Consider a closed economy in which the depreciation rate is 10% per year, the rate of population increase is 2% per year, the rate of technological progress is 1% per year Andy the households save 30% of their income. Suppose the aggregate production function is; Y=f(K, AL)= 9K^4/5 (AL)^1/5 Where Y is output, K is capital, A is the level of technology and L is labor input.

a) Derive the production function in per effective worker terms.

b) Solve for the steady-state values of capital per effective worker (k*), output per effective worker (y*), consumption per effective worker (c*), and savings per effective worker (s*).

c) Derive the equation for the growth rate of output per worker, the growth rate of capital per worker, the growth rate of output and growth rate of capital. What are the values of the growth rates derived?

In a closed economy, the consumption function is:
c = 1.15 + 0.75(y - t) billions of 1992 dollars.
The tax function is:
t = 0.1y + 0.1 billions of 1992 dollars.
Planned investment is $1 billion and planned government expenditures
are $1.5 billion. Calculate:

Explain the adjustment process to the new equilibrium. (3 points)
How does the government’s expenditures on $0.25 billion worth of
goods and services get financed? (3 points)
Show that leakages from the circular flow equal injections into it. (3
points)
Go back to the initial equilibrium expenditure. The government plans
to increase expenditures on goods and services by $0.25 billion and to
finance its expenditure by an increase in autonomous taxes. What is
the new equilibrium expenditure? (5 points)

Consider a closed economy’s market for loanable funds.

a) Write down the equation that represents the equilibrium condition for this market. What is the “price” variable in this market? How does it adjust to ensure equilibrium in this market?

b) Use the demand-supply diagram to illustrate the market for loanable funds, and make sure you label both axes and identify the equilibrium point on the graph. Use the graph to illustrate how an increase of government spending will affect savings, investments, real interest rate, and output. Explain crowding out effect using your results.

Give Examples (this is complex analysis):

(a.) First characterize open and closed sets in terms of their boundary points. Then give two examples of sets satisfying the given condition: one set that is bounded (meaning that there is some real number R > 0 such that |z| is greater than or equal to R for every z in S), and one that is not bounded. Give your answer in set builder notation. Finally, choose one of your two examples and prove that is neither open nor closed.

(b.) Give two examples of a function f: C→C that is continuous at z=0 but not differentiable at z=0 using the Cauchy-Riemann equations.

(c.) Find a cube root of -1, other than -1, in two ways: first, by using high school algebra (solve the equation z^3= -1 by factoring the polynomial z^3+1 as z+1 times a quadratic polynomial and then determine the roots of the quadratic polynomial) and second, by using the formula for computing nth roots of a complex number.

Water is contained in a closed, rigid, 0.2 m³ tank at an initial pressure of 5 bar and a quality of 50%. Heat transfer occurs until the tank contains only saturated vapor. Determine a)the final mass of vapor in the tank, in kg,b) the final pressure, in bar.

Consider the following endothermic equilibrium system in a closed container. In which direction will the equilibrium shift with each of the disturbances listed below? HCO3-(s) + HC2H3O2 (aq) ↔ C2H3O2-(aq) + H2O (aq) + CO2 (g)

a. adding CO2 (g) .

b. adding NaOH (aq) .

c. adding H+ (aq) .

d. placing it in an ice bath.

e. opening the container.

N atoms of an ideal gas are contained in a cylinder with insulating (adiabatic) walls, closed at one end by a piston.The initial volume is Vi and the initial temperature is Ti. Find the change in temperature, pressure and entropy that would occur if the volume were suddenly increased to Vf by withdrawing the piston (Vf > Vi).

Calculate the mole fraction when a sample of propane (C3H8) is placed in a closed vessel together with an amount of O2 that is 2.15 times the amount needed to completely oxidize the propane to CO2 and H2O at constant temperature. Assume all chemical species are in the gaseous phase.

Consider a closed economy in which the population grows at the rate of 1% per year. The per-worker production function is y = 6 * ((K)^0.5), where y is output per worker and k is capital per worker. The depreciation rate of capital d is 14% per year.

a. Households consume 90% of income and save the remaining 10% of income. There is no government. What are the steady-state values of capital per worker, output per worker, consumption per worker, and investment per worker?

b. Suppose that the country wants to increase its steady state value of output per worker. What steady-state value of the capital-labor ratio is needed to double the steady-state value of output per capita? What fraction of income would households have to save to achieve a steady-state level of output per worker that is twice as high as in part (a)?