Derive a Solow-Growth model and describe the intuition of the equation.

There are 4 questions on this exam.
• The total score for the Long-answer part of the exam is 40, so the score for each question is
10.
• You can use the textbook or google, but the answer must be in your own word. If I find that
more than two students have the same answer, I have to give you a zero on that question.
• You must show your work. If I don’t see any explanation for your answer, you will lose a lot
of credit. On the other hand, if you have a wrong answer because of a miscalculation even if
you take the right steps, you will get almost full credit.
• Good Luck!
Question 1. Suppose we have the following short-run model. Resource constrain is
Yt = Ct + It + Gt + EXt − IMt
Where Yt
is economy output, Ct
is consumption. It
is investment. Gt
is government purchase. EXt
is exports. IMt
is imports. Government purchase depends on the current state of the economy. For
example, when there is an economic recession, the government will spend more money to support
the economy. To incorporate this intuition, we will use the following equations. Also, we will
incorporate the consumption multiplier effects.
Ct
Y¯
t
= a¯c + x¯cY˜
t
Gt
Y¯
t
= a¯g + x¯gY˜
t
EXt = a¯exY¯
t
IMt = a¯imY¯
t
It
Y¯
t
= a¯i − ¯b(R¯
t − r¯)
a. Derive IS curve, and draw IS curve. (x-axis: Y˜ , y-axis: R(real interest rate)) What are the
slope and the y-intercept?
b. Suppose the multiplier effects for the government purchase changed(xg increase). What is the
impact of this change?
1
c. Assume, MP(monetary policy) curve is a horizontal line. Suppose, there is a negative demand
shock on consumption. (ac decrease) What is the impact of this shock? What is the response
of the federal reserve? Use the IS-MP curve graph to explain the impact of this shock.
d. Continue with the previous question. Explain how the change in inflation (∆π) changes when
there is a negative shock on consumption. (Use Philips Curve)
Question 2. Consider following Philips Curve.
∆πt = ν¯Y˜
t + o¯
a. Draw Philips curve. (x-axis: Y˜
t
, y-axis: ∆π) What is the slope and y-intercept of the curve?
b. What is the impact of the ν¯ change? Should government perform a more aggressive policy to
stabilize the inflation rate with high ν¯ or low ν¯? If so, why?
c. Suppose there is an inflation shock because a war increased oil prices. (o¯>0) What is the
impact of the oil price shock? (Use Philips curve graph)
Question 3. Consider policy rule, IS curve, and Philips curve
P olicy rule ∶ Rt − r¯ = m¯ (πt − π¯)
IS curve ∶ Y˜
t =
1
1 − x¯c
[a¯ − ¯b(Rt − r¯)]
P hilips curve ∶ πt = πt−1 + ν¯Y˜
t + o¯
a. Derive Aggregate Demand(AD) and Aggregate Supply(AS) curve, and draw AD and AS curve.
What is the slope and y-intercept for AS and AD curve?
b. Find steady state inflation, π
∗
, and output Y˜ ∗
c. Suppose there is a positive and temporary demand shock, because of the new technology
development. (a¯>0) What is the impact of this shock?
Question 4. Consider a production function
Yt = F(Kt
, Lt) = AK¯ α
t L
1−α
t
(1)
and the resource constraint
Yt = Ct + It + Gt (2)
and the capital accumulation equation
Kt+1 = It + (1 − ¯d)Kt (3)
Consumers consume a certain fraction of the output so the consumption equation is
Ct = (1 − s¯)Yt (4)
The government spending Gt
is a fraction of capital stock, so with the higher capital stock, there
is more government spending.
Gt = gK¯ t (5)
Assume there is no population growth, so Lt = Lt+1 = L¯
2
a. Derive a Solow-Growth model and describe the intuition of the equation.
b. What is the key assumption in this model
c. Find the steady state per-worker quantities of capital, output, and consumption
d. Draw the Solow model (the x-axis is Capital stock, the y-axis is output)
e. Suppose there was a big government spending. Therefore, g¯ increased. What is the new
steady state per-worker quantities of capital, output, and consumption?