Bicycle manufacturing is a monopolistic industry comprised of M firms. The aggregate utility function of a...

 

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Bicycle manufacturing is a monopolistic industry comprised of M firms. The aggregate utility function of a representative consumer is given by U(Xo) + Y where Xo represents all other goods and Y is the composite quantity index of bicycles formulated as 4/5 Y = (31/5 + y2 + + 5) From now on we will treat the U(Xo) part as a fixed constant and ignore it. Let Is denote the portion of the representative consumer's income spent on bicycles. Bicycle manufacturer i's production function is given by y = A-k; where k denotes the capital employed (at rental rate normalized to 1). A, denotes firm-specific productivity parameter. 1. Show that the demand for bicycles of firm i is of the form Ib P-4 -5 Yi = Pi where P = (P)-1/4. [6 points] 2. Find each firm's profit maximizing price and show that each firm charges the same mark-up rate over its unit cost. Also, show that the quantity produced by each firm is increasing function of its productivity. [6 points] 3. The Total Factor Productivity of Revenue for firm i (TFPR.) is defined as P: y/k. Show that at equilibrium each firm must have the same TFPR.. [6 points] 4. Now suppose that different firms face more or less favorable cost of capital - let the cost of capital for firm i be denoted by 1 + t, where t;> 0 means the firm faces less favorable cost of capital, and converse is true if t, < 0. Show that faced with these distortions, the firms now produce output and charge prices that no longer lead to the equalization of TFPR. [6 points] 5. Suppose you are assigned the task of studying the extent to which the bicycle industry suffers from low TFP due to misallocation of credit. You have acquired data about 1) price elasticity of demand (assumed to be same across all firms), 2) firm level data on revenue and capital invested. Describe step-by-step the procedure you will use to estimate the "ideal" and actual TPF in the bicycle sector. [6 points] Bicycle manufacturing is a monopolistic industry comprised of M firms. The aggregate utility function of a representative consumer is given by U(Xo) + Y where Xo represents all other goods and Y is the composite quantity index of bicycles formulated as 4/5 Y = (31/5 + y2 + + 5) From now on we will treat the U(Xo) part as a fixed constant and ignore it. Let Is denote the portion of the representative consumer's income spent on bicycles. Bicycle manufacturer i's production function is given by y = A-k; where k denotes the capital employed (at rental rate normalized to 1). A, denotes firm-specific productivity parameter. 1. Show that the demand for bicycles of firm i is of the form Ib P-4 -5 Yi = Pi where P = (P)-1/4. [6 points] 2. Find each firm's profit maximizing price and show that each firm charges the same mark-up rate over its unit cost. Also, show that the quantity produced by each firm is increasing function of its productivity. [6 points] 3. The Total Factor Productivity of Revenue for firm i (TFPR.) is defined as P: y/k. Show that at equilibrium each firm must have the same TFPR.. [6 points] 4. Now suppose that different firms face more or less favorable cost of capital - let the cost of capital for firm i be denoted by 1 + t, where t;> 0 means the firm faces less favorable cost of capital, and converse is true if t, < 0. Show that faced with these distortions, the firms now produce output and charge prices that no longer lead to the equalization of TFPR. [6 points] 5. Suppose you are assigned the task of studying the extent to which the bicycle industry suffers from low TFP due to misallocation of credit. You have acquired data about 1) price elasticity of demand (assumed to be same across all firms), 2) firm level data on revenue and capital invested. Describe step-by-step the procedure you will use to estimate the "ideal" and actual TPF in the bicycle sector. [6 points]