Question 1 A. In order to find optimal level of consumption and utility, we set MRS=MRT. These are found by:

Transcription

1 Question 1 A. In order to find optimal level of consumption and utility, we set MRS=MRT. These are found by: With x isolated, we now insert to budget constraint: Therefore, optimal consumption is: x=100, y=25, and total utility is: 250, Page 1 of 11

2 B. In order to find price elasticity, Q must be isolated: We now find the Price elasticity: Now, we insert the values and find the price elasticity of demand when P=1 and Q=100: We now conclude that price elasticity of demand =-1 C. To find out about Eric s risk preferences, we use the arrow pratt measure. It is written as minus double derivative over the derivative. We find these: They give us following arrow pratt: Page 2 of 11

3 When r>1, it means that Eric is risk seeking. When r=1, it means that Eric is risk neutral. When r<1, it means that Eric is risk averse. D. In order to find Eric s willingness to pay (WTP), we must find his expected utility (EU) from the lottery: We can now find Eric s WTP in following way: Eric is willing to pay 36 for the lottery ticket. WTP is also called certainty equivalent (CE), and in order to find Eric s risk premium, we must take expected value (EV) minus CE: Eric s risk premium for this lottery is 4. E. Eric s willingness to pay for the new lottery is found in the same way as in question D. It can be concluded that Eric is willing to pay 3600 for this lottery. Page 3 of 11

4 His risk premium is now found by finding expected value and subtracting his CE from this: Eric s risk premium for this lottery is 400. The risk premium compared to the lottery in question D: The risk premium is 396 higher in the new lottery compared to the old lottery. Question 2: A. In order to find optimal production, we must find marginal cost (MC) and set this equal to the price, since in perfect competition, MC=P. From this, we can find the optimal level of production: The optimal level of production for this firm in the short run is Q=4. B. In the long run, optimal level of production is at the minimum of average cost, since profit in long run is always 0 when the market is perfectly competitive. We first find AC: We now find the derivative of AC, set it equal to 0 and solve for Q: Page 4 of 11

5 Optimal production in the long run is therefore Q=0.71 C. Since marginal cost has the same curve as supply, we use the MC curve to find the market equilibrium quantity. In order to do this, we must inverse the demand function and set it equal to MC, which was found in question A: We now know that market equilibrium quantity is equal to 3.2. If each firm produces 0.71 unit in the long run as found in Question B, number of firms must be equal to the total market quantity produced divided by the quantity produced of the individual firm. We now conclude that in the long run there will be 4.51 firms in the market. D. A monopolist has the freedom to set the price wherever they can maximize profit. They do this by setting MC equal marginal revenue (MR). By doing so, they find optimal production. They then insert this Q into market demand function to set the price. Since profit is total revenue (P*Q) minus total cost (which is also a function of Q), we must find both P and Q in order to find the firm s profit. Therefore, in order to find profit we start by finding MC and MR, setting them equal to each other and isolating Q: Page 5 of 11

6 We can conclude that the firm s optimal level of production lies at Q=2.67. We now insert this Q into the inverse demand function to find market price: Profit can now be found by using price, quantity and the firm s cost function: We now conclude that the monopoly firm s profit is 9.63 E. Since the tax is levied on consumers, we suppose that consumers now want to pay 2 kroner less per unit, hence shifting the demand curve downwards. We therefore minus 2 from the inverse demand function to find the new demand function: Now, in order to find optimal level of production, we do exactly as in question D: Page 6 of 11

7 The optimal level of production after the tax is therefore Q=2 We now find profit, once again in the same way as in question D: We now conclude that the profit for this firm after the tax is equal to 5. Question 3: A. To find marginal product of labour (MPL) and marginal product of capital (MPK), we take the derivative of the production function with respect to each variable: B. To find optimal input, we set MRTS=-w/r. First, we find MRTS: Page 7 of 11

8 Then, we set it equal to w/r and isolate L, so we have L expressed by K: The firm s revenue must be 1000, since P*Q is 10*100. We can now insert the new L into the cost function and find the optimal demand for capital: Due to time pressure this is not done, but by inserting this into the cost function, we should be able to find the optimal level of K. After this, we can insert his level into the expression for L, and through that find the optimal level of labour. C. Since this is almost the same question as before but with other prices, the method is exactly the same. First off, we set MRTS=-w/r, and isolate for L, so we now have L expressed by K. We now substitute this new L into the cost function and isolate for K to find the optimal amount of capital demanded: We can now insert 50 into L=1.5K in order to find optimal level of L: Page 8 of 11

9 We now conclude that at these prices, optimal demand for capital is 50, while optimal demand for labour is 75. D. First off, market price and quantity is found by setting the inverse supply function equal to the inverse demand function, isolating Q and then inserting that Q into either the inverse supply or the inverse demand function: We now know that market equilibrium quantity is 20, while market equilibrium price is 10. Now we find producer surplus (PS) and consumer surplus (CS): Page 9 of 11

10 From this, it can be concluded that the consumer surplus is 20, whilst the producer surplus is 100. E. The new tax levied on producers means that the producers will now charge 3 kroner more per unit, hence shifting the supply curve upwards. New quantity and prices can now be calculated: New quantity: Page 10 of 11

11 Quantity after tax is 15. Since there is a tax, there is now a difference in the price the producers receive, and the price the consumers pay: Consumer price will be 10.5 and producer price will be 7.5 Welfare loss to consumers is the area of ABCD, whilst welfare loss to producers is equal to CBEF. Producer welfare loss is 37.5, whilst consumer welfare loss is 8.75 Page 11 of 11