Dynamic Optimal Public Finance Problem - Assignment Example

Q1. Dynamic optimal public finance problem has two solution approaches which are the dual and the primal. Although there are other types of dual problems, the Lagrangian dual problem is the most important one which is formed by making Lagrangian in which nonnegative multipliers are used. By minimizing the Lagrangian, we get the primal solution. Ramsey problems or public finance problems are fully solved through primal approach. Dual problem is complex in comparison to primal. This is basically due to the terms of equation as well as unknowns to be solved in dual solution. Primal solution is easier to solve and because of the difficulty of the dual solution, there is a lack of full numerical solutions to the dual problem. Normal or direct utility function treats utility as increasing function of goods or services consumed and decrease in the function of work being performed for it. As opposite to this, in indirect utility function increasing function of non-work income and rate of wage of work done, and decreasing function of prices of goods and services consumed is termed as utility. Q2. a)Both price and quantity can be calculated by looking at the graph. p1=4 q=120 As the price of chips in Utopia is more than the price in the world, many suppliers will be interested in providing the product in Utopia. With the existing demand and supply situation, it appears that production will mostly be done outside Utopia. Domestic production will be costly and thus not affordable. b) If the senator’s proposal is approved, there will be an increase in supply due to decreased cost, and will be an increase in demand as well. With the increased number of chips coming in market, the price will automatically decrease for the end buyer. The price will be thus down to less than 4. The quantity consumed domestically will increase so will the quantity produced domestically. The subsidy plan will cost the government a lot. It will not be a very profitable venture as all profits made through this economic strategy will be lost to the factor that imports are still allowed and subsidy is not paid on domestic sales. With the high price of chips in Utopia, even with the subsidy, Utopia will not be the cheapest producer to beat competition to the level where despite the imports and no subsidy on domestic sales this step could be profitable. c) If the tax amount is added to the whole price, it will have an impact over price of computer chips in Utopia. The price will increase on the whole. After tax p=5 Q=90. d) The Senator is correct in his objections. Considering the market price and the demand of this product, an increase of $1 tariff on imports would have resulted in the same profit collection with much less hassle. Q3) Part a) part b) As per the given data, elasticity could be calculated. The results are as follows: Elasticity E12=-0.504 E23=-0.500 E34=-0.73 E45=-0.45 E56=-0.5026 By using the above mentioned formula. Part c) Cross price elasticity of demand with natural gas is same all along the relevant schedules Part d) Income elasticity for demand for electricity is same all along the relevant schedules Q. 4 a) Qx = 8 + 1/2 Qb - Pa Qb = 2 + 1/2 Qa - Pb b) Profit funtion = revenue - Cost P = (QaPa + QbPb) - (Qx^2 - QxQb + Qb^2) Maximum Profit = P - max Where P is the profit function. c) As calculated above: P = (QaPa + QbPb) - (Qx^2 - QxQb + Qb^2) Maximum Profit = P - max d) The maximum profit equation has been derived through a set of procedures and formulas, and they show that by putting any value, profit will not go above this point. This confirms the above solution to be maximum profit. e) Direct demand function is the "quantity demanded as a function of product price" Here, assume that before sale , Qx and Qb were the quantity demanded for Skis and boots respectively. Direct demand function for Skis: Qx = a - bPa a = intercept b = Slope (change in quantity demanded / change in Price) Qx = Quantity demanded Direct demand function for Boots: Qb = a - bPb a = intercept b = Slope (change in quantity demanded / change in Price) Qb = Quantity demanded Q.5 a) The Lagrangian function to maximize utility subject to budget shall be: F= 4lnx – ½ (8-y)² + λ[P(x) x + y – 8] Now the first order derivatives would be: Fx= 4/x + λPx (Eq. 1) Fy= 8-y + λy (Eq. 2) Fλ= Px(x) + y – 8 For optimality Fx = Fy = 0 Substituting the above in equation 1 and 2. λ = -4/(x) px (Eq. 3) And λ = y -8/y (Eq. 4) Equating Equation 3 and 4: x = -4y/ Px (y-8) First order condition for optimality b) Putting ½ in the budget curve: 8 = x/2 + y and y = 8 – x/2 The values of x and y will be calculated by putting x = 0, 1 and so on. x Y 0 8 1 7.5 2 7 3 6.5 4 6 5 5.5 6 5 7 4.5 8 4 9 3.5 10 3 11 2.5 12 2 13 1.5 14 1 15 0.5 16 0 Y 8 Indifference curve X 16 c) The first order for Fish x was calculated in Eq. 1. Fx= 4/x + λPx = 0 λPx = -4/x Px = -4/xλ Or λ = -4/xPx The price elasticity would be negatively sloping line. Pd = Δθd/θd And Pd = Δx/x/ ΔPx/Px which is price elasticity d) From Eq. 2 Fy=8-y + λy = 0 y = 8/ 1- λ The above relation has no dependence of Px therefore price of fish has no effect on the demand for leisure. e) The lagrangian function with minimum expenditure is: F = 4 ln(x) – ½ (8-y)² + λ[Px(x) + y] The optimal conditions would be: Fx= 4/x + λ Px Fy= -y + λy Fx = Fy = 0 So λ = -4/ x (px) (Eq. A) And λ = 1 (Eq. B) Equating A and B x = -4/px First order condition for optimality f) The Hicksian curve varies from from Marshallian curve because Hicksian shows the variation of quantity of fish (x) with its price (px) which is evident from the result of part e. Marshallian Hiskian Read More