 2 Y − y1 X − x1       . This means that y1 X − x ( 1) x1 = Y − y1 , or y1 X − y1 x1 x1 = Y − y1 , and y1 X x1 − y1 = Y − y1 , or y1 X x1 = Y, or y1 = Y X     x1 . With this utility function we find MRS1 = MRS2 , and the contract curve is a straight line. However, if the two traders have identical preferences but different incomes, the contract curve is not a straight line when one good is inferior. 7. Give an example of conditions when the production possibilities frontier might not be concave. The production possibilities frontier is concave if at least one of the production functions exhibits decreasing returns to scale. If both production functions exhibit constant returns to scale, then the production possibilities frontier is a straight line. If both production functions exhibit increasing returns to scale, then the production function is convex. The following numerical examples can be used to illustrate this concept. Assume that L is the labor input, and X and Y are the two goods. The first example is the decreasing returns to scale case, the second example is the constant returns to scale case, and the third example is the increasing returns to scale case. Note further that it is not necessary that both products have identical production functions