Consider the full version of the Solow model with both population growth and technology: Yt = F(Kt,LtEt). We will extend this version of Solow to also explicitly include the government. The national income accounts identity becomes: Yt = Ct + It + Gt where Gt is government spending in period t. In order to fund its spending the government collects a tax Tt. Suppose for simplicity that the government runs a balanced budget Gt = Tt and that the tax collected is a constant fraction of output: Gt = Tt = Yt. The remaining disposable income for households each period is (1 )Yt. As in Solow we still assume that households save/invest a constant fraction s of their (now disposable) income. The population growth rate is n, techonology grows at g, and the depreciation rate is . (a) Assume for now that there is only private and no public investment (i.e all government purchases are spent on consumer goods and none of Gt is used to invest in capital). Write down the standard system - the equations for output, consumption, investment, and the capital accumulation equation. Define: yt = Yt/EtLt,

kt = Kt/EtLt,

it = It/EtLt,

ct = Ct/EtLt,

gt = Gt/EtLt.

(b) Transform the model from part (a) in per- effective worker form and derive the steady-state equation for capital per effective worker. Draw a graph depicting the steady state.

(c) What is the effect of higher tax rate on the steady state? Show the effect on your graph and explain the intuition for your answer.

d) Now suppose that, in addition to the case in part (a), a fraction of Tt is also invested in the capital stock, i.e. public investment equals Tt = Yt. What is total investment equal to now? Similarly to part (b) derive the steady-state equation for capital per worker and depict your answer on a graph.

(e) Show that if is sufficiently high (i.e. you will need to find a specific threshold value), then the steady state capital per effective worker will increase as a result of higher taxation.

Suppose the Beer industry (good X) is characterized by Monopolistic Competition in both the Home and Foreign country. The different producers offer differentiated products based on the tastes of their beer. Otherwise, the beer producers are in all ways identical. They each face a linear demand curve of the type described in Lesson 8: for any firm i. Fixed Costs are F=$10M, Marginal costs are constant C=$20 per keg. b=.01. The size of the home market is 10M kegs, and foreign is 25M kegs. [our Lesson 8 notes/videos for details on the terms of the equation in addition to how to solve the problem.]

How many firms will produce in the home market in autarky? How many firms will produce in the foreign market in autarky? How much output for each firm? What is the price per keg of beer? (If your answer is not a round number, round DOWN to get n. Fractions of a firm don't make sense.) Why should we only round n down in problems like this rather than follow conventional rounding rules? Explain in a short paragraph. How many firms will produce in equilibrium when the countries open for trade? How much output for each firm? What is the price per keg of beer?

the quantity of customers served by Matt - the customer demand function for Matt's restaurant - was QM = 44 - 2PM + PS and the quantity of customers served by Sean was QS = 44 - 2PS + PM. The profits for each restaurant also depend on the cost of serving each customer. Suppose that Matt is able to reduce his costs to just $2 per customer by eliminating the wait staff. Suppose that Sean continues to incur a cost of $8 per customer.

b.) What is Matt's profit function (revenue per customer multiplied by number of customers), M?

c) Sean needs to choose his price to maximize his profit function. What is Sean's best response function?

d) Matt needs to choose his price to maximize his profit function. What is Matt's best response function? Explain what this best response function means.

e) Calculate the Nash equilibrium

f) Graph the two best response functions and indicate the location of the Nash equilibrium on the graph. To match what we did in class, plot Matt's price on the vertical axis and Sona's price on the horizontal axis.

g) Look at the graph of the best response functions that we came up with in class when the cost per consumer was $8 for both Matt and Sean. Indicate which best response function in your new graph has moved and by how much. Explain why these changes occurred in your graph.

Suppose it is costly to transport exports. Let this cost be equal to some number T per unit. We are going to investigate some features of how the presence of T impacts the monopolistic competition model with trade. Assume all firms are the same in each country, and that each country is the same size. The problem is written to help guide you to the solution in a series of (hopefully) not too demanding steps. (Most of these steps I sort of do for you.)

A) Let Phh be the price a home firm charges in the home country, Phf be the price a home firm charges in the foreign country, Pfh be the price a foreign firm charges in the home country, and Pff be the price a foreign firm charges in the foreign country.

B) Also adopt notation where Xhh is the quantity of beer a home country sells at home, with other Xs defined the same way.

C) (4 points) A firm considering the prices to charge in each market will evaluate each market separately, since the trade cost introduces a friction that makes each market imperfectly integrated. When we had no transaction costs, we found in lecture that . The term equalled 1/n since we had total symmetry, which was nice. Since firms now need to charge different prices in each country, this new asymmetry will not allow for the same simplification here. Write out four MR relationships. For instance, MRhh will be

D) (4 points) Set the MRs equal to MC. Note that when a firm is exporting, the MC=C+T.

E) (4 points) Calculate the Markup over MCs for the Home firm in both markets. For instance,

F) (4 points) Will the markup be larger for the home firm when selling at home or foreign? Why? [Hint: Think about what will happen to its market share in foreign now that it has to pay a transportation cost.]

G) (2 points) Given your answer to g, write down an inequality relating how the value Phh compares to the Phf minus transport costs.

H) (4 points) The result for g describes a phenomenon called "dumping." The price charged by the home country firm in the foreign market, minus the cost of transport, is actually lower than the price the firm charges at home. "Dumping" is frowned upon by many governments. Suggest a couple reasons why. As an economist, what is your view?

Suppose it is costly to transport exports. Let this cost be equal to some number T per unit. We are going to investigate some features of how the presence of T impacts the monopolistic competition model with trade. Assume all firms are the same in each country, and that each country is the same size. The problem is written to help guide you to the solution in a series of (hopefully) not too demanding steps. (Most of these steps I sort of do for you.)

A) Let Phh be the price a home firm charges in the home country, Phf be the price a home firm charges in the foreign country, Pfh be the price a foreign firm charges in the home country, and Pff be the price a foreign firm charges in the foreign country.

B) Also adopt notation where Xhh is the quantity of beer a home country sells at home, with other Xs defined the same way.

C) (4 points) A firm considering the prices to charge in each market will evaluate each market separately, since the trade cost introduces a friction that makes each market imperfectly integrated. When we had no transaction costs, we found in lecture that . The term equalled 1/n since we had total symmetry, which was nice. Since firms now need to charge different prices in each country, this new asymmetry will not allow for the same simplification here. Write out four MR relationships. For instance, MRhh will be

D) (4 points) Set the MRs equal to MC. Note that when a firm is exporting, the MC=C+T.

E) (4 points) Calculate the Markup over MCs for the Home firm in both markets. For instance,

F) (4 points) Will the markup be larger for the home firm when selling at home or foreign? Why? [Hint: Think about what will happen to its market share in foreign now that it has to pay a transportation cost.]

G) (2 points) Given your answer to g, write down an inequality relating how the value Phh compares to the Phf minus transport costs.

H) (4 points) The result for g describes a phenomenon called "dumping." The price charged by the home country firm in the foreign market, minus the cost of transport, is actually lower than the price the firm charges at home. "Dumping" is frowned upon by many governments. Suggest a couple reasons why. As an economist, what is your view?