MEK 781: Decentralized Dynamic General Equilibrium Problem - Economics Assignment Help

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Task:

Given the following decentralized dynamic general equilibrium problem. The consumer maximizes discounted lifetime utility X1 s=t st u(cs; xs) subject to the following budget constraint ct = wtlt+r k t ktkt+1+(1k)kt(bt+1 (1 + rt) bt)+t; whereby variables have their usual meaning and moreover the consumer can invest in bond whereby bt+1 represents one-period real riskless bond, purchased in period t and maturing in period t + 1 (if bt > 0 means the consumer has a positive stocks of savings and if bt < 0 means the household has a positive stock of debt). The interest rate rt for instance is the interest rate paid on existing debt, acquired in period t1 and rt+1 represents interest rate to be paid next period on debt that the consumer chooses to take today into tomorrow. t represents proÖt (from owing Örm) distribution in the form of dividends. Let current period utility be of log form u(ct; xt) = ln ct + ln xt, where ct and xt are consumption and leisure respectively. Let the production function be of Cobb-Douglas form: yt = Atl t k 1 t where A, lt, and kt are productivity process, labour and capital respectively, consumption and investment are expected to exhaust output yt = ct +it; the law of motion of capital is given by kt+1 = (1 k)kt + it where k is the depreciation rate of capital and time endowment is shared between work and leisure 1 = xt + lt. Assume that the economy is growing at zero balanced growth path (BGP). a) Set the consumerís inÖnite utility maximization problem in recursive form using the value function. [Hint: bond is acting as an additional state varia b) Solve for the Örst order equilibrium conditions. c) Solve for the steady states of all the variables [Hint: note that in steady state, b = 0, meaning that in steady state net supply/demand of bond (debt) is zero. Moreover in steady state, A = 1]. d) Assume the following calibrated values, namely, = 0:97; k = 0:03; = 0:5; = 1 3 ; A = 1.

There are various ways to calibrate the parameters pertaining to the technology process At, namely its mean A, the persistence in the process a and the volatility of the shock and its standard deviation a. Assume that it follows a zero-mean AR(1) process in the log such that log (At) = (1 a ) log A + a log (At1) + "t reduces to log (At) = a log (At1) + "t. Having a direct measure of the shock is di¢ cult as we do not have a technology shock in the data. Actually the technology process is not observable. Two solutions have been proposed in the literature to deal with this calibration. The Örst one is to set the volatility (standard deviation) of the shock, a, so as to match the volatility of output, and to set the persistence, a , so as to match that of output. A second, more direct, approach builds a time series for the technology shock and directly estimates the process, using the Solow residual. It is this approach that you will follow. [Hint: follow notes ëThe Real Business Cycle Modelíby Harris Dellas at University of Bern, page 33 onward of the document]. The relevant data is in the Excel Öle and you can choose any software to do the regression. However, the regressions results should be clearly tabled and data transformation should be clear. In obtaining the parameters a and a, the relevant regressions and formula should be clearly stated. You will use the values for the standard deviation of the shock (a) and persistence parameter (a ) as the calibrated parameters in the codes. e) List the set of equations to go in Dynare clearly. Code the above model in Dynare in ëlevelíand let Dynare do a linearization around the level of the variables. Interprete the impulse response functions clearly to a one standard deviation shock.[Hint: provide pdf of code an not the code, in respective parts of the question] f) List the set of equations to go in Dynare clearly.

Code the above model in Dynare by providing the log of the variable to Dynare so that Dynare does a loglinearization. Interprete the impulse response functions clearly to a one standard deviation shock and mention what is the di§erence in terms of interpretation of the magnitude of responses compared to part e) above. [Hint: If you denote the level of the variable in the code in part e) as ëyí, then denote ëlyí as the log of the variable in part f)] g) Log-linearise the full model using the following formula as a guide: Et ff (X)g Et f X + @f (X) @X X jx=x x^ where X denotes the variable and X its steady state and log (X) = x and x^ denotes the log deviation from steady state, ie, x^ = log (X)log X = xx: [Hint: see page 30 of notes ëThe Real Business Cycle Modelíby Harris Dellas for full explanation of this method of log-linearizing and lecture notes and videos] h) Lastlty code part g) in Dynare and explain whether your results should match part e) and f) ab

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