# 宏观经济quiz代考 ECON 602 Assignment 3

## ECON 602 Assignment 3

### The Optimal Growth Model with Labor and Technology Growth 宏观经济quiz代考

Consider an optimal growth model with a production function

where Kt is the aggregate capital and Lt is the labor. Capital depreciates at rate 1 and the representative consumer’s preference is given by

where ctis the consumption per capita. The model also includes

 National income identity:

where Itis the aggregate investment. 宏观经济quiz代考

 Capital accumulation:

 Labor dynamics:

where n is the growth rate of population.

 Technology dynamics:

where na is the technology growth rate.
(1) Please find the expression for the per effective unit of labor form of the production function. (8 points)

(2) Please derive the budget constraint of the representative consumer at time t. (8 points)  宏观经济quiz代考

(3) Please write down the maximization problem of the representative consumer. (8 points) the maximization problem

(4) Please write down the Lagrangian and derive the first-order conditions. (8 points) The Lagrangian function is 宏观经济quiz代考

where λt is the Lagrangian multiplier. The first order conditions with respect to ct and ˜kt+1 are

(5) Please derive the Euler’s equation and discuss its implication. (8 points) Equations (2) and (3) give the Euler equation

#### The Euler equation implies the representative consumer must be indifferent between consuming today and consuming tomorrow. If this were not true, the consumer could increase utility by either consuming or saving some more for tomorrow. In other words, arbitrage is not allowed at an optimum.

(6) Please find the modified Golden rule. (8 points)
The Euler equation also implies that the competitive equilibrium steady state is characterized by 宏观经济quiz代考

(7) Suppose the consumption per effective unit of labor ˜ct and the consumption per effective unit of labor ˜kt+1 are proportional to the output per effective unit of labor ˜k α . Please find the closed form solutions for ˜ct and ˜kt+1. (You may define the relevant parameters by yourself.) (8 points)
We conjecture that ˜ct and ˜kt+1 will be proportional to ˜kα. Hence,

(9) Suppose the consumer discounts the future utility at rate 0.74. The growth rate of technology is 0.02 and the growth rate of labor is 0.02 as well. The economy starts with ˜k0 = 1, A0 = 10, and L0 = 4. Please show the response function of aggregate output when the consumer gets impatient. You may discuss what the impatience means here and formulate the question. (8 points)

Simulation based on the Excel; The question is how a decrease in β influences the economy.  宏观经济quiz代考

(10) Does the impatience affect the endogenous saving rate? Please show it using math or the excel and talk about its reason. (8 points)
The saving rate is αβ. The impatience means a smaller β and it will decrease the saving rate. The impatient consumer prefers current consumption so that she would like to save less.

### The RBC Model

(1) The RBC model suggests that the business cycle is triggered by the supply shock - technology and we did generate the business cycle based on the basic RBC model in the class. Moreover, the RBC model implies the co-movement among several macroeconomic variables. Please discuss why we should not investigate the co-movement based on the observed time series. (10 points)

The observed time series may share the similar time trend so that it looks that all of them are moving together. One has to remove the time trends and should conduct the research based on the deviations from the time trends - the cyclical components. 宏观经济quiz代考

(2) Please find the correlation coefficient between real GDP and the employment rate for the U.S. based on the filter you like. Please submit your codes as well. (10 points) (10 points)

open empirical analysis