Advance microeconomics

Question 1 (15 marks)
Tom derives his utility from consuming goods X and Y, where his utility function is
u(x,y)=40x^0.5 y^0.25.
He spends all of his wealth, w, on his purchase of goods X and Y, and he must pay prices of P_x and P_y for each unit of these goods, respectively. Assume that his wealth is 300, the unit price of goods X and Y are both $20.
Determine the amounts of goods X and Y that Tom should purchase to maximize his utility given this budget constraint and the corresponding maximum utility he receives.
Determine the value of λ^* associated with this problem and interpret this λ^* financially.
Derive Tom’s Engel curve for good X first. Then response whether good X is a normal good or an inferior good. Justify your response mathematically.

Question 2 (15 marks)
The utility function is u(x_1,x_2 )=min⁡(x_1+3x_2,〖3x〗_1+x_2) and the budget constraint is ⁡〖p_1 x〗_1+⁡〖p_2 x〗_2≤w, where p=(p_1,p_2 )≫0.
Draw the indifference curve for u(x_1,x_2 )=60. Shade the upper contour set (UCS), low contour set (LCS) and indifference set (IND) of the bundle (15,15).
For what values of ⁡p_1/p_2 , will the unique optimum be x_1=0. Also, for what values of ⁡p_1/p_2 , will the unique optimum be x_2=0
If neither x_1 nor x_2 is equal zero, and the optimum is unique, what must be the value of x_1/x_2 .
Question 3 (20 marks)
A consumer has a quasilinear utility function as
u(x_1,x_2 )=〖〖Ax_1^α x〗_2〗^(1-α)，
where α>0 and x_1,x_2∈R_+. Assume that the price vector satisfies p=(p_1,p_2 )≫0, and wealth w>0.
Write down the consumer’s utility maximization problem. Find Walrasian demand x(p_1,p_(2,),w) and the indirect utility function v(p_1,p_(2,),w)
Write down the consumer’s expenditure minimization problem. Find Hicksian demands h(p_1,p_(2,),u) and the expenditure function e(p_1,p_(2,),u).
Evaluate the Walrasian demands x(p_1,p_(2,),w) at w=e(p_1,p_(2,),u), and show that Walrasian and Hicksian Demands coincide, that is,
x(p_1,p_(2,),e(p_1,p_(2,),u))=h(p_1,p_(2,),u)
Evaluate the indirect utility function v(p_1,p_(2,),w) at w=e(p_1,p_(2,),u) and show that
v(p_1,p_(2,),e(p_1,p_(2,),u))=u.

Question 4 (15 marks)
Given a constant elasticity of substitution (CES) production function as follows:
q=f(z_1,z_2 )=〖A[〖δz〗_1^ρ+〖(1-δ)z〗_2^ρ]〗^□(γ/ρ)
Under which condition, this production function shows increasing returns to scale. Show the details of your proof.
Calculate the Marginal Rate of Technical Substitution (MRTS)
Calculate the Elasticity of Substitution (σ)

Question 5 (20 marks)
Consider a consumer with quasilinear utility function
u(x_1,x_2 )=〖ax_1^α+bx〗_2
where α>0,α>0,b>0. The budget constraint is ⁡〖p_1 x〗_1+⁡〖p_2 x〗_2≤w.
Find the Walrassian demand for Goods 1 and 2
Find the Hicksian demand for Goods 1 and 2
For simplicity, we consider the case where a=2, b=1 and α=0.5. Assume that consumer’s wealth w=$10. At time 0, the prices are
p_1^0=p_2^0=2.
At time 1, the prices has becomes p_1^1=1,p_2^1=2.
c) Compute the substitution effects, income effect and total effect.
d) Compute the welfare analysis, including CS(Consumer surplus), CV(Compensating Variation) and EV(Equivalent variation).
Question 6 (15 marks)
Consider a Cobb–Douglas production function f∶R_+^2→R_+, given by
f(z_1,z_2 )=2z_1^0.25 z_2^0.25
where z_1≥0 and z_2≥0 denote inputs in the production process.
a) Check if the production function has nonincreasing, nondecreasing, or constant returns to scale.
b) Let w∈R_+^2denote the vector of input prices and p≥0 the output price. Determine for each output level q≥0 the cost function c(w,q) and the conditional factor demand z(w,q). Also, verify Shephard’s lemma.
c) Determine the profit function π(p, w)