Exercícios de Microeconomia

1 In a market there are 8 buyers each demanding a single discrete unit Their incremental valuations are v1 20 v2 18 v3 v4 v5 v6 15 v7 v8 12 There are 5 sellers each offering a single discrete unit Their incremental costs are all equal to MC 10 a What is the BohmBawerk marginal pair in this market b Without the presence of buyer 6 what is the BohmBawerk marginal pair and the corresponding equilibrium price range c Is this market necessarily perfectly competitive in the sense of Ostroy d Consider the economy without buyer 6 In this case what is the shadow price of the maximum capacity Q 5 2 A consumer has utility ux1 xn x1β1 xnβn where β1 βn 0 and β1 βn 1 and their income is m 0 The price vector is p p1 pn 0 where 0 Rn is the null vector a Determine the Marshallian demand xip m for good i b Determine the indirect utility Vp m c If prices change from p0 p10 pn0 to p0 p11 pn1 and ki is the proportion of income spent on good i prove that the compensating variation is CV product from i1 to n pi1pi0ki 1 m 3 A monopolist firm produces its product monthly according to two different fixed proportion processes using capital and labor with technical coefficients as in the table below factor of production process I x1 process II x2 capital K 4 2 labor L 1 3 Factor prices are r w 1 The firm has a maximum capacity of Q 12 units of product and the pricing rule is linear linear pricing The demand for the product is given by P 20 12 Q Let MC be the operational marginal cost σ be the shadow price of the maximum capacity and μ be the markup Determine the price P and its decomposition P MC σ μ 4 An econometrician specified the following demand function for a good x x α µ pr αp βq where x is the quantity demanded of good x p is the price of good x q is the price of good y r is the income and α β µ 0 are parameters determined econometrically Let kx px r be the proportion of income spent on good x and ky qy r be the proportion of income spent on good y a Show that the income elasticity of good x is εx µ kx and that of good y is εy 1µ ky b Show that the direct price elasticity of good x is γxp µrβq kx c Under what conditions on the parameters will goods x and y be normal or inferior d Under what condition on the parameters will good x be a common good e Suppose the econometrician estimated the following values for the model parameters ˆα 10 ˆβ 5 ˆµ 101 Suppose also that the data show that the consumer spends 50 of their income on good y Consider these values from now on Show that the econometrician will find a negative but almost zero value for the income elasticity of good y specifically εy 002 f If the econometrician claims that good y is inferior what criticism based on the fact that econometrically estimated parameters always admit a margin of error would you make to this statement g Suppose further that they want to make a demand forecast for the next month based on the following expected values of prices and income pe 20 qe 10 re 52 i Show that the expected demand will be xe 0001 ii Show that the direct price elasticity of good x will be γxp 404 iii If the econometrician had erred in forecasting the expected income and it was actually ree 49 show that the direct price elasticity of good x would be γxp 202 h Based on the calculations made would you agree that this econome tric model is good or would you say that the econometricians conclusions need to be more cautious given the statistical uncertainty that pervades the parameter estimation 5 Let ux y be a utility function homogeneous of degree 1 Prove that the marginal utility of income is constant Hint use the firstorder condition Eulers theorem for homogeneous functions and the budget constraint 2 6 A firm wishes to hire an agent to produce a common good which is sold in a monopolistic market with linear pricing The demand function is Q αP 6 where α 0 is a constant However it faces an adverse selection problem because the agent can be of two types an efficient type with a constant marginal cost θ 20 or an inefficient type with a constant marginal cost θ 26 In the market 70 of the agents are efficient and 30 are inefficient Determine the decomposition P MC µ φ and P MC µ φ in each state of nature θ θ θ 7 A firm has two plants whose cost functions are respectively c1q1 q1 10q2 1 and c2q2 q2 5 2q2 2 Determine the firms cost function cq 3