04 Jun Question Some useful derivative rules:
Question
Some useful derivative rules:
Notation: f (x) denotes the derivative of f (x) with respect to (w.r.t.) x. fx (x, y) denotes the
derivative of f (x, y) w.r.t. x.
1. f (x) = xa f (x) = axa1
2. f (x, y) = xa y b fx (x, y) = axa1 y b
3. f (x, y) = xa y b fy (x, y) = bxa y b1
4. f (g(x)) f (g(x))g (x), e.g., f (x) = (xa + b)c f (x) = c(xa + b)c1 axa1
Questions
1. Suppose that an economy produces only one good which uses only labor, L. For this economy,
the aggregate production function is Y (L) = 10L0.75 , and total labor supply is L = 625.
(a) Calculate the total output, Y, and the equilibrium wage rate, w.
(b) What is the share of labor income in output?
(c) Are the total prots equal to zero? (Hint: Think about the remaining amount after wage
is paid to the labor.)
2. We can approximate the aggregate production function for Azerbaijan economy with a Cobb1
Douglas production function of the following form: Yt = At Kt
Nt Ht , where t is a time
subscript, A is exogenously given technology parameter, K is capital, L is unskilled labor, and
H is skilled labor.
(a) Derive an expression for marginal product of unskilled and skilled labor.
(b) Dene the wage premium as the ratio of the wage for skilled labor, wH , to wage for
unskilled labor, wN . Derive an expression for the wage premium. (Hint: Wages are going
to be equal to their respective marginal products.)
(c) The Azerbaijani government has recently increased education subsidies, which is expected
to have a big positive impact on the number of skilled labor relative to unskilled labor.
What do you think will happen to the wage premium in the future? Defend your answer
based on the expression you derived in part (b).
3. The following expressions dene an economy:
Y =C +I +G
Y = 5, 000
G = 1, 000
T = 1, 000
C = 350 + 0.75(Y T ) 20r
I = 1, 000 50r
(a) Calculate the equilibrium interest rate.
(b) Find private savings, public savings, and total savings.
4. In an economy, real GDP, Y , grows at 2%, while the money supply, M , grows at 3%. The
nominal interest, i, is 6%.
(a) Calculate the real interest rate, r.
(b) What happens to i if money supply growth rate increases by 2 percentage points?
(c) Real GDP growth rate increases to 4%. If the central bank wants to keep i = 6%, what
should be the money supply growth rate?
5. In this assignment, you will collect some labor market data for Azerbaijan. Go to the World
Banks databank as before to download the following series covering from 1998 to the latest
series available:
Employment in agriculture, industry, and services (as % of total employment)
Employment to population ratio (15+ %, modeled ILO estimate) for females, males, and
total.
Create two graphs, one for employment shares for dierent sectors and another one for employment to population ratios. Comment on the patterns you observe in both of these graphs.
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