Reference no: EM132684814

LM56 Public Economics

Exercise 1

Suppose an individual maximizes a quasi-linear utility U(X,Y ) = u(X) + Y for some increasing and concave function v, subject to the budget constraint PxX +Y = R, where R denotes income and Ps the price of X.
1. Show that the demand of X does not depend on R.
2. Right the matlab code to solve previous point.
3. Plot demand assuming that v is the natural log and the p2 is defined between [0, 1, 2]. Comment your result.
4. Government introduces a tax t = 0.5 on consumer for each unit of z. Find the new equilibrium and shows in Matlab plot the demand with and without tax.

Exercise 2
Suppose an economy with two goods (x and y) and one consumer. Consumer's preference are described by the following utility function:
U(x1,x2) = log(x1) + βlog(x2)
where β is a preference weight. Disposable income is equal to R = 3 and prices equal to p1 and p2.
1. Let qz the new price after tax of good x. Show the substitution and income effects graphically
2. Compute the Deadweight Loss due to taxation and display it with matlab
3. Which type of taxation does consumer prefer?

Exercise 3
Consider the utility function U = α log(x1) + βlog(x2) - l and budget constraint wl = q1x1 + q2x2
1. Show that the price elasticity of demand for both commodities is equal to -1.
2. Setting producer prices at pi = P2 = 1, show that the inverse elasticity rule implies t1/t2 = q1/q2
3. Letting w = 100 and a + β = 1, calculate the tax rates required to achieve revenue of R = 10.

Exercise 4
Consider the budget constraint x = b + (1 - t)wl.
1. [M] Provide an interpretation of b and plot the graph if b ∈[-2, 0, 2] [Hint: suppose that w = 1 and t = 0.5]
2. [M] Show analytically and graphically how the average rate of tax change with income
3. Let utility be given by U = x- l². How is the choice of t affected by increases in b and t? Explain these effects.

Exercise 5
Assume that utility is U = log(x) + log(1 - l).
1. Calculate the labour supply function.
2. Explain the form of this function by calculating the income effect of a wage increase.

Exercise 6
For the utility function U = r - f and two consumers of skill levels s1 and s2, s2 > s1, show that the incentive compatibility constraints imply that the income and consumption levels of the high-skill consumer cannot be lower than those of the low-skill consumer.

Exercise 7
An two-period economy is given by N = 3 households, each with a different β_t. Let indicate with Z household income. The households have the same Z. Interest rate is equal to r. Household can borrow and amount b up to b, which is the same for each household. Household's preferences over consumption c in the two periods are described by the following utility function:
U(c₁, c₂) = log(c₁) + log(c2). Let suppose also that z1 = z2, 22 = 10 and b= -1

1. Set up the household saving/borrowing problem

2. [M] Show graphically that even if the households have the same Z and face the same b heterogeneity in imples that want to borrow differenty. [Hint assume the vector of beta for the three household being equal to [0.75 0.85 0.95]
3. [MI Compute the aggregate household excess supply. This is given by B;,h = ELib*(r, 01), where ba (r, gii) is the household saving/borrwing demand at different r
4. [M] Suppose that a firm can choose capital given labor inputs. At the start of a period, a firm rents capital inputs and combines capital with labor to produce. At the end of the period, the firm sells its output and pays interest rates based on how much capital it rented, and also pays wage. Total wage bill is L•w, interest payment is K•r. Profit is denoted by Π, period interest rate (cost of capital) is r, the price of output is p, the firm makes y units of output, and the production function is Cobb-Douglas: A•Kα•Lβ. Assume also α + β < 1. Find the optimal amount of capital K*.
5. [M] If p = 1, A = 2.5, α = 0.36, β = 0.5 and L = 1, plot the aggregate demand for K*.
6. [M] Find the equilibrium interest rate: reqt", where at this interest rate: B*hh (r^equt) - K*firm(r^equt) = 0
7. [M] Suppose government decides to subsidize borrowing. Governments decide to pay for it by taxing savings with T = 0.10. Solve for the new optimal choices and equilibrium given this tax policy.
8. [M] Compares resutls with and without taxation.

Exercise 8
Suppose a good x in a perfect competitive market. Let c the marginal cost of production and x = a - bp the linear demand function.
1. [M] Compute the deadweight loss of introducing a commodity tax t as a difference bettween consumer surplus.
2. [M] How is the deadweight loss affected by changes in a and b.
3. [M] Suppose that a=3, t=0.5, c = land b ∈ [0,5]. How does a change in b affect the elasticity of demand at equilibrium without taxation? Show the change due to b in pre and post tax equilibrim quantities, consumer surplus with and without taxation, government revenu, DWL and elasticity. Comment your results.

Exercise 9
Assume that skill is uniformly distributed between 0 and 1 and total population size is normalized at 1. If utility is given by U = log(x) + log(1 - I) and the budget constraint x = b + (1- t)sf.

1. [M] Find the optimal values of b and t when zero revenue is to be raised.
2. [M] Find the optimal values of t when zero revenue is to be raised. Consider a RawLsian function.
3. [M] Consider now an utilitarian social welfare, graph utilitarian welfare against tax rate
4. Is the optimal tax system progressive

Exercise 10
A consumer has utility of income Y denoted by U = Y^1/2 . Let p the probability of detection, t the tax rate, F the fine
1. Determine the amount of income X declared to the tax authority.
2. For F = 1/2 and p = 1/2. show that the declaration X is an increasign funct ion of t.
3. Plot the optimal choice X* as a function of t if Y = 10
4. Assume that the revenue authority aims to maximize the sum of tax revenue plus fines less cost of auditing. If the latter cost is given by c(p) = p², graph the income of the revenue authority as a function of p for Y = 10, F = 1/2, t = 1/3. Then derive the optimal value of p.