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Company ABC uses two inputs, labour and capital, in its production process. Prices for these inputs are $11 per hour and $33 per unit for labour and capital, respectively. The marginal products of the two inputs are 24 for labour and 48 for capital, given the current amount of inputs. Is the firm using the cost-minimizing combination of labor and capital? If not, should ABC hire more or less labour?.....

Question 6 Dynamic Discrete Choice Models. 1. Discuss the main challenges inherent in empirical estimation of dynamic games. Pick the two or three most important, in your view. Be clear and concise. 2. Define a Markov Perfect Equilibrium. Explain why it is useful in estimation of dynamic games. Practically speaking (and in plain english), what does it imply about the beliefs of firms? 3. The next few questions relate to the Pakes, Ostrovsky, & Berry (2007) model, but touch on some common elements of dynamic estimation more generally. Below is the Bellman 2 equation for incumbent firms in the POB framework. What data are required to estimate the parameters of interest in the POB model? To which static entry/exit model is this most similar? Name the types of firms that this (static) paper examined. VC(n, =;0) = _ [i(n', ?';#) + BEsmart, VC(n', =';0)}]] P(n', ='in, z, x = 1) where w(n, z;#) is a one-period profit function, n is the number of active firms, z is a vector of exogenous profit shifters, 0 is the parameter vector, e and a are the number of entrants and exitors, respectively, $ is the sell-off (exit) value, and x is an indicator variable equal to one if an incumbent remains. 4. One of Rust's (1987) main contributions was to identify a crucial assumption, which he calls Conditional Independence, that greatly facilitates estimation in dynamic settings. Describe this assumption formally, using the relevant part of the equation above. Explain in plain english what implicit assumptions it imposes on the evolution of the state variables. 5. Describe the key observation in Hotz & Miller (1993), and how they propose exploiting it to ease the computation burden of dynamic estimation. (Be brief. A couple of concise sentences are sufficient to earn full points here.) 6. POB exploit the Hotz & Miller framework, and combine it with a parametric assumption on the distribution of $ that allows them to reach the following analytical expression, where pz is the probability of exit: Esmart, VC(n', ?';8))1 = VC(n, z;0) + ops(n, =) This allows the Bellman equation to be rewritten as follows, where M' is an estimate of the incumbents' state transition matrix: VC(n, z; 0) = M'a(n',2';0) + AM' [VC(n, =;0) + op.(n, =)] Using this expression, derive POB's closed-form solution for the value function.Section C Question 5 Production Function Estimation. Suppose that you have a random cross section of firm- level data, with information on output, labor and capital. In logs, (uni, hi, k : i = 1, 2, ..., N} You are interested in estimating the Cobb-Douglas production function: wi = auditakkitwiter 1. Discuss the issues with estimating this PF using OLS. 2. What are the two main reasons (in terms of identification, not availability) why input prices are likely poor candidates for instruments? 3. Now suppose that you have panel data and want to estimate the Olley & Pakes (1996) model. Describe clearly how to implement the two-step approach that they propose. 4. What are the three assumptions/requirements for identification of az? 5. What is the basis of the Ackerberg, Caves, & Fraser (2006) critique of the OP model? 6. What assumptions do they suggest that allow the OP approach to be salvaged? How does the two-step procedure change in this context? Question 6 Dynamic Discrete Choice Models. 1. Discuss the main challenges inherent in empirical estimation of dynamic games. Pick the two or three most important, in your view. Be clear and concise. 2. Define a Markov Perfect Equilibrium. Explain why it is useful in estimation of dynamic games. Practically speaking (and in plain english), what does it imply about the beliefs of firms? 3. The next few questions relate to the Pakes, Ostrovsky, & Berry (2007) model, but touch on some common elements of dynamic estimation more generally. Below is the Bellman 2 equation for incumbent firms in the POB framework. What data are required to estimate the parameters of interest in the POB model? To which static entry/exit model is this most similar? Name the types of firms that this (static) paper examined. VC(n, z;0) = _ [#(m', ?'; 0) + BEsmart, VC(n', =';0)}]] P(n', ='In, z, x = 1) where w(n, 2; 0) is a one-period profit function, n is the number of active firms, z is a vector of exogenous profit shifters, 0 is the parameter vector, e and z are the number of entrants and exitors, respectively, $ is the sell-off (exit) value, and x is an indicator variable equal to one if an incumbent remains.2. This question deals with Guerre, Perrigne, and Vuong, "Optimal Nonparametric Estimation of First-Price Auctions" (Econometrica, 2000), Haile and Tamer, "Inference with an Incomplete Model of English Auctions" (JPE, 2003) and Haile, Hong, and Shum, "Nonparametric Tests for Common Values in First-Price Sealed-Bid Auctions" (2005). Common Assumptions to both (a) and (b): There are / potential bidders. Assume / is exogenous and known. Bidders are symmetric and risk-neutral. Independent Private Values. Each bidder draws her private value v, from a common distribution F(v), which has a support [0, co). (a) Consider a single-object, first-price sealed-bid auction. Assume there is no reserve price for simplicity. Carefully