Consider the IS-LM closed-economy model, in which the following behavioural equations fully describe the goods market: &
Posted: Sat Feb 19, 2022 2:34 pm
Consider the IS-LM closed-economy model, in which the following behavioural equations fully
describe the goods market:
π β‘ πΆ + πΌ + πΊ
πΆ = π0 + π1π
π·
πΌ = π0 + π1π β π2π
π = π‘0 + π‘1π
where π defines aggregate demand, πΆ consumption, πΌ investment, πΊ government spending, π is the
nominal interest rate set by the central bank, π defines taxes (net of transfers), and π
π· is disposable
income, defined by π
π· = π β π; and π0, π0,π‘0 > 0 the autonomous (independent on income) parts of
consumption, investment, and taxes, respectively, π1 β (0,1) the marginal propensity to consume, π1 β
(0,1) the impact of output on investment, π‘1 β (0,1) a tax rate on income, and π2 > 0 the interest
sensitivity of investment. It is also assumed that π1 + π1 < 1. The financial markets are described by
the following equation:
ππ
πΜ
= π1π β π2π
where π
πΜ
defines real money balances, so π is money and πΜ the price level in the economy, assumed to
be constant; and π1 β (0,1) the impact of output on the demand for real money balances and π2 > 0
the interest sensitivity of the demand of real money balances. Further, assume that πΊ = πΊ0 and π = π, Μ
both given, being set by the government and the central bank, respectively.
(i) Assuming simultaneous equilibrium in the goods and the financial markets, i.e., π = π and
ππ
πΜ
=
π
πΜ
, define the endogenous variables and the exogenous variables and exogenously
~ 2 ~
given parameters in this model. [Hint: Treat the real money balances,
π
πΜ
, as one variable;
so, do not consider money, π, and the price level, πΜ , as two different variables. Moreover,
both demand, π, and disposable income, π
π·, should not be considered as variables; simply
substitute them at the beginning, and then forget about them. It is output, π, that we are
interested in.]
(10 marks)
(ii) Write the system of equations in matrix form, and define the coefficient matrix, the matrix
of endogenous variables and the matrix of the exogenous variables and the exogenously
given parameters.
(20 marks)
(iii) Solve for all the endogenous variables at equilibrium by the use of Cramerβs rule. [Hint:
When you derive the equilibrium solutions, try to gather terms in πΊ0, π,Μ π0, π0,π‘0.]
describe the goods market:
π β‘ πΆ + πΌ + πΊ
πΆ = π0 + π1π
π·
πΌ = π0 + π1π β π2π
π = π‘0 + π‘1π
where π defines aggregate demand, πΆ consumption, πΌ investment, πΊ government spending, π is the
nominal interest rate set by the central bank, π defines taxes (net of transfers), and π
π· is disposable
income, defined by π
π· = π β π; and π0, π0,π‘0 > 0 the autonomous (independent on income) parts of
consumption, investment, and taxes, respectively, π1 β (0,1) the marginal propensity to consume, π1 β
(0,1) the impact of output on investment, π‘1 β (0,1) a tax rate on income, and π2 > 0 the interest
sensitivity of investment. It is also assumed that π1 + π1 < 1. The financial markets are described by
the following equation:
ππ
πΜ
= π1π β π2π
where π
πΜ
defines real money balances, so π is money and πΜ the price level in the economy, assumed to
be constant; and π1 β (0,1) the impact of output on the demand for real money balances and π2 > 0
the interest sensitivity of the demand of real money balances. Further, assume that πΊ = πΊ0 and π = π, Μ
both given, being set by the government and the central bank, respectively.
(i) Assuming simultaneous equilibrium in the goods and the financial markets, i.e., π = π and
ππ
πΜ
=
π
πΜ
, define the endogenous variables and the exogenous variables and exogenously
~ 2 ~
given parameters in this model. [Hint: Treat the real money balances,
π
πΜ
, as one variable;
so, do not consider money, π, and the price level, πΜ , as two different variables. Moreover,
both demand, π, and disposable income, π
π·, should not be considered as variables; simply
substitute them at the beginning, and then forget about them. It is output, π, that we are
interested in.]
(10 marks)
(ii) Write the system of equations in matrix form, and define the coefficient matrix, the matrix
of endogenous variables and the matrix of the exogenous variables and the exogenously
given parameters.
(20 marks)
(iii) Solve for all the endogenous variables at equilibrium by the use of Cramerβs rule. [Hint:
When you derive the equilibrium solutions, try to gather terms in πΊ0, π,Μ π0, π0,π‘0.]