# economics / mathematics

## Project Description:

1. suppose that 10 years ago you purchased a car at \$27,000 and the car was traded in today for \$1,000. what is the depreciation rate? suppose that the car is continuously discounting its value. (show your work. this means that no work = no credit).

2. consider the production function, q = 15l3/4k1/4, where q is output, l is labor input, and k represents capital input.

2a) using natural logarithms, transform this exponential function into a linear function.
2b) now assume that l = 10 and k = 5. what is the value of ln(q)? remembering that exp(ln(q)) = q, determine the value of q.

3. given the following system equations of price (p) and quantity (q) determination in a widget market:

demand: q = 120 - 20p + 3g …..(1)
supply: q = 40 + 20p – 2n ……(2)

where the price of substitute good, g = 200, and the cost of production n =100.

3a) by using the repeated substitution method, please find p and q.

3b) if n is up by 20, show the impact of changing n on p and q.

4. consider the simplified national income model:

y = c + i…………(1)

where y is gdp (national income), c is consumption, and i is investment. consumption is determined by a behavioral equation, which in this problem takes the form
c= 3000+ 2/3 y……..(2)

where y and c are endogenous variables and investment is exogenous, and, initially we assume

i =500……………….(3)

determine the equilibrium level of national income (y) and consumption (c) by using reduced form, and matrix algebra.

5. show your work of differentiating the following 3 questions.

(this means that no work is no credit.)

y = 12x2 - 6x + 4

y = (5x + 1) * (x + 4)

y = (x + 4) / (x – 2)

6. show your work of differentiating the following question.
y = = 2 (7x2 – x) 4

7. find the price elasticity of demand from the following function:

qd = a– bp

8.consider the simplified, two-equation, national income model

y = c + i + g
c = a + b y

where national income (y) and consumption (c) are endogenous variables and investment (i) and government spending (g) are exogenous variables.

the parameters in the consumption function, where a represent the autonomous consumption expenditure and b represents the marginal propensity to consume, respectively.

2-a) set up this model with a 2 x 2 matrix of coefficients matrix, a 2 x 1 vector of endogenous variables, and a 2 x 1 vector of constants (consider i + g to be one constant).

2-b) the model can be expressed as ax = y, where a is the coefficient matrix, x is the vector of endogenous variable, and y is the vector of constants. find the solution of x.

(you must show your work. this means that no work = no credit.)
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