Please answer the following questions

.1 u .-.. ' "- 1. Consider the following short-run model: C: = 53?; I _ .. Y: = \"I -' b(Rf H T Ext = aear?! + 7(Yl _ Y!) 1M! = ile where C} is consumption at time t, It is investment, G, is govermnent purchases, EX, and IM, are exports and imports, Y, is potential output, 1', is short-run output, Rf is the real interest rate, ? is the marginal product of capital, and g, 5,, g, Em aim, and r E (0,1) are parameters. (a) Derive the equation for the IS-curve (that is, the relationship between short- run output and the real interest rate). (b) Let 7 = 0. Draw the IS-diagrarn and show graphically what happens if the marginal product of capital increases. (c) Let "iv = 0. Draw a new ISdiagram and show graphically what happens if 5 increases. (:1) Let '7 = 0. Suppose there is a sudden positive change to consumers' optimism about the state of the economy. Explain and show mathematically what would happen to the lScurve in that situation. (e) Explain why the sensitivity of short-run output with respect to the positive change in consumers' optimism about the state of the economy is different when 7 E (0, 1) compared to when I? = 0