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Department ofEconomics

EC206 Intermediate Mathematics for Economics

Spring Term, 2021-22

Take-home Exercise

Consider the IS-LM closed-economy model, in which the following behavioural equations fully describe the goods market:

≡ + +

= 0 + 1

= 0 + 1 − 2

= 0 + 1

where defines aggregate demand, consumption, investment, government spending, is the nominal interest rate set by the central bank, defines taxes (net of transfers), and is disposable income, defined by = − ; and 0, 0, 0 > 0 the autonomous (independent on income) parts of consumption, investment, and taxes, respectively, 1 ∈ (0, 1) the marginal propensity to consume, 1 ∈ (0, 1) the impact of output on investment, 1 ∈ (0, 1) a tax rate on income, and 2 > 0 the interest sensitivity of investment. It is also assumed that 1 + 1 < 1. The financial markets are described by the following equation:

= 1 − 2

defines real money balances, so is money and the price level in the economy, assumed to

be constant; and 1 ∈ (0, 1) the impact of output on the demand for real money balances and 2 > 0 the interest sensitivity of the demand of real money balances. Further, assume that = 0 and = , 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., = and

= define the endogenous variables and the exogenous variables and exogenously

given parameters in this model. [Hint: Treat the real money balances, , as one variable; so, do not consider money, , and the price level, , as two different variables. Moreover, both demand, , and disposable income, , should not be considered as variables; simply substitute them at the beginning, and then forget about them. It is output, , 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 Cramer’s rule. [Hint:

When you derive the equilibrium solutions, try to gather terms in 0, , 0, 0, 0 .]

(60 marks)

(iv) Assuming the specific numerical values that you can see in the following Table for the

exogenously-given parameters of our model, present the equilibrium solutions of the endogenous variables that you derived before as functions of the exogenous variables, alone; namely, the two policy instruments being set by the government and the central bank, respectively.