Question

Consider the following GDP equation, consumption function, exports function. imports function and investment function and use them to answer the following questions:

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Y_t = C_t + I_t +G_t + EX_t - IM_t

[I_t]/[Y[bar]_t] = a[bar]_i - b[bar]*(R_t - r[bar]); b[bar] > 0

               C = a[bar]_c* Y[bar]_t

                  G = a[bar]_g * Y[bar]_t

               EX = a[bar]_ex * Y[bar]_t

               IM = a[bar]_im * Y[bar]_t

A) Write down the formula that defines short-run output Y[median]_t in terms of actual output Y_t and potential output Y[bar]_t .

B) Derive the IS curve as a relation between short run output Y[median]_t and the real interest rate gap R_t − r[bar]. Please show all the steps.

C) The US Congress is working on passing President Biden's $1.9 trillion covid relief bill. Out of this $325 billion worth of expenditure is in supposed to be in terms of public infrastructure projects. This amount is 1.5% of potential GDP. That is a[bar]_g would increase by 0.015 if this package were to pass the Congress. Assume Ricardian equivalence does not hold and that the government finances this expenditure by borrowing, and not through taxes in the short-run. Show graphically how does this change in a[bar]_g affect the IS curve. What would happen to short-run output in this model if a[bar]_g increases by 0.015, keeping fixed real interest rate R_t? Is there a multiplier in this model? Why or why not? 

D) Congress is working on a proposal to send $520 billion in direct $1400 checks to households. According on the information presented in part C), calculate the size of this package as a percentage of potential output?