Consider the following macroeconomic model:
Y = C + I + G0
Question 1 [25 points]
a) Define concavity and convexity of a generic univariate function.
C = a + b(Y - T)
T = d + tY
Where the endogenous variables are Y, C, and T, while the exogenous variables are G and I. The parameters are such that a > 0, d > 0, 0 < b < 1, and 0 < t < 1.
b) Provide an economic example of a concave function, show that it is concave, and discuss its economic intuition. [5 points]
c) Set up the model in matrix form. [5 points]
d) Find the inverse of the matrix of parameters. [10 points]
e) Consider the following function:
f(x) = -x^3 + 4x^2 + 5x^2
Discuss its concavity and convexity over the domain x > 0. [10 points]
f) Use Cramer's rule to find equilibrium income Y* and equilibrium taxes T*. [5 points]
g) Find and discuss the impact of a rise in government spending G on equilibrium income Y*. [5 points]
h) Compute the following definite integral:
∫(2x^3) dx [10 points]
i) What does t represent in the model? What is the impact of a rise in t on equilibrium income Y* and on equilibrium taxes T*? Explain. [Suggestion: to make computation easier, feel free to group together some variables, e.g. call k = I + G] [10 points]