derive symmetric Bayesian Nash equilibrium bidding strategies, A(v.). (Consider increasing and differentiable strategies only.) (b) Consider a single-object, Milgrom-Weber "button" auction. The seller's value for the object is vo and she wants to maximize her revenue from the auction by setting a reserve price r. Write down the seller's maximization problem and derive a condition for the optimal reserve price ,* from the F.O.C. of the max problem. (c) Describe, as fully as you can, the nonparametric identification result and the two-step nonparametric estimation strategy of GPV (2000). (d) State the two axioms (or behavioral assumptions) of Haile and Tamer (2003) and construct, as fully as you can, the nonparametric (partial) identification result of Haile and Tamer (2003). Discuss the advantages and disadvantages of this incomplete approach. (e) Discuss, in general, the advantages and disadvantages of using structural models when conducting empirical research in auctions. (f) Prove the following theorem from Haile, Hong, and Shum (2005), which is the basis of their nonparametric test of common values. Theorem Under standard assumptions of symmetry, affiliation, nondegeneracy and an additional assumption of exogenous participation, v(x, x, ") is invariant to " for all x in a PV model, but strictly decreasing in n for all x in a CV model, where v(x, x', ") = E[V,| X, = x,max X, = x'].2. This question deals with Guerre, Perrigne, and Vuong, "Optimal Nonparametric Estimation of First-Price Auctions" (Econometrica, 2000), Haile and Tamer, "Inference with an Incomplete Model of English Auctions" (JPE, 2003) and Haile, Hong, and Shum, "Nonparametric Tests for Common Values in First-Price Sealed-Bid Auctions" (2005). Common Assumptions to both (a) and (b): There are / potential bidders. Assume / is exogenous and known. Bidders are symmetric and risk-neutral. Independent Private Values. Each bidder draws her private value v, from a common distribution F(v), which has a support [0, co). (a) Consider a single-object, first-price sealed-bid auction. Assume there is no reserve price for simplicity. Carefully derive symmetric Bayesian Nash equilibrium bidding strategies, A(v.). (Consider increasing and differentiable strategies only.) (b) Consider a single-object, Milgrom-Weber "button" auction. The seller's value for the object is vo and she wants to maximize her revenue from the auction by setting a reserve price r. Write down the seller's maximization problem and derive a condition for the optimal reserve price ,* from the F.O.C. of the max problem. (c) Describe, as fully as you can, the nonparametric identification result and the two-step nonparametric estimation strategy of GPV (2000). (d) State the two axioms (or behavioral assumptions) of Haile and Tamer (2003) and construct, as fully as you can, the nonparametric (partial) identification result of Haile and Tamer (2003). Discuss the advantages and disadvantages of this incomplete approach. (e) Discuss, in general, the advantages and disadvantages of using structural models when conducting empirical research in auctions. (f) Prove the following theorem from Haile, Hong, and Shum (2005), which is the basis of their nonparametric test of common values. Theorem Under standard assumptions of symmetry, affiliation, nondegeneracy and an additional assumption of exogenous participation, v(x, x, ") is invariant to " for all x in a PV model, but strictly decreasing in n for all x in a CV model, where v(x, x', ") = E[V,| X, = x,max X, = x'].2. This question deals with Guerre, Perrigne, and Vuong, "Optimal Nonparametric Estimation of First-Price Auctions" (Econometrica, 2000), Haile and Tamer, "Inference with an Incomplete Model of English Auctions" (JPE, 2003) and Haile, Hong, and Shum, "Nonparametric Tests for Common Values in First-Price Sealed-Bid Auctions" (2005). Common Assumptions to both (a) and (b): There are / potential bidders. Assume / is exogenous and known. Bidders are symmetric and risk-neutral. Independent Private Values. Each bidder draws her private value v, from a common distribution F(v), which has a support [0, co). (a) Consider a single-object, first-price sealed-bid auction. Assume there is no reserve price for simplicity. Carefully derive symmetric Bayesian Nash equilibrium bidding strategies, A(v.). (Consider increasing and differentiable strategies only.) (b) Consider a single-object, Milgrom-Weber "button" auction. The seller's value for the object is vo and she wants to maximize her revenue from the auction by setting a reserve price r. Write down the seller's maximization problem and derive a condition for the optimal reserve price ,* from the F.O.C. of the max problem. (c) Describe, as fully as you can, the nonparametric identification result and the two-step nonparametric estimation strategy of GPV (2000). (d) State the two axioms (or behavioral assumptions) of Haile and Tamer (2003) and construct, as fully as you can, the nonparametric (partial) identification result of Haile and Tamer (2003). Discuss the advantages and disadvantages of this incomplete approach. (e) Discuss, in general, the advantages and disadvantages of using structural models when conducting empirical research in auctions. (f) Prove the following theorem from Haile, Hong, and Shum (2005), which is the basis of their nonparametric test of common values. Theorem Under standard assumptions of symmetry, affiliation, nondegeneracy and an additional assumption of exogenous participation, v(x, x, ") is invariant to " for all x in a PV model, but strictly decreasing in n for all x in a CV model, where v(x, x', ") = E[V,| X, = x,max X, = x'].