Mathematical-Economics

Question 1 (6 points)

Give an example of each of the following. You can answer this by drawing rough graphs and giving an explanation in one line in each case (3 points).

y is function of x, but x is not a function of y.

y is not a function of x but x is a function of y.

y is a function of x and x is a function of y.

In each of the following cases, determine the real roots (if any) (3 points) (i) 2×2 -4x + 6 =0 (ii) x2 -2x + 1 =0 (iii) 2×2 -5x + 2=0

Question 2 (9 points)

Consider an economy with two sectors – food and manufacturing. Each sector uses labour to produce its output, and labour is mobile across both sectors, so that in aggregate labour market equilibrium, workers earn the same wage in both sectors. The supply of labour in the economy is exogenously given. In addition, food requires land and manufacturing requires capital. These factors are not mobile at all, and their quantities are exogenously given. This model can be expressed in terms of the following equations.

1. Lf = a – bw + eT0 a > 0, b > 0, e > 0 2. Lm = c – dw + gK0 c > 0, d > 0, g > 0

Lf +Lm = L0

Equation (1) describes the demand for labour in food production (Lf), where w is the wage and T is the supply of land. Similarly, equation (2) depicts the demand for labour in manufacturing production (Lm), with K standing for the supply of capital. Equation (3) is the equilibrium condition for the economy’s labour market: the total demand for labour should equal the exogenously given supply of labour (L0)

Solve for the equilibrium wage using the equilibrium condition. (3 points)

Determine the impact of an increase in the stock of capital by 2 units on the equilibrium wage. (2 points) Then find the impact on the equilibrium wage of an increase in the stock of land by 2 units. (2 points) Would the equilibrium wage increase, decrease, or remain unchanged if the quantity of capital increased by 2 units and, simultaneously, the quantity of land decreased by 2 units. Explain your answer.(2 points)

Question 3 (7 points)

Simplify each of the following expressions. (2 points)

(i) (x0.5 + x3/4x-1/4)/x0.25) (ii) x-2×3/2/x1/3 (iii) (x1/4y-2)-3 (iv) (64×9)1/3 (16y)-2

Consider the consumer demand function for blueberries: Q = P-aRb Xc where a, b and c are positive constants, Q = quantity of blueberries demanded, P =price of blueberries, R= price of raspberries, and X= consumer income.

Suppose the initial values of P, R and X are P0, R0 and X0. Now both P and R rise by 20 percent

(X remaining unchanged). Show how demand Q would change as a result? Specifically, would demand rise by more than 20 percent, less than 20 percent, or by 20 percent? Show your work. (3 points)

Economic theory tells us that if P, R and X all rise simultaneously by the same percentage, demand Q would not change? In the demand function given above, what condition would ensure this? Explain. (2 points)

Question 4 (8 points)

Find the natural logs of each of the following economic relationships:

α β

Production function in time period t: : Qt = Kt Lt , where K and L are factors of production such that

Kt = K0eut, Lt = L0evt, Q is output, and α, β, u, v, L0, and K0 are all constants. (3 points)

Revenue function: R= PQ, where P=aQ-b is the demand curve for the product, and

Q = [hKu + (1-h)Lu] r/u, where a, b, r and u are constants. (3 points)

Present value: V = y/(1+i)n , where V is the present value of a sum of money y to be received n years ahead, and i is the rate of interest. (2points)